AMC 8 · 2003 · #5

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

percentageratio-proportionfraction-arithmetic identify-subproblemsconvert-to-algebra ↑ Prerequisites: fraction-decimal-conversionlinear-equations-one-var

📏 Short solution 💡 2 insights

Problem

If $20\%$ of a number is $12$ , what is $30\%$ of the same number?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

View mode:

Toolkit + CCSS Solution

Understand

Restated: If $20\%$ of some number equals $12$, what is $30\%$ of the same number?

Givens: $20\%$ of the number is $12$; Answer choices: (A) $15$, (B) $18$, (C) $20$, (D) $24$, (E) $30$

Unknowns: The value of $30\%$ of the same number

Understand

Restated: If $20\%$ of some number equals $12$, what is $30\%$ of the same number?

Givens: $20\%$ of the number is $12$; Answer choices: (A) $15$, (B) $18$, (C) $20$, (D) $24$, (E) $30$

Plan

Primary tool: #3 Set Up an Equation

Secondary: #4 Introduce a Variable

The problem is a one-sentence percent statement with one unknown. Tool #4 (Introduce a Variable) names the mystery number $N$, and Tool #3 (Set Up an Equation) turns the sentence "$20\%$ of $N$ is $12$" directly into the equation $0.20 \, N = 12$. Once $N$ is known, finding $30\%$ of it is one more multiplication. No guess-and-check or pattern hunt is needed — the translation is immediate.

Execute — Answer: B

#4 Introduce a Variable 6.EE.A.2 Step 1

Name the number.
Let $N$ be the unknown number that the percents are taken from.

$$N = \text{the unknown number}$$

💡 Giving the mystery number a letter lets us write the percent sentence as an equation.

#3 Set Up an Equation 6.RP.A.3 Step 2

Translate "$20\%$ of $N$ is $12$" into an equation.
$20\%$ means $\dfrac{20}{100} = \dfrac{1}{5}$, so $20\%$ of $N$ is $\dfrac{N}{5}$.

$$\dfrac{1}{5} N = 12$$

💡 "Percent of" is the Grade 6 percent move: rewrite $20\%$ as the fraction $\tfrac{1}{5}$ and multiply.

#3 Set Up an Equation 6.EE.B.7 Step 3

Solve for $N$ by multiplying both sides by $5$.

$$N = 5 \times 12 = 60$$

💡 If one-fifth of the number is $12$, the whole number is five times that.

#3 Set Up an Equation 6.RP.A.3 Step 4

Now take $30\%$ of $N = 60$.
$30\%$ means $\dfrac{30}{100} = \dfrac{3}{10}$.

$$\dfrac{3}{10} \times 60 = 3 \times 6 = 18 \;\Rightarrow\; \textbf{(B)}$$

💡 Three-tenths of $60$ is three of the ten equal pieces of $60$, and each piece is $6$.

[1] #4 6.EE.A.2 Name the number. Let $N$ be the unknown number that the percents are taken from.

[2] #3 6.RP.A.3 Translate "$20\%$ of $N$ is $12$" into an equation. $20\%$ means $\dfrac{20}{100

[3] #3 6.EE.B.7 Solve for $N$ by multiplying both sides by $5$.

[4] #3 6.RP.A.3 Now take $30\%$ of $N = 60$. $30\%$ means $\dfrac{30}{100} = \dfrac{3}{10}$.

Review

Reasonableness: Check the size. Since $30\%$ is more than $20\%$, the answer should be larger than $12$. It is: $18 > 12$. Also, $30\%$ is one and a half times $20\%$, so $30\%$ of the number should be one and a half times $12$, which is $1.5 \times 12 = 18$. The trap answers map to common mistakes: (A) $15$ adds $\tfrac{1}{4}$ of $12$ instead of $\tfrac{1}{2}$, (C) $20$ confuses the answer with the number itself, (D) $24$ doubles $12$ (treating $30\%$ as twice $20\%$), and (E) $30$ uses $30$ as if it were a count, not a percent.

Alternative: Tool #5 (Find a Pattern): skip solving for $N$ by spotting the ratio between the two percents. $30\%$ is $\dfrac{30}{20} = \dfrac{3}{2}$ times as much as $20\%$, so $30\%$ of the same number is $\dfrac{3}{2}$ times as much as $20\%$ of it. That gives $\dfrac{3}{2} \times 12 = 18$, answer (B), in one step.

CCSS standards used (min grade 6)

6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers (Naming the unknown number $N$ so the percent sentence becomes an equation in one variable.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world problems, including finding a percent of a quantity (Reading "$20\%$ of $N$" as $\tfrac{1}{5}N$ and "$30\%$ of $60$" as $\tfrac{3}{10} \times 60$.)
6.EE.B.7 Solve real-world and mathematical problems by writing and solving one-variable equations of the form px = q (Solving $\tfrac{1}{5}N = 12$ by multiplying both sides by $5$ to get $N = 60$.)

⭐ When a percent of an unknown number is given, find the whole number first, then take the new percent of it. Or notice the ratio between the two percents and multiply $12$ by it directly — both paths give $18$.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 6)

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