AMC 10 · 2017 · #2
Grade 7 algebraPick an answer.
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
Two letters capture the given fact.
Giving the unknown numbers names lets you write the one fact you are told as a single equation.
6.EE.B.6Introduce A VariableWrite the target as one fraction
The target is the sum over the product.
Two fractions over a shared bottom add into one fraction, so the reciprocal sum is just the sum over the product.
Two reciprocals over a shared bottom add into one fraction: the sum over the product.
▸ Why?
Multiplying top and bottom by the same amount renames each fraction without changing its value.
▸ Why?
Once both sit over the same bottom, the two tops can be gathered into one.
Substitute and simplify
Substituting gives 4, choice (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