Q 114 - Intermediate Algebra

Question

At the end of spring break, Lucy left the beach and drove back towards home, driving at a rate of $40$ mph. Lucy’s friend left the beach for home $30$ minutes (half an hour) later, and drove $50$ mph. How long did it take Lucy’s friend to catch up to Lucy?

Step-by-Step Solution

Answer

$2$ hours take Lucy’s friend
to catch up to Lucy.

Step 1 Given information are,

Lucy towards home, driving at a rate of $40$ mph.

Lucy's friend drove towards the home after $30$ minutes at the speed of $50$ mph.

Step 2 to find:

The time taken for Lucy’s friend
to catch up to Lucy.

Let $x$ represents the length of the time of Lucy.

$y$ represents the length of the time of Lucy's friend.

Step 3 From the given information,

Let us draw a table,

Since Lucy and her friend travels in the same direction, $40x=50y$.

Step 4 It is given that,

Lucy's friend left half an hour less than lucy.

So, $y=x-\frac{1}{2}$.

Now substitute $y=x-\frac{1}{2}$ in the equation $40x=50y$.

$$\begin{gathered} 40x=50\left(x-\frac{1}{2}\right) \\ 40x=50x-25 \\ 50x-40x=25 \\ 10x=25 \\ x=\frac{25}{10} \\ x=2.5 \end{gathered}$$

Step 5 Find the value of y .

Substitute the value of $x$ in the equation $y=x-\frac{1}{2}$
$$\begin{gathered} y=2.5-\frac{1}{2} \\ y=\frac{5-1}{2} \\ y=\frac{4}{2} \\ y=2 \end{gathered}$$