AMC 10 · 2008 · #8

Grade 6 rate-ratio

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Chained Ratio Linking

ratio-proportionfraction-arithmetic convert-to-algebra ↑ Prerequisites: ratio-proportion

📏 Medium solution 💡 2 insights

Problem

Two points divide a segment in two given ratios measured from the same end. Find what fraction of the whole the middle piece takes.

Pick an answer.

(A)

$\frac {1}{36}$

(B)

$\frac {1}{13}$

(C)

$\frac {1}{10}$

(D)

$\frac {5}{36}$

(E)

$\frac {1}{5}$

How to solve

Strategy Draw a Diagram

Everything here is about positions along one straight segment, which is exactly what Tool #1 (Draw a Diagram) is for. If I mark where B and C fall on AD using fractions of the whole, the answer BC is just the gap between the two marks. Tool #4 (Introduce a Variable) supports this by letting me call the whole length AD one unit, so each ratio condition turns directly into a fraction I can plot.

1STEP 1

Call the whole segment 1

Calling the whole one makes every piece a fraction.

AD = 1

2STEP 2

Locate B

The first ratio locates one point at 4/5.

AB : BD = 4 : 1 → 5 parts → AB = 4/5, BD = 1/5

3STEP 3

Locate C

The second locates the other at 9/10.

AC : CD = 9 : 1 → 10 parts → AC = 9/10, CD = 1/10

4STEP 4

Measure the gap BC

The gap between them is 1/10, choice (C).

BC = AC - AB = 9/10 - 8/10 = 1/10 → (C)

Answer

1/10

B is at 0.8 and C is at 0.9 along a segment of length 1, so BC should be a small sliver — about a tenth. The value 1/10 matches, and it is safely between the tiny 1/36 and the larger 1/5. Both points also correctly land strictly inside the segment (0.8 and 0.9 are between 0 and 1), confirming the picture is valid.

💡Key takeaway

Turn each "4 times" or "9 times" clue into equal parts, mark both points on one line, and the middle piece is just the gap between the marks.

Call the whole segment 1
Locate B
Locate C
Measure the gap BC

Call the whole segment 1

Locate B

Locate C

Measure the gap BC

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