AMC 10 · 2007 · #13

Grade 7 rate-ratio
Archetype Meeting Time with Piecewise Rates └ Sub-archetype Piecewise-Leg Time Balance
rateratio-proportionlinear-equations-one-var convert-to-algebra ↑ Prerequisites: rate
📏 Medium solution 💡 2 insights
Problem
Yan stands somewhere on the straight road between his home and the stadium. He can either walk straight to the stadium, or walk back home first and then ride his bike to the stadium. His bike is 7 times as fast as his walking. The two routes take the same amount of time. Find the ratio of his distance from home to his distance from the stadium.

Pick an answer.

(A)
$\frac{2}{3}$
(B)
$\frac{3}{4}$
(C)
$\frac{4}{5}$
(D)
$\frac{5}{6}$
(E)
$\frac{7}{8}$

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The problem gives no actual numbers, only relationships, so Tool #4 (Introduce a Variable) is the way in: name Yan's distance from home h, his distance from the stadium s, and his walking speed w. Tool #1 (Draw a Diagram) fixes the order home-Yan-stadium on a line, which is what makes the home-to-stadium distance equal h+s. Tool #8 (Analyze the Units) supplies the one physics fact needed: time = distance ÷ speed, so each route's time is a length over a speed. Tool #13 (Convert to Algebra) then turns 'both routes take the same time' into a single equation, and because every term carries the same w, the unknown speed cancels and only the ratio survives.

1STEP 1

Draw the line and name the pieces

Line up home, Yan, stadium. Call his distance from home h, from the stadium s, so the whole road is h + s; walking speed w, biking 7w.

home h{40pt{0.4pt}} Yan s{40pt{0.4pt}} stadium, home to stadium = h + s
2STEP 2

Write each route's time

Time is distance over speed: route 1 takes sw\frac{s}{w}, while route 2 walks then bikes the full road, taking hw+h+s7w\frac{h}{w}+\frac{h+s}{7w}.

t₁ = s/w, t₂ = h/w + (h+s)/7w
3STEP 3

Set the two times equal

Both routes take the same time, so sw=hw+h+s7w\frac{s}{w}=\frac{h}{w}+\frac{h+s}{7w} — one equation tying h, s, and w together.

s/w = h/w + (h+s)/7w
4STEP 4

Clear the speed and simplify

Multiply every term by 7w; the walking speed cancels out entirely, leaving 7s = 7h + h + s, which tidies to 6s = 8h.

7s = 7h + (h+s) → 7s = 8h + s → 6s = 8h
5STEP 5

Read off the ratio

Divide 6s = 8h by 8s: the ratio of his home distance to his stadium distance is 6 to 8, that is 34\frac{3}{4} — choice (B).

6s = 8h → h/s = 6/8 = 3/4 → (B)
Answer
3/4
Test the ratio with concrete numbers. Take h = 3 and s = 4 (which give h/s = 3/4) and let walking speed w = 1. Route 1 takes s/w = 4. Route 2 walks home in h/w = 3, then bikes the full h+s = 7 at speed 7, taking 7/7 = 1, for a total of 3 + 1 = 4. Both routes take 4, exactly matching, so 3/4 is correct. It also makes sense that Yan is closer to home than to the stadium: the backtrack home only pays off because the long ride afterward is very fast.
💡Key takeaway

When a trip has no real numbers, name the distances and the speed with letters, write each time as distance over speed, and the unknown speed will cancel to leave just the ratio you want.

  • Draw the line and name the pieces
  • Write each route's time
  • Set the two times equal
  • Clear the speed and simplify
  • Read off the ratio

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