ACT Math Number Theory Problem with Solutions

ACT Math Number Theory Problem with Solutions

Today I will give a solution to yesterday’s ACT math number theory problem. Here is the question one more time followed by a full solution.

Level 3 Number Theory

What positive number when divided by its reciprocal has a result of 9/16?

A. 8/3
B. 3/8
C. 4/3
D. 3/16
E. 3/4

Solution by starting with choice C: Let’s start with 4/3 as our first guess. The reciprocal of 4/3 is 3/4, and when we divide 4/3 by 3/4, we get 16/9. This is not correct, but it is the reciprocal of what we are trying to get. So the answer is the reciprocal of 4/3, which is 3/4, choice E.

Notes: (1) The reciprocal of the fraction a/b is the fraction b/a. In other words, we get the reciprocal of the fraction by interchanging the number on top (the numerator) with the number on bottom (the denominator).

(2) We can divide 4/3 by 3/4 right in our TI-84 calculator by typing

(4 / 3) / (3 / 4) ENTER MATH ENTER ENTER

The output will be 16/9.

Pressing MATH ENTER ENTER at the end changes the decimal to a fraction.

(3) We can also do the computation by hand as follows:

$latex \frac{4}{3}\div\frac{3}{4}= \frac{4}{3}\cdot\frac{4}{3}= \frac{4\cdot 4}{3\cdot 3}=\frac{16}{9} &s=2&fg=0000ff$

(4) Let’s also just confirm that choice E is the answer. I’ll use the hand method, but you can also feel free to use your calculator.

$latex \frac{3}{4}\div\frac{4}{3}= \frac{3}{4}\cdot\frac{3}{4}= \frac{3\cdot 3}{4\cdot 4}=\frac{9}{16} &s=2&fg=0000ff$

* Algebraic solution: Let x be the positive number. We are given that

$latex x\div \frac{1}{x}=\frac{9}{16} &s=2&fg=ff0000$

$latex \frac{x\cdot x}{1}=\frac{9}{16} &s=2&fg=ff0000$

$latex x\cdot x=\frac{9}{16} &s=2&fg=ff0000$