AMC 10 · 2017 · #5

Grade 7 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

fraction-arithmeticratio-proportion symmetric-polynomialsconvert-to-algebra ↑ Prerequisites: fraction-arithmetic

📏 Short solution 💡 2 insights

Problem

Two nonzero real numbers have a sum that equals 4 times their product. Find the sum of the reciprocals of the two numbers.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

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.

1STEP 1

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.

a + b = 4ab

2STEP 2

Write the target as one fraction

Add the reciprocal sum $\frac{1}{a}$ + $\frac{1}{b}$ over the common denominator ab into the single fraction $\frac{a+b}{ab}$ .

\frac{1}{a}

\frac{1}{b}

\frac{b}{ab}

\frac{a}{ab}

\frac{a+b}{ab}

3STEP 3

Substitute and simplify

Since a+b=4ab, the numerator of $\frac{a+b}{ab}$ is 4ab; the nonzero ab cancels to leave 4 — the answer is (C).

\frac{a+b}{ab}

\frac{4ab}{ab}

=4 → (C)

Answer

Test a concrete pair. Take a=1; then 1+b=4b gives b=

\frac{1}{3}

, and indeed the product

\frac{1}{3}

times 4 is

\frac{4}{3}

, the same as the sum 1+

\frac{1}{3}

. The reciprocals are 1 and 3, summing to 4 — matching (C). The pair a=b=

\frac{1}{2}

also works and gives 2+2=4.

💡Key takeaway

The 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

Name the numbers, write the fact

Write the target as one fraction

Substitute and simplify

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