AMC 10 · 2018 · #17

Grade 7 number-theory

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Integrality / Divisibility Screen

fraction-arithmeticbound-inequality-then-enumerateextremal-constructioninterval-arithmeticlinear-diophantine extreme-principlebound-inequality-then-enumerateconvert-to-algebra ↑ Prerequisites: fraction-arithmeticinterval-arithmetic

📏 Long solution 💡 4 insights

Problem

Find positive integers forming a fraction that lies strictly between five ninths and four sevenths, choosing the pair whose denominator is as small as possible. Then report the denominator minus the numerator.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Extreme Principle

The question asks for a minimum, so the work splits into two halves: prove no small q can work, then exhibit one q that does. To prove the lower bound, clear the denominators so both conditions become statements about whole numbers. Then use the one fact that separates integers from reals — a positive integer is at least 1, not merely more than 0. Adding the two resulting inequalities with well-chosen weights makes p cancel and leaves a hard floor for q. Finally, check that the floor is actually reached, since a bound alone never proves attainability.

1STEP 1

Name what we are hunting

Give the target a name.

5/9 ＜ p/q ＜ 4/7, p, q positive integers

2STEP 2

Clear the denominators

Clear the denominators in both.

5/9 ＜ p/q ⇔ 5q ＜ 9p, p/q ＜ 4/7 ⇔ 7p ＜ 4q

3STEP 3

Each gap is at least 1

An integer gap is at least one.

9p - 5q ≥ 1, 4q - 7p ≥ 1

4STEP 4

Combine the gaps to trap q

Combining the gaps traps the denominator.

q = 7(9p - 5q) + 9(4q - 7p) ≥ 7 · 1 + 9 · 1 = 16

5STEP 5

The same trick bounds p

The same trick bounds the numerator.

p = 4(9p - 5q) + 5(4q - 7p) ≥ 4 · 1 + 5 · 1 = 9

6STEP 6

Pin down p when q is 16

At the smallest denominator the numerator is forced.

q = 16 → p ≥ 9 and 7p ＜ 64 → p ≤ 9 → p = 9

7STEP 7

Check it, then subtract

Subtracting gives 7.

5/9 ＜ 9/16 ＜ 4/7, q - p = 16 - 9 = 7

Answer

The window is thin: 4/7 - 5/9 = 1/63, so a fraction landing inside is a tight fit and a denominator in the teens is believable rather than surprising. Checking small denominators by hand agrees. At q = 7 the numerator must sit strictly between 35/9 ≈ 3.89 and 4; at q = 9 it must sit strictly between 5 and 36/7 ≈ 5.14. Both windows miss every integer, and so does every other q below 16. The winner is also unique: at q = 16 only p = 9 works, so q - p is well defined. Numerically 9/16 = 0.5625 sits comfortably between 0.5556 and 0.5714, and q - p = 7 matches choice (A).

💡Key takeaway

Whole numbers cannot beat each other by a sliver, so a narrow gap between fractions forces a large denominator.

Name what we are hunting
Clear the denominators
Each gap is at least 1
Combine the gaps to trap q
The same trick bounds p
Pin down p when q is 16
Check it, then subtract