AMC 10 · 2016 · #2

Grade 6 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
harmonic-meanfraction-arithmeticestimation convert-to-algebraestimation ↑ Prerequisites: harmonic-meanfraction-arithmetic
📏 Short solution 💡 2 insights
Problem
A rule doubles a product and divides by the sum of two very different numbers. Find the nearest whole number.

Pick an answer.

(A)
2
(B)
45
(C)
504
(D)
1008
(E)
2015
How to solve
Strategy Convert to Algebra

The only hard part is that the rule arrives as an English sentence. Tool #13 (Convert to Algebra) turns "twice their product divided by their sum" into the expression 2ab/(a+b), after which substituting 1 and 2016 is pure arithmetic. The quotient 4032/2017 is not a whole number, so tool #15 (Organize Information in More Ways) rewrites it as a whole number minus a small leftover, which shows the distance to the nearest integer directly instead of forcing a long division. Tool #3 (Eliminate Possibilities) then closes the problem: once the size of the value is known, four of the five choices are nowhere near it.

1STEP 1

Turn the sentence into a formula

The sentence becomes one short formula.

H = 2ab/(a+b)
2STEP 2

Substitute 1 and 2016

Substituting gives one awkward fraction.

H = (2 · 1 · 2016)/(1+2016) = 4032/2017
3STEP 3

Rewrite as 2 minus a scrap

It rewrites as 2 minus a tiny scrap.

4032/2017 = (4034-2)/2017 = 4034/2017 - 2/2017 = 2 - 2/2017
4STEP 4

Check the gap, then eliminate

The scrap is too small to change the rounding, choice (A).

0 ＜ 2/2017 ＜ 1/2 → 1.5 ＜ 2 - 2/2017 ＜ 2 → nearest integer = 2 → (A)
Answer
2
The size is exactly what the harmonic mean should produce. For positive a ≤ b, the sum a+b is bigger than b, so 2ab/(a+b) ＜ 2ab/b = 2a: a harmonic mean can never exceed twice the smaller number. With a=1 that caps the value at 2, which rules out 45, 504, 1008, and 2015 before any arithmetic. The same comparison run the other way, a+b ＜ 2b, gives 2ab/(a+b) ＞ a = 1, so the value is trapped strictly between 1 and 2, and the computed 2-2/2017 ≈ 1.999 lands inside that window. The tempting wrong answers are the other two means: 1008 is close to the arithmetic mean (1+2016)/2 = 1008.5, and 45 is close to the geometric mean √(2016) ≈ 44.9.
💡Key takeaway

A harmonic mean can never be more than twice the smaller number, so pairing 1 with something enormous lands just below 2 no matter how enormous it is.

  • Turn the sentence into a formula
  • Substitute 1 and 2016
  • Rewrite as 2 minus a scrap
  • Check the gap, then eliminate

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