Question 411·Hard·Math

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A student answered 75% of the questions correctly on her first practice exam. On her second practice exam, she answered 20% more questions correctly than she did on the first exam. The second exam contained 20 more questions than the first, and she answered 80% of them correctly.

How many questions were on the first practice exam?

A percent-change problem can compare correct-answer counts even when the exam totals change. Let the first total be the unknown, then find its correct answers. Apply the increase to that correct-answer count, but take the second exam's percent of its own, larger total. Set the two descriptions of the second exam's correct answers equal, then solve with a Desmos regression.

Hints

Hint 1
A percent is taken of a particular whole. If $x$ is the number of questions on the first exam, $75\%$ of $x$ gives the number she answered correctly, not the total on the second exam.
Hint 2
For a percent increase, keep the original $100\%$ and add the increase. The $20\%$ more correct answers are measured against how many she got correct on the first exam. What percent of that count did she get correct on the second?
Hint 3
The second exam's total is $x+20$ . Take $80\%$ of that total to get its correct-answer count, then set it equal to the count you found from the $20\%$ increase.

Step-by-step

Equate the two correct-answer counts
Step 1
Count correct answers on the first exam
Let $x$ be the number of questions on the first exam. A percent means a share out of $100$ , so $75\%$ of that total is the number she answered correctly:
$0.75x.$
Step 2
Write the second exam's total
The second exam contained 20 more questions than the first. So its total number of questions is
$x+20.$
Step 3
Apply the increase to correct answers
The base of the $20\%$ increase is the first exam's correct-answer count, $0.75x$ . Keeping its original $100\%$ and adding $20\%$ gives $120\%$ , or $1.20(0.75x)$ correct answers on the second exam.
Step 4
Count correct answers using the second exam's total
Its $80\%$ correct rate gives $0.80(x+20)$ correct answers.
Step 5
Equate the correct-answer counts
Both count the same answers, so
$0.80(x+20)=1.20(0.75x).$
The $20\%$ increase applies to correct answers, while the $80\%$ applies to all questions on the second exam.
Step 6
Solve for the first exam's total
Type $0.8(x_{1}+20)\sim1.2(0.75x_{1})$ in Desmos. Here $x_{1}$ stands for the first exam's total; replacing $x$ with $x_{1}$ and $=$ with $\sim$ tells Desmos to find the value that makes both sides equal. Under PARAMETERS, it shows $x_{1}=160$ . So the first practice exam had $160$ questions. Choice C.

Question 411·Hard·Math

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