AMC 10 · 2008 · #6

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

Points B and C sit on segment AD. The piece AB is 4 times as long as BD, and the piece AC is 9 times as long as CD. Find what fraction of AD the middle piece BC takes up.

Pick an answer.

(A)

$\frac{1}{36}$

(B)

$\frac{1}{13}$

(C)

$\frac{1}{10}$

(D)

$\frac{5}{36}$

(E)

$\frac{1}{5}$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

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

Both clues are ratios, so the true length of AD does not matter — set AD = 1 and every length reads off as a fraction of the whole.

AD = 1

2STEP 2

Locate B

AB = 4 · BD is a 4 to 1 split, so AD is 5 equal parts: BD is $\frac{1}{5}$ and $AB = \frac{4}{5}$ measured from A.

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

3STEP 3

Locate C

AC = 9 · CD is a 9 to 1 split, so AD is 10 equal parts: CD is $\frac{1}{10}$ and $AC = \frac{9}{10}$ measured from A.

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

4STEP 4

Measure the gap BC

Since $\frac{4}{5}$ is under $\frac{9}{10}$ , the order is A, B, C, D, so the gap BC = AC - AB = $\frac{1}{10}$ — 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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