AMC 10 · 2017 · #5
Grade 7 arithmeticPick an answer.
AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The two numbers are never told to us, so Tool #4 (Introduce a Variable) lets us name them a and b and write the single given fact as an equation. Tool #13 (Convert to Algebra) then turns the target — the sum of reciprocals — into one fraction (a+b)/ab, which lines up perfectly with the given a+b=4ab. Tool #6 (Guess and Check) confirms the result on a concrete pair of numbers.
Name the numbers, write the fact
Name the numbers a and b; "their sum is 4 times their product" becomes the single equation a+b=4ab.
Giving the unknown numbers names lets you write the one fact you are told as a single equation.
6.EE.B.6Use Matrix LogicWrite the target as one fraction
Add the reciprocal sum 1/a+1/b over the common denominator ab into the single fraction (a+b)/ab.
Two fractions over a shared bottom add into one fraction, so the reciprocal sum is just the sum over the product.
7.NS.A.1Convert To AlgebraSubstitute and simplify
Since a+b=4ab, the numerator of (a+b)/ab is 4ab; the nonzero ab cancels to leave 4 — the answer is (C).
Once the top is exactly 4 copies of the bottom, the fraction has to equal 4.
7.EE.A.2Convert To AlgebraThe sum of two reciprocals is always the sum over the product, so once the sum equals 4 times the product, the reciprocals add to 4.
- Name the numbers, write the fact
- Write the target as one fraction
- Substitute and simplify