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:

\frac{4}{3}\div\frac{3}{4}= \frac{4}{3}\cdot\frac{4}{3}= \frac{4\cdot 4}{3\cdot 3}=\frac{16}{9}

(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.

\frac{3}{4}\div\frac{4}{3}= \frac{3}{4}\cdot\frac{3}{4}= \frac{3\cdot 3}{4\cdot 4}=\frac{9}{16}

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

x\div \frac{1}{x}=\frac{9}{16} 

\frac{x\cdot x}{1}=\frac{9}{16} 

x\cdot x=\frac{9}{16} 

x^2=\frac{9}{16} 

x=\pm \sqrt{\frac{9}{16}}=\pm \frac{\sqrt{9}}{\sqrt{16}}=\pm \frac{3}{4}

Since we are given that x is a positive number, x = 3/4, choice E.

More ACT Math Problems with Explanations

If you are preparing for the ACT, you may want to take a look at the following book.

320 ACT Math Problems

And if you liked this article, please share it with your Facebook friends:

ACT Math Facebook Share Button

Get 800 SAT Math Prep Facebook Link Get 800 SAT Math Prep Twitter Link Get 800 SAT Math Prep YouTube Link Get 800 SAT Math Prep LinkedIn Link Get 800 SAT Math Prep Pinterest Link

Comments

comments