AMC 10 · 2008 · #12

Grade 7 algebra

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Chained Ratio Linking

percentageratio-proportionformula-substitution convert-to-algebra ↑ Prerequisites: percentage

📏 Medium solution 💡 1 insight

Problem

A collection has red, blue, and green marbles. There are 25% more red marbles than blue, and 60% more green marbles than red. Writing the count of red marbles as r, find the total number of marbles in terms of r.

Pick an answer.

(A)

$\ 2.85r$

(B)

$\ 3r$

(C)

$\ 3.4r$

(D)

$\ 3.85r$

(E)

$\ 4.25r$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

Everything is measured against r, the red count, so I express blue and green as multiples of r (Tool #4). The two percent phrases become equations (Tool #13); the trick is spotting which color is the base each time. Reading '% more' as 'base plus that percent of the base' (Tool #8) keeps blue and green straight. Adding the three multiples gives one coefficient, and matching it to the list (Tool #3) picks the answer.

1STEP 1

Rewrite blue from the red count

'25% more red than blue' makes blue the base, so blue is r divided by 1.25, which is 0.8r.

r = 1.25 · b → b = r/1.25 = 0.8r

2STEP 2

Rewrite green from the red count

This time red is the base, so green is r plus 0.6r, which is 1.6r — the largest pile.

g = r + 0.6r = 1.6r

3STEP 3

Add the three amounts

All three are like terms: 0.8r + r + 1.6r, and the coefficients add to give a total of 3.4r.

0.8r + r + 1.6r = (0.8 + 1 + 1.6)r = 3.4r

4STEP 4

Match the coefficient to a choice

The coefficient 3.4 sits in the list with no rounding, while 0.75r for blue would give 3.35r, so the answer is (C).

3.4r → (C)

Answer

3.4r

Test with a concrete count. Let blue = 100. Then red is 25% more: r = 125. Green is 60% more than red: 1.6 × 125 = 200. The total is 100 + 125 + 200 = 425. As a multiple of r, that is 425 / 125 = 3.4, so the total is 3.4r. This confirms (C). Note the total is a bit more than 3r because green is the biggest group, which fits.

💡Key takeaway

In '25% more A than B', B is the base you start from, so blue is r/1.25 = 0.8r; add blue, red, and green all measured in r's and you get 3.4r.

Rewrite blue from the red count
Rewrite green from the red count
Add the three amounts
Match the coefficient to a choice

Rewrite blue from the red count

Rewrite green from the red count

Add the three amounts

Match the coefficient to a choice

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