ACT math prep question

Question

30.

Suppose that x is a positive real number and 4x/6x^2 is a rational number. Which of the following statements about x MUST be true?

A. x is rational

B. x is irrational

C. x = 1

D. x = 2/3

E. x = 3/2

And why?

Answer 1

Hi John,

The definition of a rational number is any number that can be expressed as a ratio of two integers p/q where q≠0.

So 4x/(6x²) where x is a positive real number ≠0 can be written as 2/3x which is in the form p/q where p=2 and q=3x x=1,2,3....

So it looks like the answer is A.

regards

Jim

Answer 2

X is a rational number see below

Try solving for x by setting up a proportion, see if you get a unique rational number.

4x/6x²

Set up a proportion

Set this fraction equal to 1, multiply both sides by x²/x, then reduce both sides accordingly

4x/6x² = 1

x²/x(4x/6x²) = x²/x

4/6 = x²/x

Reduce

2/3 = x

Or

Set up a proportion if you start by setting the fraction equal to 1.

4x/6x² = 1

Multiply both sides of the equation by 6x² to give

4x = 6x²

Divide both sides by x

4 = 6x

Now divide both sides by 6 and reduce the final fraction

4/6 = x

2/3 = x

You can also reduce 4x/6x² first to 2/3x set it equal to 1 and solve x.

Checking

4x = 6x²

4/1 * (2/3) = 6/1 * (2/3)²

8/3 = 6/1 * (4/9) = 6/1* (4/9) = 2/1 * (4/3) Since three is common to 6 and 9

8/3 = 2/1 * (4/3)

8/3 = 8/3

This also makes x a rational number.

I hope you find this useful, if you have any questions send me a message.