Explanation:
Okay, so Linda, Sherry, June, and Connie sold cookies and earned some money. Now they want to convert all their coins into dollars, and we need to find out how much money, in cents, was left after they converted as many coins as possible into bills. Hmm, let me think.

First, I need to figure out the total amount of money they have. Let me add up each person's earnings. Let me write down the amounts:

- Linda: $27.47
- Sherry: $35.23
- June: $37.37
- Connie: $26.52

So adding these up. Let me do it step by step. Let me add Linda and Sherry first. 27.47 + 35.23. Let me add the dollars first: 27 + 35 = 62. Then the cents: 47 cents + 23 cents = 70 cents. So that's $62.70.

Now add June's amount: $37.37. So adding $62.70 + $37.37. Again, dollars: 62 + 37 = 99. Cents: 70 + 37 = 107 cents. Wait, 107 cents is equal to $1.07. So that would make the total $99 + $1.07 = $100.07. Wait, let me check that again. Maybe I should do it more carefully.

Wait, 62.70 + 37.37. Let me write it vertically:

  62.70
+ 37.37
--------
Let me add the cents first: 70 + 37 = 107 cents. So that's 1 dollar and 7 cents. Then the dollars: 62 + 37 = 99, plus the 1 dollar from the cents, so total is 100 dollars and 7 cents. So total is $100.07. Okay, that seems right.

Now add Connie's $26.52. So $100.07 + $26.52. Let me do that. Again, dollars: 100 + 26 = 126. Cents: 07 + 52 = 59 cents. So total is $126.59. Let me check that again. 100.07 + 26.52. 100 + 26 is 126, 0.07 + 0.52 is 0.59. Yes, so total is $126.59. So altogether, they have $126.59.

But wait, the question says they pooled their money together and went to the bank to convert the coins into dollars. How much money, in cents, was left after they converted as many of the coins into bills as they could?

Wait, so they have a total of $126.59. But the question is about converting coins into dollars. Wait, but if they have dollars already, maybe they have some coins? Wait, maybe the problem is that they have all their money in coins, and they want to convert them into bills? But the problem says they "pooled their money together and went to the bank to convert the coins into dollars." So maybe they had coins, and they wanted to exchange them into bills, and we need to find out how much was left in cents after converting as much as possible into dollars.

But wait, the problem says they earned certain amounts. But the amounts are given in dollars and cents. So maybe they have those amounts in cash, which includes both bills and coins? But the problem says they converted the coins into dollars. So perhaps they had some coins, and they wanted to exchange them into bills, and the question is about the leftover coins after exchanging as much as possible.

But the problem is a bit ambiguous. Wait, let me read again:

"After the sale, they pooled their money together and went to the bank to convert the coins into dollars. How much money, in cents, was left after they converted as many of the coins into bills as they could?"

So, they had money, which includes coins, and they went to the bank to convert the coins into dollars. So the total amount they have is the sum of their earnings, which is $126.59. But if they have that amount, which is in dollars and cents, but they want to convert the coins into dollars. Wait, but if they have dollars already, maybe they have some coins? Wait, but the problem says they "converted the coins into dollars", so maybe they had only coins, and they wanted to exchange them into bills. But the problem says they "pooled their money together", which is the total amount they earned, which is $126.59. But if they had that amount in coins, then converting them into bills would mean exchanging all coins into bills, and there would be no leftover coins. But that can't be, because the question is asking how much was left in cents. Therefore, there must be some coins that couldn't be converted into bills. Wait, but how?

Wait, maybe the problem is that they had their money in coins, and they wanted to convert as much as possible into bills, and the leftover coins would be the cents. But if they have a total amount of money, which is $126.59, which is 126 dollars and 59 cents. If they convert all the coins into bills, then they would have 126 dollars and 59 cents. But if they have only coins, then they would have to exchange the coins into bills. But if they have 59 cents, that's the leftover. Wait, but the question says "how much money, in cents, was left after they converted as many of the coins into bills as they could?"

Wait, maybe the problem is that they had all their money in coins, and they wanted to convert as much as possible into bills, and the leftover would be the coins that couldn't be converted. But if they have a total amount of money, which is $126.59, which is 126 dollars and 59 cents. If they convert all the coins into bills, then they would have 126 dollars and 59 cents. But if they have only coins, then they would have to exchange the coins into bills. However, if they have 59 cents, that's the leftover. But the question is in cents, so 59 cents. But that seems too straightforward. However, maybe the problem is that they had coins, but the total amount is in dollars and cents, so maybe they have some coins that can't be converted into bills? Wait, but bills are dollars, so if you have coins, you can exchange them into bills. For example, if you have 100 cents, you can exchange them into 1 dollar bill. If you have 59 cents, you can't exchange them into a bill, so that's the leftover. Therefore, the answer would be 59 cents. But that seems too easy. However, maybe the problem is more complicated. Let me check again.

Wait, the problem says they "converted the coins into dollars". So maybe they had coins, and they wanted to convert them into dollars. But the total amount they have is $126.59. But if they have that amount in coins, then converting them into bills would result in 126 dollars and 59 cents. But the question is asking how much was left after converting as many coins into bills as possible. So if they have 59 cents, that's the leftover. Therefore, the answer is 59 cents. But maybe I need to check if there's something else here.

Wait, but maybe the problem is that they had their money in coins, but the amounts they earned are given in dollars and cents. So maybe they have the total amount in coins, which is $126.59, and they want to convert as much as possible into bills. But if they have coins, then the total amount is in cents. Wait, maybe the problem is that they have all their money in coins, and they want to convert as much as possible into bills, and the leftover is the coins that couldn't be converted. But if they have a total amount of money, which is $126.59, which is 12659 cents. Then, converting as many coins into bills as possible would mean converting all the coins into bills, except for the cents. Wait, but if you have 12659 cents, you can convert 126 dollars (which is 12600 cents) into bills, and you have 59 cents left. Therefore, the answer is 59 cents. So maybe that's the answer. But why is the problem giving individual amounts? Maybe there's something else here.

Wait, maybe the problem is that they had their money in coins, but each person had different coins, and when they pooled their money, they had some coins that couldn't be converted? Wait, but the problem says they "converted the coins into dollars", so maybe they had coins, and they wanted to exchange them into dollars. But if they have a total amount of money, which is $126.59, then converting all coins into bills would leave 59 cents. But maybe the problem is that they had coins, but the total amount is in dollars and cents, so they have to convert all coins into bills, and the leftover is the cents. Therefore, the answer is 59 cents. But maybe I need to check if there's a trick here.

Wait, let me think again. Maybe the problem is that they had their money in coins, but when they pooled their money, they had different coins, and when they went to the bank, they could exchange coins into bills, but the bank might not give them exact change? Wait, but the problem says "convert as many of the coins into bills as they could". So if they have coins, they can exchange them into bills. For example, if you have 100 cents, you can exchange it into 1 dollar bill. If you have 59 cents, you can't exchange it into a bill, so that's the leftover. Therefore, the total amount in cents is 126.59 dollars, which is 12659 cents. If you convert as many as possible into bills, you can convert 126 dollars (which is 12600 cents) into bills, leaving 59 cents. Therefore, the answer is 59 cents. But why is the problem giving individual amounts? Maybe there's something else here.

Wait, maybe the problem is that they had their money in coins, but the amounts they earned are in dollars and cents, so maybe they had some coins and some bills? But the problem says they "converted the coins into dollars", so maybe they had only coins, and they wanted to convert them into bills. Therefore, the total amount is $126.59, which is 12659 cents. Converting as much as possible into bills would mean converting all the coins into bills, except for the cents. Therefore, the leftover is 59 cents. Therefore, the answer is 59 cents. But maybe the problem is more complicated. Let me check again.

Alternatively, maybe the problem is that they had their money in coins, but when they pooled their money, they had different denominations, and when they went to the bank, they could only exchange coins into bills in certain ways, and the leftover is the coins that couldn't be converted. But unless there's a specific denomination constraint, which is not mentioned here, the standard way is that you can exchange any number of coins into bills. For example, if you have 100 cents, you can exchange it into a dollar bill. If you have 59 cents, you can't exchange it into a bill, so that's the leftover. Therefore, the answer is 59 cents.

But maybe the problem is that they had their money in coins, but the total amount is $126.59, which is 126 dollars and 59 cents. If they convert all the coins into bills, they would have 126 bills of $1 and 59 cents. But the question is asking how much money, in cents, was left after converting as many coins into bills as possible. So if they had 59 cents, that's the leftover. Therefore, the answer is 59 cents. But maybe the problem is expecting me to consider that they had coins, and when they converted them into bills, they had to give up some coins? Wait, but the problem says "convert the coins into dollars", so they are exchanging coins for dollars. So if they have coins, they can exchange them for bills. The leftover would be the coins that couldn't be converted. But if they have 59 cents, that's the leftover. Therefore, the answer is 59 cents.

But maybe I need to check if the total amount is actually in cents. Wait, the total amount is $126.59, which is 12659 cents. If they convert as many coins as possible into bills, then they can convert 126 dollars (which is 12600 cents) into bills, and have 59 cents left. Therefore, the answer is 59 cents. Therefore, the answer is 59 cents. But maybe the problem is more complicated. Let me check again.

