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2007 AMC 10A Problem 13

2007 AMC 10A Problem 13AMC 10 Step by Step

Name: Date: September 13, 2026

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

Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium?

A)
$\frac 23$
B)
$\frac 34$
C)
$\frac 45$
D)
$\frac 56$
E)
$\frac 78$

Answer

Key insight

Measure time in walking units: walking b directly equals walking a home plus biking a + b at one-seventh the cost, so 7b = 8a.

Solution

Let Yan be $a$ from home and $b$ from the stadium, and let his walking speed be $1$ (so biking speed is $7$ ).

Option 1, walk straight to the stadium: time $b$ .

Option 2, walk home then bike the whole way: walking $a$ takes time $a$ ; biking the full distance $a + b$ at speed $7$ takes $\frac{a+b}{7}$ . Total $a + \frac{a+b}{7}$ .

Set the times equal and clear the fraction:
$b = a + \frac{a+b}{7} \quad\Longrightarrow\quad 7b = 7a + a + b \quad\Longrightarrow\quad 6b = 8a \quad\Longrightarrow\quad \frac{a}{b} = \frac{3}{4}.$

The answer is $\boxed{\textbf{(B)}\ \frac 34}$ .

Why this works

Only ratios matter here, so the walking speed can be set to $1$ and the distances kept as variables. Equal times give a single linear equation, and the answer is the ratio $a/b$ rather than the individual distances. Choosing units that make one rate equal to $1$ is a standard simplification in speed problems.

The trap

Forgetting that the bike ride covers the full home-to-stadium distance a + b, not just the part Yan was already past.

Common mistakes

Forgetting that the bike ride covers the full home-to-stadium distance $a + b$ , not just the part Yan was already past.
Inverting the ratio and answering $\frac43$ (not offered) or misreading which distance is the numerator.

Techniques

Set up the equation/formula and compute; no special trick needed