Question

Last week, Shelly rode her bike a total of 30 miles over a three-day period. On the second day, she rode LaTeX: \frac{4}{5}45 the distance she rode on the first day. On the third day, she rode LaTeX: \frac{3}{2}32 the distance she rode on the second day

Answer 1

We make expressions for each afirmation

Where X is the first day, Y second day and Z the third

1. the sum of the 3 days gives us 30

`X+Y+Z=30`

2. Second day is 4/5 of the first day

`Y=(4)/(5)X`

3.Third day is 3/2 of the second day

`Z=(3)/(2)Y`

Whit the expressions I try to represent everything as a function of X

I must represent Z in function of X, for this I can replace Y of the second expression in the third expression

`\begin{gathered} Z=(3)/(2)((4)/(5)X) \\ Z=(12)/(10)X \\ Z=(6)/(5)X \end{gathered}`

So I have:

`\begin{gathered} Y=(4)/(5)X \\ Z=(6)/(5)X \\ \end{gathered}`

And I can replace on the first expression

`\begin{gathered} X+Y+Z=30 \\ X+((4)/(5)X)+((6)/(5)X)=30 \end{gathered}`

I must find X

`\begin{gathered} (1+(4)/(5)+(6)/(5))X=30 \\ 3X=30 \\ X=(30)/(3) \\ X=10 \end{gathered}`

So, if I have X I can replace on this expressions to find de value:

`\begin{gathered} Y=(4)/(5)X \\ Z=(6)/(5)X \end{gathered}`

Where X is 10

`\begin{gathered} Y=(4)/(5)*10 \\ Y=(40)/(5)=8 \\ \\ Z=(6)/(5)*10 \\ Z=(60)/(5)=12 \end{gathered}`

To check:

`\begin{gathered} X+Y+Z=30 \\ (10)+(8)+(12)=30 \\ 30=30 \\ \end{gathered}`

The result is correct, therefore:

`\begin{gathered} X=10 \\ Y=8 \\ Z=12 \end{gathered}`