Wait, maybe the problem is that they had their money in coins, but the amounts they earned are given in dollars and cents, so maybe they had some coins and some bills? But the problem says they "converted the coins into dollars", so maybe they had only coins. Therefore, the total amount is $126.59, which is 12659 cents. Converting as many as possible into bills would mean converting all the coins into bills, except for the cents. Therefore, the leftover is 59 cents. Therefore, the answer is 59 cents. But maybe the problem is expecting me to do something else. Let me check the problem again.

Wait, the problem says: "After the sale, they pooled their money together and went to the bank to convert the coins into dollars. How much money, in cents, was left after they converted as many of the coins into bills as they could?"

So, they had their money, which is the total of $126.59. But the key is that they had coins. So, if they had coins, they wanted to convert them into dollars. But if they had bills already, they wouldn't need to convert them. Wait, but the problem says they "converted the coins into dollars", so maybe they had some coins and some bills, and they wanted to convert the coins into dollars. But the problem says they "pooled their money together", so maybe they had all their money in coins. Therefore, converting the coins into dollars would mean exchanging all coins into bills, and the leftover would be the cents. Therefore, the answer is 59 cents. But maybe the problem is that they had coins, but the total amount is in dollars and cents, so they have to convert all coins into bills, and the leftover is the cents. Therefore, 59 cents.

Alternatively, maybe the problem is that they had their money in coins, but when they went to the bank, they could only exchange coins into bills in certain denominations, and the leftover is the coins that couldn't be converted. But unless there's a specific denomination constraint, which is not mentioned here, the standard way is that you can exchange any number of coins into bills. For example, if you have 100 cents, you can exchange it into a dollar bill. If you have 59 cents, you can't exchange it into a bill, so that's the leftover. Therefore, the answer is 59 cents.

But maybe the problem is expecting me to consider that they had coins, and when they converted them into bills, they had to give up some coins? Wait, but the problem says "convert the coins into dollars", so they are exchanging coins for dollars. Therefore, the leftover would be the coins that couldn't be converted. But if they have 59 cents, that's the leftover. Therefore, the answer is 59 cents.

Alternatively, maybe the problem is that they had their money in coins, but the total amount is $126.59, which is 126 dollars and 59 cents. If they convert all the coins into bills, they would have 126 bills of $1 and 59 cents. But the question is asking how much money, in cents, was left after converting as many coins into bills as they could. So if they had 59 cents, that's the leftover. Therefore, the answer is 59 cents.

But maybe I need to check if there's a different interpretation. For example, maybe they had coins, and they wanted to convert them into dollars, but the bank only gives them whole dollars, so they have to give up coins to get dollars. For example, if you have 100 cents, you can exchange it for 1 dollar. If you have 59 cents, you can't exchange it for a dollar, so you have 59 cents left. Therefore, the answer is 59 cents.

Alternatively, maybe the problem is that they had coins, and they wanted to convert them into bills, but the total amount is in dollars and cents, so they have to convert all coins into bills, and the leftover is the cents. Therefore, the answer is 59 cents.

But maybe the problem is more complicated. Let me think again. Maybe the problem is that they had their money in coins, but when they pooled their money, they had different coins, and when they went to the bank, they could only exchange coins into bills in certain ways, and the leftover is the coins that couldn't be converted. But unless there's a specific denomination constraint, which is not mentioned here, the answer should be 59 cents.

Alternatively, maybe the problem is that they had their money in coins, but the total amount is $126.59, which is 12659 cents. If they convert as many coins as possible into bills, they can convert 126 dollars (which is 12600 cents) into bills, leaving 59 cents. Therefore, the answer is 59 cents.

But maybe the problem is expecting me to consider that they had coins, and when they converted them into bills, they had to give up some coins, but the question is about the leftover coins. However, if they have 59 cents, that's the leftover. Therefore, the answer is 59 cents.

Alternatively, maybe the problem is that they had their money in coins, but the amounts they earned are given in dollars and cents, so maybe they had some coins and some bills, and they wanted to convert the coins into bills. But the problem says they "pooled their money together", so maybe they had all their money in coins. Therefore, converting all coins into bills would leave 59 cents.

Alternatively, maybe the problem is that they had their money in coins, but when they went to the bank, they had to give up coins to get bills, and the leftover is the coins that couldn't be converted. But again, if they have 59 cents, that's the leftover.

Therefore, after all this thinking, I think the answer is 59 cents. Therefore, the answer is \boxed{59}.

Wait, but let me check again. Maybe I made a mistake in adding up the total amount. Let me check the total again.

Linda: $27.47

Sherry: $35.23

June: $37.37

Connie: $26.52

Adding them up:

First, add Linda and Sherry: 27.47 + 35.23. Let me do this again:

27.47 + 35.23:

27 + 35 = 62

0.47 + 0.23 = 0.70

Total: 62.70

Then add June's 37.37:

62.70 + 37.37:

62 + 37 = 99

0.70 + 0.37 = 1.07

So total is 99 + 1.07 = 100.07

Then add Connie's 26.52:

100.07 + 26.52:

100 + 26 = 126

0.07 + 0.52 = 0.59

Total: 126.59. Yes, that's correct. So total is $126.59, which is 126 dollars and 59 cents. Therefore, converting as many coins as possible into bills would leave 59 cents. Therefore, the answer is 59 cents.

But maybe the problem is that they had coins, but the total amount is in dollars and cents, so they have to convert all coins into bills, and the leftover is the cents. Therefore, 59 cents.

Alternatively, maybe the problem is that they had coins, and when they converted them into bills, they had to give up coins for bills, but the question is about the leftover coins. But if they have 59 cents, that's the leftover.

Therefore, I think the answer is 59 cents. Therefore, the answer is \boxed{59}.

**Final Answer**
\boxed{59}

To solve the problem, we begin by calculating the total amount of money Linda, Sherry, June, and Connie earned from selling cookies. Each person's earnings are given in dollars and cents, so we sum them up:

- Linda: \$27.47
- Sherry: \$35.23
- June: \$37.37
- Connie: \$26.52

---

### Step 1: Add the amounts

We add the amounts step by step:

1. **Linda + Sherry**
   $27.47 + 35.23 = 62.70$

2. **Add June**
   $62.70 + 37.37 = 100.07$

3. **Add Connie**
   $100.07 + 26.52 = 126.59$

So, the total amount of money they pooled together is **\$126.59**.

---

### Step 2: Convert the total into cents

To work with the problem in terms of **cents**, we convert the total amount:

$$
126.59 \text{ dollars} = 126 \times 100 + 59 = 12659 \text{ cents}
$$

---

### Step 3: Convert as many coins as possible into bills

The problem states that they went to the bank to **convert the coins into dollars**. This implies that they had **only coins** (not bills) and wanted to exchange them into **bills**.

When you convert coins into bills, you can exchange **any number of cents** into **whole dollars**. For example:

- 100 cents = 1 dollar
- 50 cents = 0.50 dollars (but not a whole bill)
- 59 cents = 0.59 dollars (not a whole bill)

So, the **maximum number of whole dollars** they can convert is:

$$
\left\lfloor \frac{12659}{100} \right\rfloor = 126 \text{ dollars}
$$

This corresponds to:

$$
126 \times 100 = 12600 \text{ cents}
$$

---

### Step 4: Calculate the leftover cents

Subtract the amount converted into bills from the total:

$$
12659 - 12600 = 59 \text{ cents}
$$

These 59 cents **cannot** be converted into a whole dollar bill, so they remain as **leftover coins**.

---

### Final Answer

$$
\boxed{59}
$$

[Assistant]
Explanation:
Okay, so Linda, Sherry, June, and Connie sold cookies and earned some money. Now they want to convert all their coins into dollars, and we need to find out how much money, in cents, was left after they converted as many coins as possible into bills. Hmm, let me think.

First, I need to figure out the total amount of money they have. Let me add up each person's earnings. Let me write down the amounts:

- Linda: $27.47
- Sherry: $35.23
- June: $37.37
- Connie: $26.52

So adding these up. Let me do it step by step. Let me add Linda and Sherry first. 27.47 + 35.23. Let me add the dollars first: 27 + 35 = 62. Then the cents: 47 cents + 23 cents = 70 cents. So that's $62.70.

Now add June's amount: $37.37. So adding $62.70 + $37.37. Again, dollars: 62 + 37 = 99. Cents: 70 + 37 = 107 cents. Wait, 107 cents is equal to $1.07. So that would make the total $99 + $1.07 = $100.07. Wait, let me check that again. Maybe I should do it more carefully.

Wait, 62.70 + 37.37. Let me write it vertically:

  62.70
+ 37.37
--------
Let me add the cents first: 70 + 37 = 107 cents. So that's 1 dollar and 7 cents. Then the dollars: 62 + 37 = 99, plus the 1 dollar from the cents, so total is 100 dollars and 7 cents. So total is $100.07. Okay, that seems right.

Now add Connie's $26.52. So $100.07 + $26.52. Let me do that. Again, dollars: 100 + 26 = 126. Cents: 07 + 52 = 59 cents. So total is $126.59. Let me check that again. 100.07 + 26.52. 100 + 26 is 126, 0.07 + 0.52 is 0.59. Yes, so total is $126.59. So altogether, they have $126.59.

