AMC 10 · 2008 · #15

Grade 7 rate-ratio
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
ratesystems-of-equationslinear-equations-two-var convert-to-algebradimensional-analysis ↑ Prerequisites: ratesystems-of-equations
📏 Medium solution 💡 2 insights
Problem
Three people drove yesterday: Ian, Han, and Jan. Compared to Ian, Han drove 1 hour longer at a speed 5 mph faster, and Jan drove 2 hours longer at a speed 10 mph faster. Han's trip was 70 miles longer than Ian's. Find how many more miles Jan drove than Ian.

Pick an answer.

(A)
$\ 120$
(B)
$\ 130$
(C)
$\ 140$
(D)
$\ 150$
(E)
$\ 160$

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

How to solve
Strategy Introduce a Variable

Nothing gives Ian's actual speed or time, so name them with variables and write every distance in terms of them. When the two extra-distance expressions are expanded, the unknown product cancels and the same combination shows up in both. So the plan is to build the expressions, then shift focus to that shared combination instead of chasing the individual variables.

1STEP 1

Distance equals speed times time

Every driver's distance is speed times time — the one relationship that links all the numbers here.

distance = speed × time
2STEP 2

Name Ian's speed and time

Let s be Ian's speed and t his time. Then Ian drove s·t, Han (s+5)(t+1), and Jan (s+10)(t+2) miles.

Ian: s t, Han: (s+5)(t+1), Jan: (s+10)(t+2)
3STEP 3

Use Han's 70-mile clue

Expanding (s+5)(t+1) - s·t cancels s·t and leaves s + 5t + 5 = 70, so s + 5t = 65.

(s+5)(t+1) - s t = s + 5t + 5 = 70 → s + 5t = 65
4STEP 4

Write Jan's gap the same way

Likewise (s+10)(t+2) - s·t gives 2s + 10t + 20, which is exactly 2(s + 5t) + 20 — the same block again.

(s+10)(t+2) - s t = 2s + 10t + 20 = 2(s + 5t) + 20
5STEP 5

Substitute and finish

Putting s + 5t = 65 into Jan's expression gives 2(65) + 20 = 150 extra miles. Answer (D).

2(65) + 20 = 130 + 20 = 150
Answer
150
Jan drove more hours and more speed above Ian than Han did, so Jan's gap should exceed Han's 70 miles, and 150 does. A quick concrete check confirms it: if Ian went 45 mph for 4 hours, then s + 5t = 45 + 20 = 65 holds; Ian drove 180, Han (50)(5)=250 which is 70 more, and Jan (55)(6)=330 which is 150 more. Answer (D) fits.
💡Key takeaway

You don't always need each unknown by itself; sometimes finding one combination, like s + 5t, is enough to answer the question.

  • Distance equals speed times time
  • Name Ian's speed and time
  • Use Han's 70-mile clue
  • Write Jan's gap the same way
  • Substitute and finish

More like this

Same archetype — closest grade level first.