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 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)
0
(B)
1
(C)
2
(D)
3
(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 11/10.

(x+1)/(y+1) = 11/10·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, 6/12=1/2 = 1.1·5/11
Answer
1
Plug (5,11) back in: 5/11 is about 0.4545 and 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 ×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

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