# Rate-Time-Distance Problems

When you have a rate of motion, such as $x$ feet per $y$ seconds, or $x$ miles per $y$ hours, you can use the important relationship

rate × time = distance

to solve many problems.

Example 1:

If you ride your cycle at a speed of $15$ miles an hour in a straight line, how far will you be from your starting place after $3$ hours?

The rate is $\frac{15\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{miles}}{1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{hour}}$ , and the time is $3$ hours.

Multiply:

$\frac{15\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{miles}}{1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\overline{)\text{hour}}}\cdot 3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\overline{)\text{hours}}=45\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{miles}$

So, you will be $45$ miles from your starting place.

Example 2:

Light leaving the Sun takes about $8$ minutes to reach the Earth, traveling a distance of approximately $93,000,000$ miles. Find the speed of light in miles per second.

The rate is unknown. The time is $8$ minutes and the distance is $93,000,000$ miles.

Set up an equation.

$r\cdot 8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{minutes}=93,000,000\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{miles}$

Divide both sides by $8$ minutes.

$r=\frac{93000000\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{miles}}{8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{minutes}}$

Divide $93,000,000$ by $8$ to reduce the fraction.

$r=\frac{11625000\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{miles}}{\text{minute}}$

Divide by $60$ to get miles per second.

$r\approx \frac{193,750\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{miles}}{\text{second}}$

So the speed of light is about $190,000$ miles per second.