Measuring Mathematical Problem Solving With the MATH Dataset
https://arxiv.org/abs/2103.03874
Many intellectual endeavors require mathematical problem solving, but this skill remains beyond the capabilities of computers. To measure this ability in machine learning models, we introduce MATH, a new dataset of 12,500 challenging competition mathematics problems. Each problem in MATH has a full step-by-step solution which can be used to teach models to generate answer derivations and explanations.
NOTE: The few-shot and the generated answer extraction is based on the [Minerva](https://arxiv.org/abs/2206.14858) paper and exact match equivalence is calculated using the `sympy` library
Homepage: https://github.com/hendrycks/math
## Citation
```
@article{hendrycksmath2021,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Dan Hendrycks and Collin Burns and Saurav Kadavath and Akul Arora and Steven Basart and Eric Tang and Dawn Song and Jacob Steinhardt},
journal={NeurIPS},
year={2021}
}
```
### Groups, Benchmarks and Tasks
#### Benchmarks
-`hendrycks_math`
#### Groups
-`math_word_problems`
-`greedy_until`
#### Tasks
-`math_algebra`
-`math_counting_and_prob`
-`math_geometry`
-`math_intermediate_algebra`
-`math_num_theory`
-`math_prealgebra`
-`math_precalc`
### Checklist
The checklist is the following:
For adding novel benchmarks/datasets to the library:
* [x] Is the task an existing benchmark in the literature?
* [x] Have you referenced the original paper that introduced the task?
* [x] If yes, does the original paper provide a reference implementation? If so, have you checked against the reference implementation and documented how to run such a test?
* The implementation in the original paper is one where the model is first fine-tuned on the data. They do have a few-shot evaluation for GPT-3, however the few-shot context used here is sourced from [Lewkowycz et al](https://arxiv.org/abs/2206.14858). The achieved accuracy on Llama-2 models is comparable to that provided in the paper, though not identical.
If other tasks on this dataset are already supported:
* [ ] Is the "Main" variant of this task clearly denoted?
* [ ] Have you provided a short sentence in a README on what each new variant adds / evaluates?
* [ ] Have you noted which, if any, published evaluation setups are matched by this variant?
Find the domain of the expression $\frac{\sqrt{x-2}}{\sqrt{5-x}}$.}
Solution:
The expressions inside each square root must be non-negative. Therefore, $x-2 \ge 0$, so $x\ge2$, and $5 - x \ge 0$, so $x \le 5$. Also, the denominator cannot be equal to zero, so $5-x>0$, which gives $x<5$. Therefore, the domain of the expression is $\boxed{[2,5)}$.
Final Answer: The final answer is $[2,5)$. I hope it is correct.
Problem:
If $\det \mathbf{A} = 2$ and $\det \mathbf{B} = 12,$ then find $\det (\mathbf{A} \mathbf{B}).$
Solution:
We have that $\det (\mathbf{A} \mathbf{B}) = (\det \mathbf{A})(\det \mathbf{B}) = (2)(12) = \boxed{24}.$
Final Answer: The final answer is $24$. I hope it is correct.
Problem:
Terrell usually lifts two 20-pound weights 12 times. If he uses two 15-pound weights instead, how many times must Terrell lift them in order to lift the same total weight?
Solution:
If Terrell lifts two 20-pound weights 12 times, he lifts a total of $2\cdot 12\cdot20=480$ pounds of weight. If he lifts two 15-pound weights instead for $n$ times, he will lift a total of $2\cdot15\cdot n=30n$ pounds of weight. Equating this to 480 pounds, we can solve for $n$:
\begin{align*}
30n&=480\\
\Rightarrow\qquad n&=480/30=\boxed{16}
\end{align*}
Final Answer: The final answer is $16$. I hope it is correct.
Problem:
If the system of equations
\begin{align*}
6x-4y&=a,\\
6y-9x &=b.
\end{align*}has a solution $(x, y)$ where $x$ and $y$ are both nonzero,
find $\frac{a}{b},$ assuming $b$ is nonzero.
Solution:
If we multiply the first equation by $-\frac{3}{2}$, we obtain
$$6y-9x=-\frac{3}{2}a.$$Since we also know that $6y-9x=b$, we have