But wait, the question says they pooled their money together and went to the bank to convert the coins into dollars. How much money, in cents, was left after they converted as many of the coins into bills as they could?

Wait, so they have a total of $126.59. But the question is about converting coins into dollars. Wait, but if they have dollars already, maybe they have some coins? Wait, maybe the problem is that they have all their money in coins, and they want to convert them into bills? But the problem says they "pooled their money together and went to the bank to convert the coins into dollars." So maybe they had coins, and they wanted to exchange them into bills, and we need to find out how much was left in cents after converting as much as possible into dollars.

But wait, the problem says they earned certain amounts. But the amounts are given in dollars and cents. So maybe they have those amounts in cash, which includes both bills and coins? But the problem says they converted the coins into dollars. So perhaps they had some coins, and they wanted to exchange them into bills, and the question is about the leftover coins after exchanging as much as possible.

But the problem is a bit ambiguous. Wait, let me read again:

"After the sale, they pooled their money together and went to the bank to convert the coins into dollars. How much money, in cents, was left after they converted as many of the coins into bills as they could?"

So, they had money, which includes coins, and they went to the bank to convert the coins into dollars. So the total amount they have is the sum of their earnings, which is $126.59. But if they have that amount, which is in dollars and cents, but they want to convert the coins into dollars. Wait, but if they have dollars already, maybe they have some coins? Wait, but the problem says they "converted the coins into dollars", so maybe they had only coins, and they wanted to exchange them into bills. But the problem says they "pooled their money together", which is the total amount they earned, which is $126.59. But if they had that amount in coins, then converting them into bills would mean exchanging all coins into bills, and there would be no leftover coins. But that can't be, because the question is asking how much was left in cents. Therefore, there must be some coins that couldn't be converted into bills. Wait, but how?

Wait, maybe the problem is that they had their money in coins, and they wanted to convert as much as possible into bills, and the leftover coins would be the cents. But if they have a total amount of money, which is $126.59, which is 126 dollars and 59 cents. If they convert all the coins into bills, then they would have 126 dollars and 59 cents. But if they have only coins, then they would have to exchange the coins into bills. But if they have 59 cents, that's the leftover. Wait, but the question says "how much money, in cents, was left after they converted as many of the coins into bills as they could?"

Wait, maybe the problem is that they had all their money in coins, and they wanted to convert as much as possible into bills, and the leftover would be the coins that couldn't be converted. But if they have a total amount of money, which is $126.59, which is 126 dollars and 59 cents. If they convert all the coins into bills, then they would have 126 dollars and 59 cents. But if they have only coins, then they would have to exchange the coins into bills. However, if they have 59 cents, that's the leftover. But the question is in cents, so 59 cents. But that seems too straightforward. However, maybe the problem is that they had coins, but the total amount is in dollars and cents, so maybe they have some coins that can't be converted into bills? Wait, but bills are dollars, so if you have coins, you can exchange them into bills. For example, if you have 100 cents, you can exchange them into 1 dollar bill. If you have 59 cents, you can't exchange them into a bill, so that's the leftover. Therefore, the answer would be 59 cents. But that seems too easy. However, maybe the problem is more complicated. Let me check again.

Wait, the problem says they "converted the coins into dollars". So maybe they had coins, and they wanted to convert them into dollars. But the total amount they have is $126.59. But if they have that amount in coins, then converting them into bills would result in 126 dollars and 59 cents. But the question is asking how much was left after converting as many coins into bills as possible. So if they have 59 cents, that's the leftover. Therefore, the answer is 59 cents. But maybe I need to check if there's something else here.

Wait, but maybe the problem is that they had their money in coins, but the amounts they earned are given in dollars and cents. So maybe they have the total amount in coins, which is $126.59, and they want to convert as much as possible into bills. But if they have coins, then the total amount is in cents. Wait, maybe the problem is that they have all their money in coins, and they want to convert as much as possible into bills, and the leftover is the coins that couldn't be converted. But if they have a total amount of money, which is $126.59, which is 12659 cents. Then, converting as many coins into bills as possible would mean converting all the coins into bills, except for the cents. Wait, but if you have 12659 cents, you can convert 126 dollars (which is 12600 cents) into bills, and you have 59 cents left. Therefore, the answer is 59 cents. So maybe that's the answer. But why is the problem giving individual amounts? Maybe there's something else here.

Wait, maybe the problem is that they had their money in coins, but each person had different coins, and when they pooled their money, they had some coins that couldn't be converted? Wait, but the problem says they "converted the coins into dollars", so maybe they had coins, and they wanted to exchange them into dollars. But if they have a total amount of money, which is $126.59, then converting all coins into bills would leave 59 cents. But maybe the problem is that they had coins, but the total amount is in dollars and cents, so they have to convert all coins into bills, and the leftover is the cents. Therefore, the answer is 59 cents. But maybe I need to check if there's a trick here.

Wait, let me think again. Maybe the problem is that they had their money in coins, but when they pooled their money, they had different coins, and when they went to the bank, they could exchange coins into bills, but the bank might not give them exact change? Wait, but the problem says "convert as many of the coins into bills as they could". So if they have coins, they can exchange them into bills. For example, if you have 100 cents, you can exchange it into 1 dollar bill. If you have 59 cents, you can't exchange it into a bill, so that's the leftover. Therefore, the total amount in cents is 126.59 dollars, which is 12659 cents. If you convert as many as possible into bills, you can convert 126 dollars (which is 12600 cents) into bills, leaving 59 cents. Therefore, the answer is 59 cents. But why is the problem giving individual amounts? Maybe there's something else here.

Wait, maybe the problem is that they had their money in coins, but the amounts they earned are in dollars and cents, so maybe they had some coins and some bills? But the problem says they "converted the coins into dollars", so maybe they had only coins, and they wanted to convert them into bills. Therefore, the total amount is $126.59, which is 12659 cents. Converting as much as possible into bills would mean converting all the coins into bills, except for the cents. Therefore, the leftover is 59 cents. Therefore, the answer is 59 cents. But maybe the problem is more complicated. Let me check again.

Alternatively, maybe the problem is that they had their money in coins, but when they pooled their money, they had different denominations, and when they went to the bank, they could only exchange coins into bills in certain ways, and the leftover is the coins that couldn't be converted. But unless there's a specific denomination constraint, which is not mentioned here, the standard way is that you can exchange any number of coins into bills. For example, if you have 100 cents, you can exchange it into a dollar bill. If you have 59 cents, you can't exchange it into a bill, so that's the leftover. Therefore, the answer is 59 cents.

But maybe the problem is that they had their money in coins, but the total amount is $126.59, which is 126 dollars and 59 cents. If they convert all the coins into bills, they would have 126 bills of $1 and 59 cents. But the question is asking how much money, in cents, was left after converting as many coins into bills as possible. So if they had 59 cents, that's the leftover. Therefore, the answer is 59 cents. But maybe the problem is expecting me to consider that they had coins, and when they converted them into bills, they had to give up some coins? Wait, but the problem says "convert the coins into dollars", so they are exchanging coins for dollars. So if they have coins, they can exchange them for bills. The leftover would be the coins that couldn't be converted. But if they have 59 cents, that's the leftover. Therefore, the answer is 59 cents.

But maybe I need to check if the total amount is actually in cents. Wait, the total amount is $126.59, which is 12659 cents. If they convert as many coins as possible into bills, then they can convert 126 dollars (which is 12600 cents) into bills, and have 59 cents left. Therefore, the answer is 59 cents. Therefore, the answer is 59 cents. But maybe the problem is more complicated. Let me check again.

Wait, maybe the problem is that they had their money in coins, but the amounts they earned are given in dollars and cents, so maybe they had some coins and some bills? But the problem says they "converted the coins into dollars", so maybe they had only coins. Therefore, the total amount is $126.59, which is 12659 cents. Converting as many as possible into bills would mean converting all the coins into bills, except for the cents. Therefore, the leftover is 59 cents. Therefore, the answer is 59 cents. But maybe the problem is expecting me to do something else. Let me check the problem again.

Wait, the problem says: "After the sale, they pooled their money together and went to the bank to convert the coins into dollars. How much money, in cents, was left after they converted as many of the coins into bills as they could?"

So, they had their money, which is the total of $126.59. But the key is that they had coins. So, if they had coins, they wanted to convert them into dollars. But if they had bills already, they wouldn't need to convert them. Wait, but the problem says they "converted the coins into dollars", so maybe they had some coins and some bills, and they wanted to convert the coins into dollars. But the problem says they "pooled their money together", so maybe they had all their money in coins. Therefore, converting the coins into dollars would mean exchanging all coins into bills, and the leftover would be the cents. Therefore, the answer is 59 cents. But maybe the problem is that they had coins, but the total amount is in dollars and cents, so they have to convert all coins into bills, and the leftover is the cents. Therefore, 59 cents.

Alternatively, maybe the problem is that they had their money in coins, but when they went to the bank, they could only exchange coins into bills in certain denominations, and the leftover is the coins that couldn't be converted. But unless there's a specific denomination constraint, which is not mentioned here, the standard way is that you can exchange any number of coins into bills. For example, if you have 100 cents, you can exchange it into a dollar bill. If you have 59 cents, you can't exchange it into a bill, so that's the leftover. Therefore, the answer is 59 cents.

