AMC 8 · 2003 · #3

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition

percentagefraction-arithmeticcomplementary-counting identify-subproblemscomplementary-counting ↑ Prerequisites: fraction-decimal-conversionmulti-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

A burger at Ricky C's weighs $120$ grams, of which $30$ grams are filler.
What percent of the burger is not filler?

$\mathrm{(A)}\ 60\% \qquad\mathrm{(B)}\ 65\% \qquad\mathrm{(C)}\ 70\% \qquad\mathrm{(D)}\ 75\% \qquad\mathrm{(E)}\ 90\%$

Pick an answer.

(A)

$60\%$

(B)

$65\%$

(C)

$70\%$

(D)

$75\%$

(E)

$90\%$

View mode:

Toolkit + CCSS Solution

Understand

Restated: A burger at Ricky C's weighs $120$ grams total, and $30$ grams of that is filler. What percent of the burger is **not** filler?

Givens: Total burger weight $= 120$ grams; Filler weight $= 30$ grams; Answer choices: (A) $60\%$, (B) $65\%$, (C) $70\%$, (D) $75\%$, (E) $90\%$

Unknowns: The percent of the burger that is not filler

Understand

Restated: A burger at Ricky C's weighs $120$ grams total, and $30$ grams of that is filler. What percent of the burger is **not** filler?

Givens: Total burger weight $= 120$ grams; Filler weight $= 30$ grams; Answer choices: (A) $60\%$, (B) $65\%$, (C) $70\%$, (D) $75\%$, (E) $90\%$

Plan

Primary tool: #3 Set Up an Equation

Secondary: #9 Solve an Easier Problem

The question asks for a percent, which is a part-over-whole equation. Tool #3 (Set Up an Equation) gives the formula directly: percent $=$ (non-filler weight) $\div$ (total weight) $\times 100\%$. The trap is reading the problem too fast and computing the filler percent ($30/120 = 25\%$) instead of the non-filler percent. Tool #9 (Solve an Easier Problem) handles that by first finding the non-filler weight as its own sub-problem, then plugging it into the percent equation.

Execute — Answer: D

#9 Solve an Easier Problem 6.NS.B.3 Step 1

Find the non-filler weight.
Subtract the filler from the total — this is the easier sub-problem.

$$\text{non-filler} = 120 - 30 = 90 \text{ grams}$$

💡 Splitting the burger into filler and non-filler is the Grade 6 part-part-whole move on whole numbers.

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

Set up the percent equation.
The non-filler is the part, the full burger is the whole.

$$\text{percent} = \dfrac{\text{part}}{\text{whole}} \times 100\% = \dfrac{90}{120} \times 100\%$$

💡 The Grade 6 percent formula reads any fraction as a count out of $100$.

#3 Set Up an Equation 6.RP.A.1 Step 3

Simplify the fraction before multiplying.
Divide top and bottom by $30$.

$$\dfrac{90}{120} = \dfrac{3}{4}$$

💡 Reducing first turns the messy $\tfrac{90}{120}$ into the familiar $\tfrac{3}{4}$, whose percent is memorized.

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

Convert $\tfrac{3}{4}$ to a percent and match the choice.

$$\dfrac{3}{4} \times 100\% = 75\% \;\Rightarrow\; \textbf{(D)}$$

💡 $\tfrac{3}{4}$ of $100$ is $75$, so $\tfrac{3}{4}$ of the burger is $75\%$.

[1] #9 6.NS.B.3 Find the non-filler weight. Subtract the filler from the total — this is the eas

[2] #3 6.RP.A.3 Set up the percent equation. The non-filler is the part, the full burger is the

[3] #3 6.RP.A.1 Simplify the fraction before multiplying. Divide top and bottom by $30$.

[4] #3 6.RP.A.3 Convert $\tfrac{3}{4}$ to a percent and match the choice.

Review

Reasonableness: Cross-check by computing the filler percent instead: $\tfrac{30}{120} = \tfrac{1}{4} = 25\%$. Filler $+$ non-filler must total $100\%$, and $25\% + 75\% = 100\%$. The numbers also pass a feel test — only one-quarter of the burger is filler, so three-quarters is the real thing, which lands on (D). Choice (E) $90\%$ is the trap that comes from reading $90$ grams as a percent directly; (A) $60\%$ and (C) $70\%$ are common "round number" distractors that ignore the actual ratio.

Alternative: Tool #9 (Solve an Easier Problem) all the way: instead of computing a fraction, ask "how many groups of $\tfrac{1}{4}$ of a burger does $90$ grams equal?" Since $\tfrac{1}{4}$ of $120$ is $30$ grams, the non-filler $90$ grams equals three of those quarter-pieces. Three quarters $= 75\%$, so the answer is (D).

CCSS standards used (min grade 6)

6.NS.B.3 Fluently add, subtract, multiply, and divide multi-digit decimals (Subtracting $120 - 30 = 90$ grams to isolate the non-filler weight before the percent step.)
6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship (Reducing $\tfrac{90}{120}$ to $\tfrac{3}{4}$ as a part-to-whole ratio.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Applying the percent formula $\tfrac{\text{part}}{\text{whole}} \times 100\%$ to get $75\%$.)

⭐ Percent questions ask "part out of the whole" — pin down which weight is the part, simplify the fraction, then read off the percent.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 6)

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