AMC 10 · 2019 · #2

Grade 7 rate-ratio

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Percent / Fraction Recovery

percentageratio-proportionlinear-equations-two-var convert-to-algebradimensional-analysis ↑ Prerequisites: percentageratio-proportion

📏 Short solution 💡 2 insights

Problem

Two quantities a and b are linked by the statement that a is 150 percent of b. Using that link, express 3b as a percent of a.

Pick an answer.

(A)

(B)

$66+\frac{2}{3}$

(C)

150

(D)

200

(E)

450

How to solve

Strategy Convert to Algebra

Both sentences here are percent sentences in words, and each one is really an equation in disguise, so Tool #13 (Convert to Algebra) is the workhorse: "a is 150% of b" becomes a=3/2b, and the question becomes an equation too. The percent asked for has no name yet, so Tool #4 (Introduce a Variable) calls it p and writes 3b=p/100a. Finally Tool #8 (Analyze the Units) handles the last conversion, reading a plain multiplier as a percent, since "percent" only means "per hundred". The whole difficulty of the problem is that the base switches from b in the given to a in the question, and writing both sentences as equations makes that switch impossible to miss.

1STEP 1

Turn the given percent into an equation

Turn the given percent into an equation.

a=150/100 b=3/2 b

2STEP 2

Name the percent being asked for

Give the wanted percent a name.

3b=p/100 a

3STEP 3

Write 3b in terms of a

Rewrite three b in terms of a.

a=3/2 b → 2a=3b

4STEP 4

Read the multiplier as a percent

Reading the multiplier as a percent gives 200.

p/100 a=2a → p/100=2 → p=200 → (D)

Answer

200

Check the size first: a is only 1.5 times b, but the question triples b, so 3b must overshoot a and the percent has to be more than 100. That alone kills (A) 50 and (B) 66+2/3. Now test concrete numbers: with b=2 the given makes a=3, and 3b=6, so 6/3=2=200%, which matches. The wrong choices are all recognisable traps: (B) is the answer to "what percent of a is b?" since b/a=2/3, (C) just echoes the 150 printed in the problem, and (E) comes from multiplying 3 × 150 instead of comparing 3b with a.

💡Key takeaway

In any percent sentence the quantity right after the word "of" is the base you divide by, so "what percent of a is 3b" means compute 3b/a and then multiply by 100.

Turn the given percent into an equation
Name the percent being asked for
Write 3b in terms of a
Read the multiplier as a percent

Turn the given percent into an equation

Name the percent being asked for

Write 3b in terms of a

Read the multiplier as a percent

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