AMC 10 · 2015 · #15

Grade 7 rate-ratio

Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration

linear-diophantinegcdfraction-arithmetic convert-to-algebrasimons-favorite-factoring-trick ↑ Prerequisites: fraction-arithmeticgcd

📏 Medium solution 💡 3 insights

Problem

Look at every fraction

\frac{x}{y}

where x and y are positive integers with no common factor. Add 1 to the top and 1 to the bottom. We want the new fraction to be exactly 10% larger than the old one. Count how many such fractions exist.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

infinitely many

AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

The fraction already gives us names x and y, so turn the words '10% larger' into an equation between x and y. The equation mixes x and y in a product, so a factoring rearrangement turns it into 'two whole numbers multiply to 110.' Then list the factor pairs and throw out the ones that are not relatively prime.

1STEP 1

Write the 10% condition

A 10% increase means multiplying by $\frac{11}{10}$ .

\frac{x+1}{y+1}

\frac{11}{10}

\frac{x}{y}

2STEP 2

Clear fractions

Cross-multiplying clears the fractions, leaving one equation in x and y.

10y(x+1) = 11x(y+1) → 10xy+10y = 11xy+11x → xy+11x-10y = 0

3STEP 3

Factor into a product

Rearranging and adding a constant turns the equation into a product equal to 110.

xy+11x-10y = 0 → (10-x)(y+11) = 110

4STEP 4

List the factor pairs

Matching divisors of 110 give three candidate pairs: (5,11), (8,44), (9,99).

110 = 5· 22 = 2· 55 = 1· 110

5STEP 5

Keep only the coprime fraction

Only (5,11) is coprime, so exactly one fraction works: answer (B).

gcd(5,11)=1,

\frac{6}{12}

\frac{1}{2}

= 1.1·

\frac{5}{11}

Answer

Plug (5,11) back in:

\frac{5}{11}

is about 0.4545 and

\frac{6}{12}

= 0.5, which is exactly a 10% jump (0.4545 x 1.1 = 0.5). The other factor pairs failed only because of the coprime rule, not the equation, so they confirm the algebra was right while leaving exactly one valid fraction.

💡Key takeaway

Turn '10% bigger' into × $\frac{11}{10}$ , factor into a product equal to 110, and keep only the coprime factor pair.

Write the 10% condition
Clear fractions
Factor into a product
List the factor pairs
Keep only the coprime fraction

Write the 10% condition

Clear fractions

Factor into a product

List the factor pairs

Keep only the coprime fraction

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