Interesting but still failing.

Algebra Level 2

$\text{\_\_\_\_\_\_\_\_\_\_}\text{ wrong}\qquad ;\qquad \text{\_\_\_\_\_\_\_\_\_\_}\text{ \% right}$

The above is the way a teacher summarizes the result of a test with $n$ problems of equal weights.

For example, if $n=20$ and a student gets $x=3$ problems wrong, the teacher writes "3 wrong; 85% right."

What is the least integer $x$ such that the two numbers written in the blanks could be exactly the same?

1 solution

Jeremy Galvagni
Apr 11, 2018

On a $n=25$ question test, getting $x=\boxed{20}$ wrong means getting $5$ right and $\frac{5}{25}=\boxed{20}\%$

On an $n$ problem test, getting $x$ wrong means getting $\frac{100(n-x)}{n}\%$ right.

Setting these equal and solving for $x$ gives

$x=\frac{100n}{n+100}$

A quick table gives the solution as the first value of $n$ with a whole number for $x$ .