Competition · AMC preparation · step 4 of 4

AMC 10 · 2019B · #18

Grade 8 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Multi-Segment Path or Count Sum

recursive-sequencesequences-geometriclinear-equations-two-varsystems-of-equations identify-subproblemswork-backwards ↑ Prerequisites: linear-equations-two-varsequences-geometric

📏 Medium solution 💡 3 insights

Problem

Henry's home is at 0 km, his gym at 2 km. He walks

\frac{3}{4}

of the way to the gym, then

\frac{3}{4}

of the way back home from there, then

\frac{3}{4}

of the way to the gym from there, and so on, always turning around after covering

\frac{3}{4}

of the remaining distance. In the long run his back-and-forth motion approaches two limit points A (closer to home) and B (closer to gym), measured in km from home. Find |A - B|.

Pick an answer.

(A)

$\frac{2}{3}$

(B)

(C)

$1 \frac{1}{5}$

(D)

$1 \frac{1}{4}$

(E)

$1 \frac{1}{2}$

AMC 10 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Convert to Algebra

Tool #1 (Diagram): a number line from 0 to 2 with A and B marked makes the geometry obvious. Tool #13 (Algebra): the limit condition gives two clean linear equations in A and B. Tool #11 (Work Backwards): instead of simulating forward from the first walk, use the fixed-point relations A and B must satisfy at the limit and solve directly.

1STEP 1

Draw the walk on a number line

On a line from 0 (home) to 2 (gym), from B walking $\frac{3}{4}$ toward home leaves $\frac{1}{4}$ of the way, landing at $\frac{1}{4}$ B — that point is A.

A = 1/4B

2STEP 2

Write the second walk

From A, walking $\frac{3}{4}$ toward the gym at 2 lands at A + $\frac{3}{4}$ (2 - A) = $\frac{1}{4}$ A + $\frac{3}{2}$ , which is B.

B = 1/4A + 3/2

3STEP 3

Substitute one into the other

Substitute A = $\frac{1}{4}$ B into B = $\frac{1}{4}$ A + $\frac{3}{2}$ , giving B = $\frac{1}{16}$ B + $\frac{3}{2}$ , so $\frac{15}{16}$ B = $\frac{3}{2}$ .

15/16B = 3/2

4STEP 4

Solve for both positions

Divide: B = $\frac{3}{2}$ · $\frac{16}{15}$ = $\frac{8}{5}$ , then A = $\frac{1}{4}$ B = $\frac{2}{5}$ .

B = 8/5, A = 2/5

5STEP 5

Find the distance between them

The gap: |A - B| = | $\frac{2}{5}$ - $\frac{8}{5}$ | = $\frac{6}{5}$ = 1 $\frac{1}{5}$ — choice (C).

|A - B| = 6/5 = 1 1/5

Answer

1 1/5

Sanity-check the limit points: from B =

\frac{8}{5}

, walk

\frac{3}{4}

toward home covers (

\frac{3}{4}

)(

\frac{8}{5}

) =

\frac{6}{5}

km, landing at

\frac{8}{5}

\frac{6}{5}

\frac{2}{5}

= A. ✓. From A =

\frac{2}{5}

, walk

\frac{3}{4}

toward gym covers (

\frac{3}{4}

)(2 -

\frac{2}{5}

) = (

\frac{3}{4}

)(

\frac{8}{5}

) =

\frac{6}{5}

km, landing at

\frac{2}{5}

\frac{6}{5}

\frac{8}{5}

= B. ✓. The gap

\frac{6}{5}

= 1.2 km is between the choices 1 and 1.5, matching choice (C).

💡Key takeaway

This AMC 10 problem only needs Grade 8 linear-equation skills you already know — at the limit, Henry's two turning points A and B satisfy A = $\frac{B}{4}$ and B = $\frac{A}{4}$ + $\frac{3}{2}$ , giving B = $\frac{8}{5}$ and A = $\frac{2}{5}$ , so the gap is $\frac{6}{5}$ = 1 $\frac{1}{5}$ . The answer is (C).

Draw the walk on a number line
Write the second walk
Substitute one into the other
Solve for both positions
Find the distance between them

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Draw the walk on a number line

Write the second walk

Substitute one into the other

Solve for both positions

Find the distance between them

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