Question

Emma rode 8 times as fast as Emily . In fact , she rode 48 kilometers in 4 hours less than it took Emily to ride 36 miles . How fast did each of them ride ? How long did they ride ?

Answer 1

Let M be Emma's speed and E be Emily's speed. Since Emma rode 8 times as fast as Emily, we can write it as

$M=8\times E$

Now, we know that she rode 48 km in 4 hours less than it took Emily to ride 36 miles. This means that

$\begin{gathered} \\ \\ 48km=M\times(t-4hours) \end{gathered}$

and

$36\text{miles}=E\times(t)$

where t denotes the time in which they rode the same distance.

From the second equation,we have

$\begin{gathered} \\ (t-4)=\frac{48}{M}\ldots(A) \end{gathered}$

and for the third equation, we have

$\begin{gathered} \\ t=\frac{36}{E}= \end{gathered}$

Now, we need to convert units, from miles to kilometers. We know that 1miles is equal to 1.6km, then we get

$\begin{gathered} \\ 36\text{miles}=36\text{miles(}\frac{1.6\operatorname{km}}{1mile})=57.6\text{ km} \end{gathered}$

then, our last equation is equivalent to

$t=\frac{57.6}{E}\ldots(B)$

By substituting equation B into equation A, we have

$\frac{57.6}{E}-4=\frac{48}{M}\ldots(C)$

Then, we have 2 equations in 2 unknows, that is, our first equation and equation C. Then, By substituting our first equation into equation C, we have

$\begin{gathered} \frac{57.6}{E}-4=\frac{48}{8\cdot E} \\ or \\ \frac{57.6}{E}-4=\frac{6}{E} \end{gathered}$

By moving the right hand side to the left hand side and -4 to the right hand side, we have

$\frac{1}{E}(57.6-6)=4$

which gives

$\begin{gathered} \\ \frac{51.6}{E}=4 \\ E=\frac{51.6}{4} \\ E=12.9(\frac{\operatorname{km}}{hour}) \end{gathered}$

with this result, we can find M by substituting this values into our first equation, that is,

$\begin{gathered} M=8\times E \\ M=8\times12.9 \\ M=103.2(\frac{\operatorname{km}}{hour}) \end{gathered}$

Then, for the first question How fast did each of them ride ? The answer is

$\begin{gathered} \text{Emma's sp}eed=103.2\text{ }\frac{\operatorname{km}}{hour} \\ \text{Emily's sp}eed=12.9\text{ }\frac{\operatorname{km}}{hour} \\ \end{gathered}$

Now, in order to find how long they ride, we need to find the time t. We can find it by substituting E into equation B, that is

$\begin{gathered} t=\frac{57.6}{E} \\ t=\frac{57.6}{12.9} \\ t=4.46\text{ hour} \end{gathered}$

Then, the distance is

$103.2(\frac{km}{\text{hour}})(4.46\text{hour)}=460.8\text{ km}$

then, How long did they ride ? 460.8 kilometers