But maybe the problem is expecting me to consider that they had coins, and when they converted them into bills, they had to give up some coins? Wait, but the problem says "convert the coins into dollars", so they are exchanging coins for dollars. Therefore, the leftover would be the coins that couldn't be converted. But if they have 59 cents, that's the leftover. Therefore, the answer is 59 cents.

Alternatively, maybe the problem is that they had their money in coins, but the total amount is $126.59, which is 126 dollars and 59 cents. If they convert all the coins into bills, they would have 126 bills of $1 and 59 cents. But the question is asking how much money, in cents, was left after converting as many coins into bills as they could. So if they had 59 cents, that's the leftover. Therefore, the answer is 59 cents.

But maybe I need to check if there's a different interpretation. For example, maybe they had coins, and they wanted to convert them into dollars, but the bank only gives them whole dollars, so they have to give up coins to get dollars. For example, if you have 100 cents, you can exchange it for 1 dollar. If you have 59 cents, you can't exchange it for a dollar, so you have 59 cents left. Therefore, the answer is 59 cents.

Alternatively, maybe the problem is that they had coins, and they wanted to convert them into bills, but the total amount is in dollars and cents, so they have to convert all coins into bills, and the leftover is the cents. Therefore, the answer is 59 cents.

But maybe the problem is more complicated. Let me think again. Maybe the problem is that they had their money in coins, but when they pooled their money, they had different coins, and when they went to the bank, they could only exchange coins into bills in certain ways, and the leftover is the coins that couldn't be converted. But unless there's a specific denomination constraint, which is not mentioned here, the answer should be 59 cents.

Alternatively, maybe the problem is that they had their money in coins, but the total amount is $126.59, which is 12659 cents. If they convert as many coins as possible into bills, they can convert 126 dollars (which is 12600 cents) into bills, leaving 59 cents. Therefore, the answer is 59 cents.

But maybe the problem is expecting me to consider that they had coins, and when they converted them into bills, they had to give up some coins, but the question is about the leftover coins. However, if they have 59 cents, that's the leftover. Therefore, the answer is 59 cents.

Alternatively, maybe the problem is that they had their money in coins, but the amounts they earned are given in dollars and cents, so maybe they had some coins and some bills, and they wanted to convert the coins into bills. But the problem says they "pooled their money together", so maybe they had all their money in coins. Therefore, converting all coins into bills would leave 59 cents.

Alternatively, maybe the problem is that they had their money in coins, but when they went to the bank, they had to give up coins to get bills, and the leftover is the coins that couldn't be converted. But again, if they have 59 cents, that's the leftover.

Therefore, after all this thinking, I think the answer is 59 cents. Therefore, the answer is \boxed{59}.

Wait, but let me check again. Maybe I made a mistake in adding up the total amount. Let me check the total again.

Linda: $27.47

Sherry: $35.23

June: $37.37

Connie: $26.52

Adding them up:

First, add Linda and Sherry: 27.47 + 35.23. Let me do this again:

27.47 + 35.23:

27 + 35 = 62

0.47 + 0.23 = 0.70

Total: 62.70

Then add June's 37.37:

62.70 + 37.37:

62 + 37 = 99

0.70 + 0.37 = 1.07

So total is 99 + 1.07 = 100.07

Then add Connie's 26.52:

100.07 + 26.52:

100 + 26 = 126

0.07 + 0.52 = 0.59

Total: 126.59. Yes, that's correct. So total is $126.59, which is 126 dollars and 59 cents. Therefore, converting as many coins as possible into bills would leave 59 cents. Therefore, the answer is 59 cents.

But maybe the problem is that they had coins, but the total amount is in dollars and cents, so they have to convert all coins into bills, and the leftover is the cents. Therefore, 59 cents.

Alternatively, maybe the problem is that they had coins, and when they converted them into bills, they had to give up coins for bills, but the question is about the leftover coins. But if they have 59 cents, that's the leftover.

Therefore, I think the answer is 59 cents. Therefore, the answer is \boxed{59}.

**Final Answer**
\boxed{59}

To solve the problem, we begin by calculating the total amount of money Linda, Sherry, June, and Connie earned from selling cookies. Each person's earnings are given in dollars and cents, so we sum them up:

- Linda: \$27.47
- Sherry: \$35.23
- June: \$37.37
- Connie: \$26.52

---

### Step 1: Add the amounts

We add the amounts step by step:

1. **Linda + Sherry**
   $27.47 + 35.23 = 62.70$

2. **Add June**
   $62.70 + 37.37 = 100.07$

3. **Add Connie**
   $100.07 + 26.52 = 126.59$

So, the total amount of money they pooled together is **\$126.59**.

---

### Step 2: Convert the total into cents

To work with the problem in terms of **cents**, we convert the total amount:

$$
126.59 \text{ dollars} = 126 \times 100 + 59 = 12659 \text{ cents}
$$

---

### Step 3: Convert as many coins as possible into bills

The problem states that they went to the bank to **convert the coins into dollars**. This implies that they had **only coins** (not bills) and wanted to exchange them into **bills**.

When you convert coins into bills, you can exchange **any number of cents** into **whole dollars**. For example:

- 100 cents = 1 dollar
- 50 cents = 0.50 dollars (but not a whole bill)
- 59 cents = 0.59 dollars (not a whole bill)

So, the **maximum number of whole dollars** they can convert is:

$$
\left\lfloor \frac{12659}{100} \right\rfloor = 126 \text{ dollars}
$$

This corresponds to:

$$
126 \times 100 = 12600 \text{ cents}
$$

---

### Step 4: Calculate the leftover cents

Subtract the amount converted into bills from the total:

$$
12659 - 12600 = 59 \text{ cents}
$$

These 59 cents **cannot** be converted into a whole dollar bill, so they remain as **leftover coins**.

---

### Final Answer

$$
\boxed{59}
$$

Solution:
Instead of adding large numbers together, we can find the residue for each person for easier computation. We convert the amount they earned to cents and find the modulo $100$ for each.  \begin{align*}
2747 &\equiv 47 \pmod{100},\\
3523 &\equiv 23 \pmod{100},\\
3737 &\equiv 37 \pmod{100},\\
2652 &\equiv 52 \pmod{100}
\end{align*}We want to find the modulo $100$ of the total number of cents. We can add the separate residues to get $$47+23+37+52 \equiv 159 \equiv 59 \pmod{100}$$Therefore, they have $\boxed{59}$ cents left after converting as much of the money into bills as possible.


Explanation:
Okay, so I need to figure out for how many values of x the expression (x - 5)/(x² - 9) is not defined. Hmm, let me recall. When is a rational expression not defined? Well, it's undefined when the denominator is zero because division by zero is undefined. So, I need to find the values of x that make the denominator zero. The denominator here is x² - 9.

Alright, so let me set the denominator equal to zero and solve for x. That would be x² - 9 = 0. Let me solve this equation.

x² - 9 = 0
Adding 9 to both sides: x² = 9
Taking the square root of both sides: x = ±√9
Which simplifies to x = 3 or x = -3.

Wait, so the denominator is zero when x is 3 or -3. Therefore, the expression is undefined at these two values. So, does that mean there are two values of x where the expression is not defined?

Hold on, let me double-check. Maybe I made a mistake in solving the equation. Let me verify.

Original denominator: x² - 9. This is a difference of squares, right? So, it factors into (x - 3)(x + 3). So, setting that equal to zero, we have (x - 3)(x + 3) = 0. Therefore, the solutions are x = 3 and x = -3. Yep, that's correct.

So, those are the two values where the denominator is zero. Therefore, the expression is undefined at x = 3 and x = -3.

But wait, let me check if the numerator is also zero at these points. Because if both numerator and denominator are zero, then it's an indeterminate form, but technically, the expression is still undefined. However, the question is just asking for when the expression is not defined, regardless of whether it's an indeterminate form or not.

So, let me check the numerator at x = 3 and x = -3.

Numerator is x - 5.

At x = 3: 3 - 5 = -2 ≠ 0.

At x = -3: -3 - 5 = -8 ≠ 0.

So, neither of these make the numerator zero. Therefore, the expression is undefined at x = 3 and x = -3, and there are no other points where the denominator is zero.

Therefore, there are two values of x where the expression is not defined.

Wait, but let me make sure there are no other factors or something. The denominator is x² - 9, which we factored into (x - 3)(x + 3). There are no other factors. So, the denominator is zero only when x is 3 or -3.

Therefore, the answer should be 2.

But let me think again. Maybe there's something else? For example, sometimes people get confused between the expression being undefined and the expression being undefined for certain reasons. But in this case, the only reason the expression is undefined is when the denominator is zero.

So, unless there's some other restriction on x, but the problem doesn't mention any other restrictions. So, the expression is a rational function, and the domain is all real numbers except where the denominator is zero. Therefore, the expression is undefined at x = 3 and x = -3.

Therefore, the number of values is 2.

Wait, but maybe I need to check if there are any other points where the expression is undefined? For example, if there was a variable in the denominator with some other exponent or something. But here, the denominator is x² - 9, which is a quadratic. So, it's only zero at two points.

