Question

# Jamal hiked on two trails. The first trail was 3 1/3 miles long, and the second trail was 1 3/5 times as long as the first trail. How many miles did Jamal hiked ? Answer as a mixed number in simplest form.

Fractions
Jamal hiked on two trails. The first trail was $$\displaystyle{3}\frac{{1}}{{3}}$$ miles long, and the second trail was 1 $$\displaystyle\frac{{3}}{{5}}$$ times as long as the first trail. How many miles did Jamal hiked ? Answer as a mixed number in simplest form.

2021-02-14
We know how long the first trail was so we need to find how long the second trail is before we can find the total number of miles that Jamal hiked.
It is given that the first trail is $$\displaystyle{3}{\left(\frac{{1}}{{3}}\right)}$$ miles and that the second trail is $$\displaystyle{1}{\left(\frac{{3}}{{5}}\right)}$$ times as long as the first trail. The second trail is then $$\displaystyle{3}{\left(\frac{{1}}{{3}}\right)}×{1}{\left(\frac{{3}}{{5}}\right)}.$$
To multiply two mixed numbers, we must first rewrite them as improper fractions. This gives $$\displaystyle{3}{\left(\frac{{1}}{{3}}\right)}×{1}{\left(\frac{{3}}{{5}}\right)}=\frac{{10}}{{3}}×\frac{{8}}{{5}}.$$
Now that the mixed numbers are written as improper fractions, we can multiply by multiplying the numerators and denominators. Therefore $$\displaystyle\frac{{10}}{{3}}×\frac{{8}}{{5}}={10}×\frac{{8}}{{3}}×{5}={8015}.$$
Since 5 is a factor of 80 and 15, we can reduce the fraction by dividing the numerator and denominator by 5. This gives $$\displaystyle\frac{{80}}{{15}}={80}÷\frac{{5}}{{15}}÷{5}=\frac{{16}}{{3}}.$$
Converting $$\displaystyle\frac{{16}}{{3}}$$ to a mixed number gives $$\displaystyle\frac{{16}}{{3}}={5}{\left(\frac{{1}}{{3}}\right)}$$ so the second trail is $$\displaystyle{5}{\left(\frac{{1}}{{3}}\right)}$$ miles.
The total number of miles that Jamal hiked is the length of the first trail plus the length of the second trail. The total number of miles he hiked is then $$\displaystyle{3}{\left(\frac{{1}}{{3}}\right)}+{5}{\left(\frac{{1}}{{3}}\right)}$$ miles.
To add mixed numbers with a common denominator, we just need to add the whole numbers and add the numerators. Therefore $$\displaystyle{3}{\left(\frac{{1}}{{3}}\right)}+{5}{\left(\frac{{1}}{{3}}\right)}={\left({3}+{5}\right)}+\frac{{{1}+{1}}}{{3}}={8}{\left(\frac{{2}}{{3}}\right)}.$$
The total distance that Jamal hiked as a mixed number is then $$\displaystyle{8}{\left(\frac{{2}}{{3}}\right)}$$ miles.