A student answered $86$ problems correctly, which was $81.9\%$ of the total number of problems on the test. How many problems did the student answer incorrectly? $\newline$ Choose $1$ answer: $\newline$ (A) $4$ $\newline$ (B) $19$ $\newline$ (C) $86$ $\newline$ (D) $105$

Full solution

Q. A student answered

86

problems correctly, which was

81.9\%

of the total number of problems on the test. How many problems did the student answer incorrectly?

\newline

Choose

1

answer:

\newline

(A)

4

\newline

(B)

19

\newline

(C)

86

\newline

(D)

105

Find Total Problems: First, we need to find the total number of problems on the test. We know that $86$ problems represent $81.9\%$ of the total. To find the total number of problems, we can set up the equation: $\newline$ $86 = 81.9\% \text{ of Total Problems}$ $\newline$ To find the Total Problems, we divide $86$ by $81.9\%$ (or $0.819$ in decimal form). $\newline$ Total Problems = $\frac{86}{0.819}$
Calculate Total: Now, let's perform the division to find the total number of problems. $\newline$ Total Problems = $\frac{86}{0.819} \approx 105$ (rounded to the nearest whole number since we can't have a fraction of a problem)
Find Incorrect Problems: Next, we need to find out how many problems the student answered incorrectly. Since the student answered $86$ problems correctly, and there are $105$ problems in total, we subtract the number of correct answers from the total number of problems. $\newline$ Incorrect Problems $=$ Total Problems $-$ Correct Problems $\newline$ Incorrect Problems $=$ $105 - 86$
Subtract to Find Incorrect: Let's do the subtraction to find the number of incorrect answers. $\newline$ Incorrect Problems = $105 - 86 = 19$