Alternatively, maybe the question is trying to trick me? Let me check again.

Original expression: (x - 5)/(x² - 9).

So, denominator is x² - 9. So, solving x² - 9 = 0 gives x = 3 and x = -3.

Therefore, two values.

Alternatively, maybe the question is in a different context? For example, if x is a complex number? But the question doesn't specify. Usually, unless stated otherwise, we assume x is real.

Therefore, in real numbers, there are two values.

Therefore, the answer is 2.

But let me check once more. Maybe I can graph the function or think about it.

The denominator x² - 9 is a parabola opening upwards, crossing the x-axis at -3 and 3. So, the expression is undefined at those two points.

Yes, so there are two values.

Therefore, I think the answer is 2.

**Final Answer**
The expression is not defined for \boxed{2} values of $ x $.

To determine for how many values of $ x $ the expression
$$
\frac{x - 5}{x^2 - 9}
$$
is **not defined**, we need to analyze when the expression becomes undefined.

---

### Step 1: Understand When a Rational Expression is Undefined

A rational expression is undefined when its **denominator is zero**, because division by zero is not defined in mathematics. Therefore, we need to find the values of $ x $ that make the denominator zero.

---

### Step 2: Set the Denominator Equal to Zero

The denominator of the expression is:
$$
x^2 - 9
$$

Set it equal to zero:
$$
x^2 - 9 = 0
$$

This is a difference of squares, so we can factor it:
$$
(x - 3)(x + 3) = 0
$$

Solving for $ x $, we get:
$$
x = 3 \quad \text{or} \quad x = -3
$$

---

### Step 3: Check for Indeterminate Forms

It's important to note that even if the numerator were also zero at these points, the expression would still be undefined (it would be an indeterminate form like $ \frac{0}{0} $), but the question is only asking for when the expression is **not defined**, not when it is undefined due to indeterminate forms.

Let’s check the numerator at these points:

- At $ x = 3 $: $ x - 5 = 3 - 5 = -2 \neq 0 $
- At $ x = -3 $: $ x - 5 = -3 - 5 = -8 \neq 0 $

So, the numerator is not zero at either of these points, and the expression is **undefined** at both $ x = 3 $ and $ x = -3 $.

---

### Step 4: Confirm There Are No Other Points of Undefined Behavior

The denominator is a quadratic expression, and we have already found all real solutions to $ x^2 - 9 = 0 $. Since this is a degree-2 polynomial, it can have at most two real roots, and we have found both of them.

There are no other values of $ x $ that make the denominator zero, and the numerator does not introduce any additional undefined points.

---

### Final Conclusion

The expression is **not defined** for exactly **two** values of $ x $: $ x = 3 $ and $ x = -3 $.

$$
\boxed{2}
$$

[Assistant]
Explanation:
Okay, so I need to figure out for how many values of x the expression (x - 5)/(x² - 9) is not defined. Hmm, let me recall. When is a rational expression not defined? Well, it's undefined when the denominator is zero because division by zero is undefined. So, I need to find the values of x that make the denominator zero. The denominator here is x² - 9.

Alright, so let me set the denominator equal to zero and solve for x. That would be x² - 9 = 0. Let me solve this equation.

x² - 9 = 0
Adding 9 to both sides: x² = 9
Taking the square root of both sides: x = ±√9
Which simplifies to x = 3 or x = -3.

Wait, so the denominator is zero when x is 3 or -3. Therefore, the expression is undefined at these two values. So, does that mean there are two values of x where the expression is not defined?

Hold on, let me double-check. Maybe I made a mistake in solving the equation. Let me verify.

Original denominator: x² - 9. This is a difference of squares, right? So, it factors into (x - 3)(x + 3). So, setting that equal to zero, we have (x - 3)(x + 3) = 0. Therefore, the solutions are x = 3 and x = -3. Yep, that's correct.

So, those are the two values where the denominator is zero. Therefore, the expression is undefined at x = 3 and x = -3.

But wait, let me check if the numerator is also zero at these points. Because if both numerator and denominator are zero, then it's an indeterminate form, but technically, the expression is still undefined. However, the question is just asking for when the expression is not defined, regardless of whether it's an indeterminate form or not.

So, let me check the numerator at x = 3 and x = -3.

Numerator is x - 5.

At x = 3: 3 - 5 = -2 ≠ 0.

At x = -3: -3 - 5 = -8 ≠ 0.

So, neither of these make the numerator zero. Therefore, the expression is undefined at x = 3 and x = -3, and there are no other points where the denominator is zero.

Therefore, there are two values of x where the expression is not defined.

Wait, but let me make sure there are no other factors or something. The denominator is x² - 9, which we factored into (x - 3)(x + 3). There are no other factors. So, the denominator is zero only when x is 3 or -3.

Therefore, the answer should be 2.

But let me think again. Maybe there's something else? For example, sometimes people get confused between the expression being undefined and the expression being undefined for certain reasons. But in this case, the only reason the expression is undefined is when the denominator is zero.

So, unless there's some other restriction on x, but the problem doesn't mention any other restrictions. So, the expression is a rational function, and the domain is all real numbers except where the denominator is zero. Therefore, the expression is undefined at x = 3 and x = -3.

Therefore, the number of values is 2.

Wait, but maybe I need to check if there are any other points where the expression is undefined? For example, if there was a variable in the denominator with some other exponent or something. But here, the denominator is x² - 9, which is a quadratic. So, it's only zero at two points.

Alternatively, maybe the question is trying to trick me? Let me check again.

Original expression: (x - 5)/(x² - 9).

So, denominator is x² - 9. So, solving x² - 9 = 0 gives x = 3 and x = -3.

Therefore, two values.

Alternatively, maybe the question is in a different context? For example, if x is a complex number? But the question doesn't specify. Usually, unless stated otherwise, we assume x is real.

Therefore, in real numbers, there are two values.

Therefore, the answer is 2.

But let me check once more. Maybe I can graph the function or think about it.

The denominator x² - 9 is a parabola opening upwards, crossing the x-axis at -3 and 3. So, the expression is undefined at those two points.

Yes, so there are two values.

Therefore, I think the answer is 2.

**Final Answer**
The expression is not defined for \boxed{2} values of $ x $.

To determine for how many values of $ x $ the expression
$$
\frac{x - 5}{x^2 - 9}
$$
is **not defined**, we need to analyze when the expression becomes undefined.

---

### Step 1: Understand When a Rational Expression is Undefined

A rational expression is undefined when its **denominator is zero**, because division by zero is not defined in mathematics. Therefore, we need to find the values of $ x $ that make the denominator zero.

---

### Step 2: Set the Denominator Equal to Zero

The denominator of the expression is:
$$
x^2 - 9
$$

Set it equal to zero:
$$
x^2 - 9 = 0
$$

This is a difference of squares, so we can factor it:
$$
(x - 3)(x + 3) = 0
$$

Solving for $ x $, we get:
$$
x = 3 \quad \text{or} \quad x = -3
$$

---

### Step 3: Check for Indeterminate Forms

It's important to note that even if the numerator were also zero at these points, the expression would still be undefined (it would be an indeterminate form like $ \frac{0}{0} $), but the question is only asking for when the expression is **not defined**, not when it is undefined due to indeterminate forms.

Let’s check the numerator at these points:

- At $ x = 3 $: $ x - 5 = 3 - 5 = -2 \neq 0 $
- At $ x = -3 $: $ x - 5 = -3 - 5 = -8 \neq 0 $

So, the numerator is not zero at either of these points, and the expression is **undefined** at both $ x = 3 $ and $ x = -3 $.

---

### Step 4: Confirm There Are No Other Points of Undefined Behavior

The denominator is a quadratic expression, and we have already found all real solutions to $ x^2 - 9 = 0 $. Since this is a degree-2 polynomial, it can have at most two real roots, and we have found both of them.

There are no other values of $ x $ that make the denominator zero, and the numerator does not introduce any additional undefined points.

---

### Final Conclusion

The expression is **not defined** for exactly **two** values of $ x $: $ x = 3 $ and $ x = -3 $.

$$
\boxed{2}
$$

Solution: The expression is not defined when the denominator is equal to zero. Thus, we need to find the number of values of $x$ such that $x^2-9=0$. Rearranging the equation and taking the square root of both sides, we have $x^2-9=0\Rightarrow x^2=9 \Rightarrow x=\pm3$. Thus, there are $\boxed{2}$ different values of $x$ that make the given expression undefined.


Explanation:
Okay, so I need to find the area of the shaded triangle in the diagram. Let me try to visualize it first. The Asymptote code draws a rectangle with length 10 cm on the bottom and top, and height 3 cm on the sides. Then there's a triangle connecting the bottom-left corner (0,0) to the point (2,3) and then to the bottom-right corner (10,0). The shaded area is this triangle.

First, let me recall that the area of a triangle is (base * height)/2. But I need to figure out the base and the height of the shaded triangle. However, since the triangle isn't necessarily aligned with the axes, maybe I need to use coordinates to calculate it?

Alternatively, maybe I can use coordinates to find the area. The coordinates of the triangle are (0,0), (2,3), and (10,0). So if I can use the coordinates, I can apply the shoelace formula or something like that. Let me try that.

