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2008 AMC 10A Problem 15

2008 AMC 10A Problem 15AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 3 of 5 · about 4 min · position P11-15

Yesterday Han drove 1 hour longer than Ian at an average speed 5 miles per hour faster than Ian. Jan drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Han drove 70 miles more than Ian. How many more miles did Jan drive than Ian?

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

Answer

Key insight

Expanding Han's distance gives v + 5t = 65 for Ian's speed v and time t; Jan's excess 2v + 10t + 20 is exactly twice that plus 20.

Solution

Let Ian drive for $t$ hours at $v$ miles per hour, covering $vt$ miles.

Han drove $(v + 5)(t + 1) = vt + v + 5t + 5$ miles, which is $70$ more than Ian's $vt$ :
$v + 5t + 5 = 70 \quad\Longrightarrow\quad v + 5t = 65.$

Jan drove $(v + 10)(t + 2) = vt + 2v + 10t + 20$ miles, so Jan's excess over Ian is
$2v + 10t + 20 = 2(v + 5t) + 20 = 2 \cdot 65 + 20 = 150.$

The answer is $\boxed{\textbf{(D)}\ 150}$ .

Why this works

The problem looks under-determined (two unknowns, one equation), but the quantity asked for is a linear combination of the same expression the given data determines. Expand both products, cancel the common $vt$ , and compare the leftover linear parts. This "you do not need the individual values" pattern is common on the AMC.

Alternative approach

Any pair with $v + 5t = 65$ works, so take $t = 1$ , $v = 60$ : Ian $60$ miles, Han $65 \cdot 2 = 130$ (check: $70$ more), Jan $70 \cdot 3 = 210$ , which is $150$ more than Ian.

The trap

Trying to solve for v and t individually; there is only one equation, but the target expression depends only on the combination v + 5t.

Common mistakes

Trying to solve for v and t individually; there is only one equation, but the target expression depends only on the combination v + 5t.
Guessing "twice 70" $= 140$ because Jan's increases are double Han's, forgetting the extra cross term $20$ .

Techniques

Set up the equation/formula and compute; no special trick needed · Test small/specific values or special cases to find or verify the answer