AMC 10 · 2020 · #2

Grade 7 algebra
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
difference-of-squarespolynomial-factoringfraction-multiplication identify-subproblemspattern-recognition ↑ Prerequisites: difference-of-squaresfraction-multiplication
📏 Short solution 💡 2 insights
Problem
One numerical expression is a product of two fractions. The first has a difference of squares on top and on the bottom; the second is built from the same factors with their places swapped. Find the number this product equals.

Pick an answer.

(A)
1
(B)
$\frac{9951}{9950}$
(C)
$\frac{4780}{4779}$
(D)
$\frac{108}{107}$
(E)
$\frac{81}{80}$
How to solve
Strategy Organize Information in More Ways

The two factors are written in two different formats — squares subtracted in one, difference-times-sum in the other — which hides how closely related they are. Tool #15 (Organize Information in More Ways) puts both into the same format, and once they match, the relationship is visible without any multiplication. Tool #5 (Look for a Pattern) is what triggers that move: the same two number pairs keep reappearing with top and bottom swapped, which is a designed structure, not a coincidence. Tool #4 (Introduce a Variable) justifies the rewrite once and for all with letters a and b, so it does not have to be re-checked for each pair. Tool #3 (Eliminate Possibilities) closes the loop at the end: the wrong choices are all built from the leftovers of the computation, so knowing where each one comes from confirms the reading was right.

1STEP 1

Spot the repeated number pairs

Spot the repeated number pairs.

(100²-7²)/(70²-11²)·(70-11)(70+11)/(100-7)(100+7)
2STEP 2

Factor each difference of squares

Turn each difference of squares into a product.

(a-b)(a+b)=a²-b² → 100²-7²=(100-7)(100+7), 70²-11²=(70-11)(70+11)
3STEP 3

The second factor is the reciprocal

The second is revealed as the reciprocal.

(100-7)(100+7)/(70-11)(70+11)·(70-11)(70+11)/(100-7)(100+7)=(100-7)(100+7)(70-11)(70+11)/(70-11)(70+11)(100-7)(100+7)=1
4STEP 4

Check with the actual numbers

Everything cancels to 1.

9951/4779·4779/9951=1 → (A)
Answer
1
Two independent routes agree on 1: the structural cancellation and the direct arithmetic 9951/4779·4779/9951. A third check needs no multiplication at all — once the squares are factored, the second factor is literally the first one turned upside down, and a positive number times its reciprocal can never be anything but 1, so no choice larger than 1 can be right. That the four wrong choices each sit within about 1% above 1 is itself a signal: they are the results of small arithmetic slips on the correct computation, not the results of a different method.
💡Key takeaway

When the same numbers keep showing up on opposite floors of two fractions, rewrite until the two look alike — a number times its own reciprocal is 1, so there is nothing left to multiply.

  • Spot the repeated number pairs
  • Factor each difference of squares
  • The second factor is the reciprocal
  • Check with the actual numbers

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