The shoelace formula for three points (x1,y1), (x2,y2), (x3,y3) is:

Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))/2|

Let me plug in the coordinates:

x1 = 0, y1 = 0

x2 = 2, y2 = 3

x3 = 10, y3 = 0

So plugging into the formula:

Area = |(0*(3 - 0) + 2*(0 - 0) + 10*(0 - 3))/2|

Calculating each term:

First term: 0*(3 - 0) = 0*3 = 0

Second term: 2*(0 - 0) = 2*0 = 0

Third term: 10*(0 - 3) = 10*(-3) = -30

Adding them up: 0 + 0 - 30 = -30

Take absolute value and divide by 2: |-30| / 2 = 30 / 2 = 15

So the area is 15 cm²? Wait, but let me check if that's correct. Let me think again.

Alternatively, maybe I can divide the figure into parts. The rectangle has area 10 cm * 3 cm = 30 cm². The shaded triangle is part of this rectangle. If I can find the area of the triangle by subtracting other areas?

But maybe not. Alternatively, since the triangle is formed by points (0,0), (2,3), (10,0). Let me try to find the base and height. If I take the base as the side from (0,0) to (10,0), which is 10 cm. Then the height would be the vertical distance from the point (2,3) to this base. But since the base is along the x-axis, the height is just the y-coordinate of the point (2,3), which is 3 cm. Wait, but then area would be (10 * 3)/2 = 15 cm². Which matches the shoelace formula. So that seems correct.

But wait, hold on. If the base is 10 cm and the height is 3 cm, then the area is 15 cm². But is that accurate? Because the triangle isn't necessarily having the base as the entire length of the rectangle. Wait, but the triangle is connecting (0,0) to (2,3) to (10,0). So if I take the base as the base from (0,0) to (10,0), which is 10 cm, then the height would be the perpendicular distance from (2,3) to this base. Since the base is horizontal, the vertical distance is indeed 3 cm. So that would make the area 15 cm².

But wait, but if I imagine the triangle, it's not a right triangle. Wait, but if I take the base as 10 cm, then the height is 3 cm. But is that correct? Let me think. If the triangle is between (0,0), (10,0), and (2,3), then the height is not necessarily 3 cm. Wait, no. Because the height is the perpendicular distance from the base to the opposite vertex. Since the base is along the x-axis from (0,0) to (10,0), then the height is the vertical distance from (2,3) to the x-axis, which is indeed 3 cm. So that's correct. Therefore, area is 1/2 * base * height = 1/2 * 10 * 3 = 15 cm².

But let me check with another method. Let me use coordinates again. The coordinates are (0,0), (2,3), (10,0). Let me use vectors or something else. Alternatively, maybe divide the triangle into two parts?

Alternatively, think of the triangle as a large triangle minus some smaller triangles? Wait, but maybe not necessary. Since both methods gave me 15 cm², maybe that's correct. But let me check with the shoelace formula again.

Shoelace formula:

List the coordinates in order: (0,0), (2,3), (10,0), (0,0)

Compute sum of products going one way:

(0*3) + (2*0) + (10*0) = 0 + 0 + 0 = 0

Compute sum of products going the other way:

(0*2) + (3*10) + (0*0) = 0 + 30 + 0 = 30

Subtract the two: |0 - 30| = 30, divide by 2: 15. Yep, same result. So that's correct.

But wait, the problem is presented with a diagram. Let me check if the diagram is as I imagine. The Asymptote code draws a rectangle from (0,0) to (10,0) to (10,3) to (0,3) and back. Then draws the triangle from (0,0) to (2,3) to (10,0). So the shaded triangle is inside the rectangle.

But maybe there's a trick here? For example, maybe the triangle is not as straightforward? Let me think again.

Alternatively, maybe the base is not 10 cm? Wait, the base from (0,0) to (10,0) is 10 cm. The height is 3 cm. So area is 15. But maybe the problem is expecting something else? Let me check the labels. The diagram labels the top side as 10 cm, the bottom side as 10 cm, and the left and right sides as 3 cm. So the rectangle is 10 cm by 3 cm.

But then the triangle is connecting (0,0) to (2,3) to (10,0). So the point (2,3) is on the top side of the rectangle. Wait, the top side is from (0,3) to (10,3). So (2,3) is 2 cm from the left on the top side. So the triangle is formed by the bottom-left corner, a point 2 cm from the left on the top, and the bottom-right corner.

Alternatively, maybe I can use coordinates to find the area. But I already did that.

Alternatively, maybe using coordinates, find the equations of the lines and integrate or something? Probably overkill.

Alternatively, think of the triangle as two smaller triangles? For example, from (0,0) to (2,3) to (10,0). Maybe split it into two triangles? But that might complicate.

Alternatively, use determinants. Which is essentially what the shoelace formula is.

Alternatively, think of the area as the area of the rectangle minus the areas of the other parts. Let me try that. The rectangle is 10*3=30. The shaded triangle is part of it. If I can find the areas of the other regions and subtract them.

Looking at the diagram, the triangle divides the rectangle into three parts: the shaded triangle, and two other regions. Let me try to figure out those regions.

From the rectangle, if we draw the triangle from (0,0) to (2,3) to (10,0), then the rectangle is divided into the shaded triangle and two quadrilaterals? Or maybe two triangles and a quadrilateral? Let me think.

Alternatively, maybe the triangle can be split into two triangles. For example, from (0,0) to (2,3) to (10,0). If I draw a line from (0,0) to (2,3), and from (2,3) to (10,0). So maybe if I consider the area of the triangle as the area of the triangle (0,0), (2,3), (10,0).

Alternatively, think of the triangle as a large triangle with base 10 cm and height 3 cm, but adjusted because the apex is not directly above the center? Wait, but earlier calculations show that the area is 15 cm², which is exactly half of the rectangle. Wait, the rectangle is 30 cm², and the triangle is 15 cm². So that would mean the triangle is exactly half of the rectangle. Is that possible?

Wait, if you connect (0,0) to (2,3) to (10,0), is that triangle half of the rectangle? Let me think. If you connect (0,0) to (10,0) to (0,3) to (0,0), that would be a triangle with area 15 cm² as well. But in this case, the triangle is different.

Wait, but maybe there's some symmetry? Let me check. If the triangle is connecting (0,0) to (2,3) to (10,0), then maybe the area is indeed half of the rectangle? But why?

Alternatively, maybe not. Let me think. If I imagine the rectangle, and the triangle is cutting off a portion. If I can find the area of the triangle by coordinates, which we did as 15 cm², which is half of the rectangle. But maybe that's a coincidence?

Alternatively, maybe there's another way to see it. For example, the line from (0,0) to (2,3) has a slope of (3-0)/(2-0) = 3/2. Then the line from (2,3) to (10,0) has a slope of (0 - 3)/(10 - 2) = (-3)/8.

But how does that help? Maybe not.

Alternatively, think of the triangle as a combination of two triangles. For example, from (0,0) to (2,3) to (0,3) to (0,0). That would be a triangle with base 2 cm and height 3 cm, area (2*3)/2 = 3 cm². Then from (0,0) to (10,0) to (2,3) to (0,0). Wait, no. Maybe not.

Alternatively, think of the triangle as the area between the two lines. Maybe not helpful.

Alternatively, use coordinates again. Since I already have two methods giving me 15 cm², and the rectangle is 30 cm², which is double. Maybe that's just a coincidence.

Wait, but if I think of the triangle with vertices at (0,0), (10,0), (2,3). If I use the formula for area with coordinates, it's 15. So maybe that's correct.

Alternatively, maybe the problem is trying to trick me? Let me check the Asymptote code again. The code draws the rectangle with length 10 cm on the bottom and top, and 3 cm on the sides. Then draws the triangle from (0,0) to (2,3) to (10,0). The labels are 10 cm on the top and bottom, 3 cm on the left and right. So the rectangle is 10 cm by 3 cm.

But maybe the triangle is not as straightforward? Wait, if I imagine the triangle, it's a large triangle with base 10 cm and height 3 cm, but the apex is not at the top center. However, the area formula still holds because the height is measured perpendicular to the base.

Wait, but if the apex is not directly above the base, does that affect the area? No, because the area of a triangle is always 1/2 * base * height, regardless of where the apex is, as long as the height is the perpendicular distance from the base to the apex.

So in this case, the base is from (0,0) to (10,0), which is 10 cm. The apex is at (2,3). The height is the vertical distance from this apex to the base, which is 3 cm. Therefore, area is 1/2 * 10 * 3 = 15 cm².

Alternatively, if I take a different base. For example, take the base as from (0,0) to (2,3). Then the height would be the distance from (10,0) to the line connecting (0,0) to (2,3). That might be more complicated, but let me try.

First, find the equation of the line from (0,0) to (2,3). The slope is 3/2, so the equation is y = (3/2)x.

Then, the distance from the point (10,0) to this line. The formula for distance from a point (x0,y0) to the line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a² + b²).

First, write the line equation in standard form. y = (3/2)x can be rewritten as 3x - 2y = 0.

So a = 3, b = -2, c = 0.

Distance from (10,0):

|3*10 + (-2)*0 + 0| / sqrt(3² + (-2)^2) = |30| / sqrt(9 + 4) = 30 / sqrt(13).

