Printable

2017 AMC 10B Problem 5

2017 AMC 10B Problem 5AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 1 of 5 · about 2 min · position P1-10

Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating $10$ pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have?

A)
$10$
B)
$20$
C)
$30$
D)
$40$
E)
$50$

Answer

Key insight

Let c be the cherry count; then 2c - 10 = 3(c - 10) gives c = 20, and the blueberry count is 2c = 40.

Solution

Let $c$ be the original number of cherry jelly beans, so there were $2c$ blueberry ones. After eating $10$ of each, the counts are $c - 10$ and $2c - 10$ , and the new ratio is $3 : 1$ :
$2c - 10 = 3(c - 10) = 3c - 30.$
So $c = 20$ , and the original blueberry count is $2c = 40$ . Check: $30$ blueberry and $10$ cherry remain, a $3:1$ ratio.

The answer is $\boxed{\textbf{(D)}\ 40}$ .

Why this works

Two ratio statements about the same pair of quantities, before and after a change, give one linear equation in one unknown. Name the smaller quantity, express the larger through the first ratio, then apply the second ratio to the changed amounts. Always re-read the question to see which quantity is requested.

Alternative approach

Test the choices: with $40$ blueberry and $20$ cherry, eating $10$ of each leaves $30$ and $10$ , exactly three to one. The other choices fail this check.

The trap

Reporting the cherry count 20 (choice B) instead of the blueberry count asked for.

Common mistakes

Reporting the cherry count 20 (choice B) instead of the blueberry count asked for.
Subtracting $10$ from only one kind, or subtracting $10$ from the ratio instead of from the counts.

Techniques

Set up the equation/formula and compute; no special trick needed