1Cademy - Finding Walking and Biking Speeds Using Equal Distances

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Finding Walking and Biking Speeds Using Equal Distances

Apply the distance, rate, and time problem-solving strategy to find two unknown speeds when both trips cover the same distance but use different modes of travel, requiring a unit conversion from minutes to hours and clearing fractions to solve.

Problem: When Naoko walks to school, it takes her $30$ minutes. If she rides her bike, it takes her $15$ minutes. Her biking speed is $3$ miles per hour faster than her walking speed. What are her walking speed and her biking speed?

Read and draw: Sketch the route from home to school. The walking path ( $30$ minutes) and the biking path ( $15$ minutes, $3$ mph faster) cover the same distance. Create a rate–time–distance table.
Identify: The walking speed and the biking speed.
Name: Let $r$ = walking speed in mph. Then the biking speed is $r + 3$ . Because the rate is in miles per hour, convert the times into hours: $30$ minutes is $\frac{30}{60}$ or $\frac{1}{2}$ hour, and $15$ minutes is $\frac{15}{60}$ or $\frac{1}{4}$ hour. Multiply rate by time to fill in the distance column:

	Rate (mph)	Time (hrs)	Distance (miles)
Walk	$r$	$\frac{1}{2}$	$\frac{1}{2}r$
Bike	$r + 3$	$\frac{1}{4}$	$\frac{1}{4}(r + 3)$

Translate: The distance from home to school is the same whether Naoko walks or bikes:

$\frac{1}{2}r = \frac{1}{4}(r + 3)$

Solve: Clear the fractions by multiplying both sides by the LCD of $2$ and $4$ , which is $4$ :

$4 \cdot \frac{1}{2}r = 4 \cdot \frac{1}{4}(r + 3)$

$2r = r + 3$

Subtract $r$ from both sides: $r = 3$ . The walking speed is $3$ mph. Her biking speed is $3 + 3 = 6$ mph.

Check: Walk: $3 \cdot 0.5 = 1.5$ miles. Bike: $6 \cdot 0.25 = 1.5$ miles. Both distances are $1.5$ miles. $\checkmark$
Answer: Naoko's walking speed is $3$ mph and her biking speed is $6$ mph.

This example extends the equal-distance uniform motion scenario by requiring a time conversion before setup and solving an algebraic equation with fractional coefficients.

Updated 2026-04-22

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