Then, area would be 1/2 * base * height. The base here is the length from (0,0) to (2,3), which is sqrt((2-0)^2 + (3-0)^2) = sqrt(4 + 9) = sqrt(13).

So area would be 1/2 * sqrt(13) * (30 / sqrt(13)) ) = 1/2 * 30 = 15 cm².

Same result! So regardless of which base I take, the area is 15 cm². Therefore, that confirms it.

Therefore, the area of the shaded triangle is 15 cm².

But let me check once more. Maybe there's a misunderstanding in the diagram? The Asymptote code draws the rectangle with (0,0) to (10,0) to (10,3) to (0,3). Then draws the triangle (0,0)--(2,3)--(10,0). So the shaded triangle is indeed the one with those three points.

Alternatively, maybe the diagram is different? For example, maybe the triangle is not connecting (0,0) to (2,3) to (10,0), but something else? But according to the Asymptote code, it's fill((0,0)--(2,3)--(10,0)--cycle,gray); so yes, that's the triangle.

Therefore, I think my answer is correct. The area is 15 cm².

But just to be thorough, let me think if there's another way someone might approach this problem and possibly get a different answer. For example, if someone thought the base was 2 cm and the height was 3 cm, they might calculate 3 cm², but that's not the case. Or if they thought the triangle was right-angled, but it's not.

Alternatively, someone might think that since the rectangle is 10x3, and the triangle is somehow related to that. But since the triangle's area is half of the rectangle, that's a possible way to see it.

Alternatively, if you consider that the line from (0,0) to (2,3) divides the rectangle into two parts, and the line from (2,3) to (10,0) divides it again. But without more information, it's hard to see.

Alternatively, maybe using coordinate geometry to find the area. For example, using vectors. The vectors from (0,0) to (2,3) is (2,3), and from (0,0) to (10,0) is (10,0). The area of the parallelogram formed by these vectors is the absolute value of the cross product, which is |2*0 - 3*10| = |0 - 30| = 30. Therefore, the area of the triangle is half of that, which is 15.

Yes, that's another method. So that's consistent.

Therefore, I think I can be confident that the area is 15 cm².

**Final Answer**
The area of the shaded triangle is \boxed{15} square centimeters.

To determine the area of the shaded triangle in the given diagram, we begin by analyzing the coordinates of its vertices. The triangle is defined by the points:

- $ A = (0, 0) $
- $ B = (2, 3) $
- $ C = (10, 0) $

---

### Step 1: Use the Shoelace Formula

The **shoelace formula** for the area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is:

$$
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
$$

Substituting the coordinates:

$$
\text{Area} = \frac{1}{2} \left| 0(3 - 0) + 2(0 - 0) + 10(0 - 3) \right|
= \frac{1}{2} \left| 0 + 0 - 30 \right|
= \frac{1}{2} \times 30 = 15
$$

---

### Step 2: Confirm with Base and Height

Alternatively, we can compute the area using the standard triangle area formula:

$$
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
$$

- The **base** is the horizontal segment from $ (0, 0) $ to $ (10, 0) $, which has length $ 10 $ cm.
- The **height** is the vertical distance from the point $ (2, 3) $ to the base (which lies along the x-axis), which is simply the y-coordinate of the point: $ 3 $ cm.

$$
\text{Area} = \frac{1}{2} \times 10 \times 3 = 15
$$

---

### Step 3: Cross-Verification with Vector Method

We can also use the **vector cross product** method. The vectors from the origin to the other two points are:

- $ \vec{AB} = (2, 3) $
- $ \vec{AC} = (10, 0) $

The area of the parallelogram formed by these vectors is the absolute value of the determinant:

$$
\text{Area of parallelogram} = |2 \cdot 0 - 3 \cdot 10| = |0 - 30| = 30
$$

So the area of the triangle is half of that:

$$
\text{Area} = \frac{1}{2} \times 30 = 15
$$

---

### Final Conclusion

All methods consistently yield the same result. The shaded triangle has an area of:

$$
\boxed{15}
$$

square centimeters.

[Assistant]
Explanation:
Okay, so I need to find the area of the shaded triangle in the diagram. Let me try to visualize it first. The Asymptote code draws a rectangle with length 10 cm on the bottom and top, and height 3 cm on the sides. Then there's a triangle connecting the bottom-left corner (0,0) to the point (2,3) and then to the bottom-right corner (10,0). The shaded area is this triangle.

First, let me recall that the area of a triangle is (base * height)/2. But I need to figure out the base and the height of the shaded triangle. However, since the triangle isn't necessarily aligned with the axes, maybe I need to use coordinates to calculate it?

Alternatively, maybe I can use coordinates to find the area. The coordinates of the triangle are (0,0), (2,3), and (10,0). So if I can use the coordinates, I can apply the shoelace formula or something like that. Let me try that.

The shoelace formula for three points (x1,y1), (x2,y2), (x3,y3) is:

Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))/2|

Let me plug in the coordinates:

x1 = 0, y1 = 0

x2 = 2, y2 = 3

x3 = 10, y3 = 0

So plugging into the formula:

Area = |(0*(3 - 0) + 2*(0 - 0) + 10*(0 - 3))/2|

Calculating each term:

First term: 0*(3 - 0) = 0*3 = 0

Second term: 2*(0 - 0) = 2*0 = 0

Third term: 10*(0 - 3) = 10*(-3) = -30

Adding them up: 0 + 0 - 30 = -30

Take absolute value and divide by 2: |-30| / 2 = 30 / 2 = 15

So the area is 15 cm²? Wait, but let me check if that's correct. Let me think again.

Alternatively, maybe I can divide the figure into parts. The rectangle has area 10 cm * 3 cm = 30 cm². The shaded triangle is part of this rectangle. If I can find the area of the triangle by subtracting other areas?

But maybe not. Alternatively, since the triangle is formed by points (0,0), (2,3), (10,0). Let me try to find the base and height. If I take the base as the side from (0,0) to (10,0), which is 10 cm. Then the height would be the vertical distance from the point (2,3) to this base. But since the base is along the x-axis, the height is just the y-coordinate of the point (2,3), which is 3 cm. Wait, but then area would be (10 * 3)/2 = 15 cm². Which matches the shoelace formula. So that seems correct.

But wait, hold on. If the base is 10 cm and the height is 3 cm, then the area is 15 cm². But is that accurate? Because the triangle isn't necessarily having the base as the entire length of the rectangle. Wait, but the triangle is connecting (0,0) to (2,3) to (10,0). So if I take the base as the base from (0,0) to (10,0), which is 10 cm, then the height would be the perpendicular distance from (2,3) to this base. Since the base is horizontal, the vertical distance is indeed 3 cm. So that would make the area 15 cm².

But wait, but if I imagine the triangle, it's not a right triangle. Wait, but if I take the base as 10 cm, then the height is 3 cm. But is that correct? Let me think. If the triangle is between (0,0), (10,0), and (2,3), then the height is not necessarily 3 cm. Wait, no. Because the height is the perpendicular distance from the base to the opposite vertex. Since the base is along the x-axis from (0,0) to (10,0), then the height is the vertical distance from (2,3) to the x-axis, which is indeed 3 cm. So that's correct. Therefore, area is 1/2 * base * height = 1/2 * 10 * 3 = 15 cm².

But let me check with another method. Let me use coordinates again. The coordinates are (0,0), (2,3), (10,0). Let me use vectors or something else. Alternatively, maybe divide the triangle into two parts?

Alternatively, think of the triangle as a large triangle minus some smaller triangles? Wait, but maybe not necessary. Since both methods gave me 15 cm², maybe that's correct. But let me check with the shoelace formula again.

Shoelace formula:

List the coordinates in order: (0,0), (2,3), (10,0), (0,0)

Compute sum of products going one way:

(0*3) + (2*0) + (10*0) = 0 + 0 + 0 = 0

Compute sum of products going the other way:

(0*2) + (3*10) + (0*0) = 0 + 30 + 0 = 30

Subtract the two: |0 - 30| = 30, divide by 2: 15. Yep, same result. So that's correct.

But wait, the problem is presented with a diagram. Let me check if the diagram is as I imagine. The Asymptote code draws a rectangle from (0,0) to (10,0) to (10,3) to (0,3) and back. Then draws the triangle from (0,0) to (2,3) to (10,0). So the shaded triangle is inside the rectangle.

But maybe there's a trick here? For example, maybe the triangle is not as straightforward? Let me think again.

Alternatively, maybe the base is not 10 cm? Wait, the base from (0,0) to (10,0) is 10 cm. The height is 3 cm. So area is 15. But maybe the problem is expecting something else? Let me check the labels. The diagram labels the top side as 10 cm, the bottom side as 10 cm, and the left and right sides as 3 cm. So the rectangle is 10 cm by 3 cm.

But then the triangle is connecting (0,0) to (2,3) to (10,0). So the point (2,3) is on the top side of the rectangle. Wait, the top side is from (0,3) to (10,3). So (2,3) is 2 cm from the left on the top side. So the triangle is formed by the bottom-left corner, a point 2 cm from the left on the top, and the bottom-right corner.

Alternatively, maybe I can use coordinates to find the area. But I already did that.

Alternatively, maybe using coordinates, find the equations of the lines and integrate or something? Probably overkill.

