Competition · AMC preparation · step 4 of 4

AMC 10 · 2020A · #3

Grade 7 algebra
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
fraction-arithmeticpattern-recognitionpolynomial-factoring pattern-recognition ↑ Prerequisites: fraction-arithmetic
📏 Short solution 💡 2 insights
Problem
Simplify the product a−35−c\frac{a-3}{5-c} · b−43−a\frac{b-4}{3-a} · c−54−b\frac{c-5}{4-b}, given that none of the denominators are zero.

Pick an answer.

(A)
${-}1$
(B)
1
(C)
$\frac{abc}{60}$
(D)
$\frac{1}{abc} - \frac{1}{60}$
(E)
$\frac{1}{60} - \frac{1}{abc}$

AMC 10 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

Tool #5 (Look for a Pattern): every numerator (a-3), (b-4), (c-5) is the negative of some denominator -(3-a), -(4-b), -(5-c). That single repeated structure is the whole problem. Tool #15 (Reorganize) — re-pair the fractions so each numerator sits over its own negative: (a-3)/(3-a) · (b-4)/(4-b) · (c-5)/(5-c). Tool #6 (Guess and Check) is an even faster sanity path: pick easy numbers (a=b=c=0) and just compute. Tool #3 (Eliminate) confirms — the expression must be a constant (no a,b,c left), so (C), (D), (E) all involve abc and are out.

1STEP 1

Spot the opposite pairs

Each numerator is the opposite of a denominator: (a-3) = -(3-a), same for the others.

a - 3 = -(3 - a), b - 4 = -(4 - b), c - 5 = -(5 - c)
2STEP 2

Pair each numerator with its match

Reorder the factors so each numerator sits over its own opposite denominator.

(a-3)/(5-c) · (b-4)/(3-a) · (c-5)/(4-b) = (a-3)/(3-a) · (b-4)/(4-b) · (c-5)/(5-c)
3STEP 3

Multiply the three negative ones

Each re-paired fraction is thing−thing\frac{thing}{-thing} = -1, and (-1)(-1)(-1) = -1 → (A).

(-1)(-1)(-1) = -1 → (A)
4STEP 4

Test with real numbers

Check with a=b=c=0: −35\frac{-3}{5} · −43\frac{-4}{3} · −54\frac{-5}{4} = −6060\frac{-60}{60} = -1.

-3/5 · -4/3 · -5/4 = -60/60 = -1 → (A)
5STEP 5

Rule out the variable choices

(C)(D)(E) contain abc but our result is a constant, so they're out; the sign check picks (A) over (B).

constant result → choice is (A) or (B); sign check picks (A)
Answer
-1
Try a second concrete triple to be sure. With a = 1, b = 1, c = 1: product = −24\frac{-2}{4} · −32\frac{-3}{2} · −43\frac{-4}{3} = (−2)(−3)(−4)(4⋅2⋅3)\frac{(-2)(-3)(-4)}{(4 · 2 · 3)} = −2424\frac{-24}{24} = -1. Same answer, supporting (A).
💡Key takeaway

This AMC 10 problem only needs Grade 7 "multiplying signed numbers" you already know — each numerator is the opposite of one denominator, so each pair gives -1, and three of those multiply to -1.

  • Spot the opposite pairs
  • Pair each numerator with its match
  • Multiply the three negative ones
  • Test with real numbers
  • Rule out the variable choices

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