Question

samantha drove 168 miles in $3\frac{1}{2}$ hours. she then drove the next $2\frac{1}{4}$ hours at a rate of 5 miles an hour faster than the first rate. how many miles did samantha drive during the $5\frac{3}{4}$ hours?

Explanation:

Step1: Find first driving rate

First, convert $3\frac{1}{2}$ to an improper fraction: $3\frac{1}{2}=\frac{7}{2}$ hours.
Rate is distance divided by time, so:
$$\text{Rate}_1 = \frac{168}{\frac{7}{2}} = 168 \times \frac{2}{7} = 48 \text{ mph}$$

Step2: Calculate second driving rate

The second rate is 5 mph faster:
$$\text{Rate}_2 = 48 + 5 = 53 \text{ mph}$$

Step3: Find distance of second trip

Convert $2\frac{1}{4}$ to an improper fraction: $2\frac{1}{4}=\frac{9}{4}$ hours.
Distance is rate multiplied by time:
$$\text{Distance}_2 = 53 \times \frac{9}{4} = \frac{477}{4} = 119.25 \text{ miles}$$

Step4: Total distance driven

Add the two distances together:
$$\text{Total Distance} = 168 + 119.25 = 287.25$$
Or as a fraction: $168 + \frac{477}{4} = \frac{672}{4} + \frac{477}{4} = \frac{1149}{4} = 287\frac{1}{4}$

Answer:

$287\frac{1}{4}$ miles (or 287.25 miles)