AMC 10 · 2003 · #13

Grade 7 algebra
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
systems-of-equationslinear-equations-one-var convert-to-algebraidentify-subproblems ↑ Prerequisites: systems-of-equations
📏 Medium solution 💡 2 insights
Problem
Three numbers add up to 20. The first equals four times the sum of the other two, and the second equals seven times the third. Find the product of the three numbers.

Pick an answer.

(A)
28
(B)
40
(C)
100
(D)
400
(E)
800

AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

Each sentence in the problem describes a number in terms of the others, so Tool #4 (Introduce a Variable) lets us name the unknowns and turn each sentence into an equation. The smart move is to not solve for all three at once. Tool #7 (Identify Subproblems) splits the work: first treat 'the other two' as a single chunk to pin down the first number, then break that chunk apart to get the second and third. Tool #13 (Convert to Algebra) carries each English condition into a clean equation we can solve one step at a time.

1STEP 1

Name the numbers and write the conditions

Call them a, b, c. The sentences become a+b+c=20, a=4(b+c), and b=7c — three facts for three unknowns.

a+b+c=20, a=4(b+c), b=7c
2STEP 2

Treat the other two as one chunk to find the first

Substituting a=4(b+c) into the total gives 5(b+c)=20, so the chunk b+c=4 and the first number is 16.

4(b+c)+(b+c)=5(b+c)=20 → b+c=4, a=16
3STEP 3

Break the chunk apart for the second and third

Feeding b=7c into b+c=4 gives 8c=4, so the third number is 0.5 and the second is 3.5.

7c+c=8c=4 → c=1/2, b=7/2=3.5, a=16
4STEP 4

Multiply the three numbers

Pair the decimals first: 3.5 × 0.5 = 1.75, and 16 × 1.75 = 28, which is choice (A).

16 × 3.5 × 0.5 = 16 × 1.75 = 28 → (A)
Answer
28
The three numbers 16, 3.5, 0.5 pass every condition: they sum to 20, the first (16) is four times the other two combined (4 × 4), and the second (3.5) is seven times the third (0.5). Their product 28 is the smallest answer choice, which fits because two of the numbers are less than 1-and-a-half, dragging the product down; the larger choices like 400 or 800 would need all three numbers to be big, but only the first one is.
💡Key takeaway

When one number is four times all the rest, the total splits into five equal parts, so peel off the big number first, then split what is left by the given ratio.

  • Name the numbers and write the conditions
  • Treat the other two as one chunk to find the first
  • Break the chunk apart for the second and third
  • Multiply the three numbers

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