AMC 10 · 2009 · #7

Grade 7 arithmetic
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Percent / Fraction Recovery
percentagelinear-equations-one-var convert-to-algebra ↑ Prerequisites: percentage
📏 Medium solution 💡 2 insights
Problem
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.

Pick an answer.

(A)
$\ \frac{12}{5}$
(B)
$\ 3$
(C)
$\ \frac{10}{3}$
(D)
$\ 38$
(E)
$\ 42$

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Convert to Algebra

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.

1STEP 1

Turn "40% less" into "60% of"

Whole milk is 100% of itself, so taking 40% away leaves 60% — the 2% carton holds 60% of the whole-milk fat.

100%-40%=60%, 2 = 60% of the whole-milk fat
2STEP 2

Name the unknown and write the equation

Let w be the whole-milk fat percentage, so "2 is 60% of w" becomes one equation, 0.6w = 2.

let w = whole-milk fat %, 0.6 w = 2
3STEP 3

Solve for the whole-milk percentage

Divide both sides by 0.6: w = 2 ÷ 0.6 = 2 × 10/6 = 10/3, about 3.33.

w = 2/0.6 = 2 × 10/6 = 20/6 = 10/3 ≈ 3.33
4STEP 4

Match and eliminate the traps

60% of 10/3 is exactly 2, while (A) gives 1.44 and (B) gives 1.8, and 38 or 42 is absurd for milk — so (C).

0.6 × 10/3 = 2 ✓, 0.6×12/5=1.44, 0.6 × 3 = 1.8 → (C)
Answer
10/3
Whole milk should have more fat than the reduced carton, and 10/3≈ 3.33 is indeed bigger than 2, which passes the smell test. The exact check also holds: reducing 10/3 by 40% removes 0.4 × 10/3 = 4/3, leaving 10/3-4/3 = 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.
💡Key takeaway

"40% less" doesn't mean subtract 40 — it means keep 60%, so scale by 0.6 and work backward to the full amount.

  • Turn "40% less" into "60% of"
  • Name the unknown and write the equation
  • Solve for the whole-milk percentage
  • Match and eliminate the traps

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