Alternatively, think of the triangle as two smaller triangles? For example, from (0,0) to (2,3) to (10,0). Maybe split it into two triangles? But that might complicate.

Alternatively, use determinants. Which is essentially what the shoelace formula is.

Alternatively, think of the area as the area of the rectangle minus the areas of the other parts. Let me try that. The rectangle is 10*3=30. The shaded triangle is part of it. If I can find the areas of the other regions and subtract them.

Looking at the diagram, the triangle divides the rectangle into three parts: the shaded triangle, and two other regions. Let me try to figure out those regions.

From the rectangle, if we draw the triangle from (0,0) to (2,3) to (10,0), then the rectangle is divided into the shaded triangle and two quadrilaterals? Or maybe two triangles and a quadrilateral? Let me think.

Alternatively, maybe the triangle can be split into two triangles. For example, from (0,0) to (2,3) to (10,0). If I draw a line from (0,0) to (2,3), and from (2,3) to (10,0). So maybe if I consider the area of the triangle as the area of the triangle (0,0), (2,3), (10,0).

Alternatively, think of the triangle as a large triangle with base 10 cm and height 3 cm, but adjusted because the apex is not directly above the center? Wait, but earlier calculations show that the area is 15 cm², which is exactly half of the rectangle. Wait, the rectangle is 30 cm², and the triangle is 15 cm². So that would mean the triangle is exactly half of the rectangle. Is that possible?

Wait, if you connect (0,0) to (2,3) to (10,0), is that triangle half of the rectangle? Let me think. If you connect (0,0) to (10,0) to (0,3) to (0,0), that would be a triangle with area 15 cm² as well. But in this case, the triangle is different.

Wait, but maybe there's some symmetry? Let me check. If the triangle is connecting (0,0) to (2,3) to (10,0), then maybe the area is indeed half of the rectangle? But why?

Alternatively, maybe not. Let me think. If I imagine the rectangle, and the triangle is cutting off a portion. If I can find the area of the triangle by coordinates, which we did as 15 cm², which is half of the rectangle. But maybe that's a coincidence?

Alternatively, maybe there's another way to see it. For example, the line from (0,0) to (2,3) has a slope of (3-0)/(2-0) = 3/2. Then the line from (2,3) to (10,0) has a slope of (0 - 3)/(10 - 2) = (-3)/8.

But how does that help? Maybe not.

Alternatively, think of the triangle as a combination of two triangles. For example, from (0,0) to (2,3) to (0,3) to (0,0). That would be a triangle with base 2 cm and height 3 cm, area (2*3)/2 = 3 cm². Then from (0,0) to (10,0) to (2,3) to (0,0). Wait, no. Maybe not.

Alternatively, think of the triangle as the area between the two lines. Maybe not helpful.

Alternatively, use coordinates again. Since I already have two methods giving me 15 cm², and the rectangle is 30 cm², which is double. Maybe that's just a coincidence.

Wait, but if I think of the triangle with vertices at (0,0), (10,0), (2,3). If I use the formula for area with coordinates, it's 15. So maybe that's correct.

Alternatively, maybe the problem is trying to trick me? Let me check the Asymptote code again. The code draws the rectangle with length 10 cm on the bottom and top, and 3 cm on the sides. Then draws the triangle from (0,0) to (2,3) to (10,0). The labels are 10 cm on the top and bottom, 3 cm on the left and right. So the rectangle is 10 cm by 3 cm.

But maybe the triangle is not as straightforward? Wait, if I imagine the triangle, it's a large triangle with base 10 cm and height 3 cm, but the apex is not at the top center. However, the area formula still holds because the height is measured perpendicular to the base.

Wait, but if the apex is not directly above the base, does that affect the area? No, because the area of a triangle is always 1/2 * base * height, regardless of where the apex is, as long as the height is the perpendicular distance from the base to the apex.

So in this case, the base is from (0,0) to (10,0), which is 10 cm. The apex is at (2,3). The height is the vertical distance from this apex to the base, which is 3 cm. Therefore, area is 1/2 * 10 * 3 = 15 cm².

Alternatively, if I take a different base. For example, take the base as from (0,0) to (2,3). Then the height would be the distance from (10,0) to the line connecting (0,0) to (2,3). That might be more complicated, but let me try.

First, find the equation of the line from (0,0) to (2,3). The slope is 3/2, so the equation is y = (3/2)x.

Then, the distance from the point (10,0) to this line. The formula for distance from a point (x0,y0) to the line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a² + b²).

First, write the line equation in standard form. y = (3/2)x can be rewritten as 3x - 2y = 0.

So a = 3, b = -2, c = 0.

Distance from (10,0):

|3*10 + (-2)*0 + 0| / sqrt(3² + (-2)^2) = |30| / sqrt(9 + 4) = 30 / sqrt(13).

Then, area would be 1/2 * base * height. The base here is the length from (0,0) to (2,3), which is sqrt((2-0)^2 + (3-0)^2) = sqrt(4 + 9) = sqrt(13).

So area would be 1/2 * sqrt(13) * (30 / sqrt(13)) ) = 1/2 * 30 = 15 cm².

Same result! So regardless of which base I take, the area is 15 cm². Therefore, that confirms it.

Therefore, the area of the shaded triangle is 15 cm².

But let me check once more. Maybe there's a misunderstanding in the diagram? The Asymptote code draws the rectangle with (0,0) to (10,0) to (10,3) to (0,3). Then draws the triangle (0,0)--(2,3)--(10,0). So the shaded triangle is indeed the one with those three points.

Alternatively, maybe the diagram is different? For example, maybe the triangle is not connecting (0,0) to (2,3) to (10,0), but something else? But according to the Asymptote code, it's fill((0,0)--(2,3)--(10,0)--cycle,gray); so yes, that's the triangle.

Therefore, I think my answer is correct. The area is 15 cm².

But just to be thorough, let me think if there's another way someone might approach this problem and possibly get a different answer. For example, if someone thought the base was 2 cm and the height was 3 cm, they might calculate 3 cm², but that's not the case. Or if they thought the triangle was right-angled, but it's not.

Alternatively, someone might think that since the rectangle is 10x3, and the triangle is somehow related to that. But since the triangle's area is half of the rectangle, that's a possible way to see it.

Alternatively, if you consider that the line from (0,0) to (2,3) divides the rectangle into two parts, and the line from (2,3) to (10,0) divides it again. But without more information, it's hard to see.

Alternatively, maybe using coordinate geometry to find the area. For example, using vectors. The vectors from (0,0) to (2,3) is (2,3), and from (0,0) to (10,0) is (10,0). The area of the parallelogram formed by these vectors is the absolute value of the cross product, which is |2*0 - 3*10| = |0 - 30| = 30. Therefore, the area of the triangle is half of that, which is 15.

Yes, that's another method. So that's consistent.

Therefore, I think I can be confident that the area is 15 cm².

**Final Answer**
The area of the shaded triangle is \boxed{15} square centimeters.

To determine the area of the shaded triangle in the given diagram, we begin by analyzing the coordinates of its vertices. The triangle is defined by the points:

- $ A = (0, 0) $
- $ B = (2, 3) $
- $ C = (10, 0) $

---

### Step 1: Use the Shoelace Formula

The **shoelace formula** for the area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is:

$$
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
$$

Substituting the coordinates:

$$
\text{Area} = \frac{1}{2} \left| 0(3 - 0) + 2(0 - 0) + 10(0 - 3) \right|
= \frac{1}{2} \left| 0 + 0 - 30 \right|
= \frac{1}{2} \times 30 = 15
$$

---

### Step 2: Confirm with Base and Height

Alternatively, we can compute the area using the standard triangle area formula:

$$
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
$$

- The **base** is the horizontal segment from $ (0, 0) $ to $ (10, 0) $, which has length $ 10 $ cm.
- The **height** is the vertical distance from the point $ (2, 3) $ to the base (which lies along the x-axis), which is simply the y-coordinate of the point: $ 3 $ cm.

$$
\text{Area} = \frac{1}{2} \times 10 \times 3 = 15
$$

---

### Step 3: Cross-Verification with Vector Method

We can also use the **vector cross product** method. The vectors from the origin to the other two points are:

- $ \vec{AB} = (2, 3) $
- $ \vec{AC} = (10, 0) $

The area of the parallelogram formed by these vectors is the absolute value of the determinant:

$$
\text{Area of parallelogram} = |2 \cdot 0 - 3 \cdot 10| = |0 - 30| = 30
$$

So the area of the triangle is half of that:

$$
\text{Area} = \frac{1}{2} \times 30 = 15
$$

---

### Final Conclusion

All methods consistently yield the same result. The shaded triangle has an area of:

$$
\boxed{15}
$$

square centimeters.

Solution: The shaded triangle has a base of length $10\text{ cm}.$ Since the triangle is enclosed in a rectangle of height $3\text{ cm},$ then the height of the triangle is $3\text{ cm}.$ (We know that the enclosing shape is a rectangle, because any figure with four sides, including two pairs of equal opposite sides, and two right angles must be a rectangle.) Therefore, the area of the triangle is $$\frac{1}{2}\times 3 \times 10 = \boxed{15\mbox{ cm}^2}.$$