Problem Solving

AAPL wrote: ↑

Mon Sep 21, 2020 6:20 pm

Magoosh

\(\dfrac{\dfrac{a}{b}+1}{\dfrac{c}{b}}\)

In the expression above, a, b, and c are different numbers and each is one of the numbers 2, 3 or 5. What is the greatest possible value of the expression?

A. \(8/3\)
B. \(4\)
C. \(9/2\)
D. \(5\)
E. \(6\)

OA B

Solution:

Since we want the greatest possible value of the expression, we want the numerator a/b + 1 to be as large as possible and the denominator c/b to be as small as possible. Therefore, a has to 5. However, b and c could be 2 and 3, respectively, or they could be 3 and 2, respectively. Let’s check both options.

If b = 2 and c = 3:

(5/2 + 1)/(3/2) = 7/2 x 2/3 = 7/3

If b = 3 and c = 2:

(5/3 + 1)/(2/3) = 8/3 x 3/2 = 8/2 = 4

Since 4 > 7/3, the correct answer is 4.

Alternate Solution:

Let’s use the “flip and multiply” rule for division of two fractions, which turns the division problem into a multiplication problem:

(a/b + 1) / c/b = (a/b + 1) x b/c = (a + b) / c

In order for the fraction (a + b) / c to be as large as possible, we want the two numbers in the numerator, which are a and b, to be as large as possible, and we want the denominator c to be as small as possible. Thus, from the given numbers, we choose a and b to be 3 and 5, and we choose c to be 2. Thus, we have:

(a + b) / c = (3 + 5) / 2 = 8/2 = 4.



Answer: B