AMC 10 · 2009 · #7
Grade 7 arithmeticA carton contains milk that is 2% fat, an amount that is 40% less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk?
Pick an answer.
AMC 10 2009 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.
Toolkit + CCSS Solution
Understand
Restated: A carton of milk is $2\%$ fat. That $2\%$ is $40\%$ less than the fat percentage of whole milk. Find the fat percentage of whole milk.
Givens: The lower-fat milk is $2\%$ fat; That $2\%$ is $40\%$ less than the whole-milk fat percentage; Answer choices: (A) $\frac{12}{5}$, (B) $3$, (C) $\frac{10}{3}$, (D) $38$, (E) $42$
Unknowns: The fat percentage of whole milk
Understand
Restated: A carton of milk is $2\%$ fat. That $2\%$ is $40\%$ less than the fat percentage of whole milk. Find the fat percentage of whole milk.
Givens: The lower-fat milk is $2\%$ fat; That $2\%$ is $40\%$ less than the whole-milk fat percentage; Answer choices: (A) $\frac{12}{5}$, (B) $3$, (C) $\frac{10}{3}$, (D) $38$, (E) $42$
Plan
Primary tool: #13 Convert to Algebra
Secondary: #4 Introduce a Variable, #3 Eliminate Possibilities
The trap in this problem is the phrase "$40\%$ less than," so Tool #13 (Convert to Algebra) is the spine: it forces "$40\%$ less" to become the exact multiplier "$60\%$ of," which is where most wrong answers come from. Tool #4 (Introduce a Variable) names the whole-milk percentage so the sentence becomes one clean equation to solve. Tool #3 (Eliminate Possibilities) is the finishing check: the answer is an unusual fraction, so testing it and the near-miss choices against the "$40\%$ less" rule confirms which one truly fits.
Execute — Answer: C
7.RP.A.3 Step 1 Turn "40% less" into "60% of"
- Whole milk is the full amount, which counts as $100\%$ of itself.
- Taking $40\%$ away leaves $100\%-40\%=60\%$ of the whole-milk fat.
- So the $2\%$ carton holds $60\%$ of whatever whole milk holds.
- The key move is refusing to subtract $40$ from anything: "$40\%$ less" scales by $60\%$, it does not subtract $40$.
💡 Cutting something by $40\%$ keeps $60\%$ of it, so "less by a percent" is really "times a smaller percent."
6.EE.B.7 Step 2 Name the unknown and write the equation
- Let $w$ stand for the fat percentage of whole milk.
- "$2$ is $60\%$ of $w$" becomes $0.6\,w = 2$.
- Writing $60\%$ as the decimal $0.6$ turns the sentence into a single equation of the form $px=q$, ready to solve.
💡 Giving the unknown a name lets the English sentence become one equation you can actually solve.
6.NS.B.3 Step 3 Solve for the whole-milk percentage
- Undo the multiplication by dividing both sides by $0.6$: $w = 2 \div 0.6$.
- Dividing by $0.6$ is the same as dividing by $\frac{6}{10}$, i.e.
- multiplying by $\frac{10}{6}$, so $w = 2 \times \frac{10}{6} = \frac{20}{6} = \frac{10}{3}$.
- As a decimal that is about $3.33$.
💡 To undo "times $0.6$," divide by $0.6$ — that pulls the whole amount back out of its shrunken version.
7.RP.A.3 Step 4 Match and eliminate the traps
- $\frac{10}{3}$ is exactly choice (C).
- Check it against the rule: $60\%$ of $\frac{10}{3}$ is $0.6 \times \frac{10}{3} = 2$, correct.
- The traps fail this test: $60\%$ of (A) $\frac{12}{5}$ is $1.44$, and $60\%$ of (B) $3$ is $1.8$ — neither is $2$ — while (D) $38$ and (E) $42$ are absurd fat levels for milk.
- Only (C) survives, so the answer is (C).
💡 The right whole shrinks back to exactly $2$ under a $40\%$ cut; the trap choices shrink to the wrong number.
7.RP.A.3 Whole milk is the full amount, which counts as $100\%$ of itself. Taking $40\%$ 6.EE.B.7 Let $w$ stand for the fat percentage of whole milk. "$2$ is $60\%$ of $w$" becom 6.NS.B.3 Undo the multiplication by dividing both sides by $0.6$: $w = 2 \div 0.6$. Divid 7.RP.A.3 $\frac{10}{3}$ is exactly choice (C). Check it against the rule: $60\%$ of $\fra Review
Reasonableness: Whole milk should have more fat than the reduced carton, and $\frac{10}{3}\approx 3.33$ is indeed bigger than $2$, which passes the smell test. The exact check also holds: reducing $\frac{10}{3}$ by $40\%$ removes $0.4 \times \frac{10}{3} = \frac{4}{3}$, leaving $\frac{10}{3}-\frac{4}{3} = \frac{6}{3} = 2$, matching the given $2\%$. The big choices $38$ and $42$ are red herrings built to tempt someone who multiplies instead of scaling down.
Alternative: Skip the decimal and reason in thirds of the whole. If $60\%$ (three-fifths) of the whole equals $2$, then one-fifth is $\frac{2}{3}$, so the full five-fifths is $5 \times \frac{2}{3} = \frac{10}{3}$ — the same answer without ever writing an equation.
CCSS standards used (min grade 7)
7.RP.A.3Use proportional relationships to solve multi-step ratio and percent problems (Reading "$40\%$ less than" as "$60\%$ of" and testing each choice against that percent-decrease rule.)6.EE.B.7Solve real-world problems by writing and solving equations of the form px = q (Naming the whole-milk percentage $w$ and forming the equation $0.6\,w = 2$.)6.NS.B.3Fluently add, subtract, multiply, and divide multi-digit decimals (Dividing $2 \div 0.6$ to solve for $w = \frac{10}{3}$.)
⭐ "$40\%$ less" doesn't mean subtract $40$ — it means keep $60\%$, so scale by $0.6$ and work backward to the full amount.
⭐ "$40\%$ less" doesn't mean subtract $40$ — it means keep $60\%$, so scale by $0.6$ and work backward to the full amount.
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