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$72
Given:
Abbie spent ⁵/₈ of her money and saved the rest.
She spent $45.
Question:
How much money did she have at first?
The Process:
The meaning of "Abbie spent ⁵/₈ of her money" is that she has spent 5 of the 8 parts of the money she had at first.
Let us make a pattern of 8 units for the amount of money that Abbie had at first.
[tex]\boxed{\cdot}\boxed{\cdot}\boxed{\cdot}\boxed{\cdot}\boxed{\cdot}\boxed{\cdot}\boxed{\cdot}\boxed{\cdot}[/tex]
Since Abbie spent ⁵/₈ of her money, then we create a pattern of 5 units containing $9 each. Recall that [tex]\boxed{\$ 45 \div 5 = \$ 9}[/tex]
[tex]\boxed{\$ 9}\boxed{\$ 9}\boxed{\$ 9}\boxed{\$ 9}\boxed{\$ 9} = \$ 45[/tex]
From the pattern above, we can see that [tex]\boxed{ \ 1 \ unit = \$ 9 \ }.[/tex]
Let us calculate how much money Abbie had at first.
[tex]\boxed{ \ 1 \ unit = \$ 9 \ }[/tex]
[tex]\boxed{ \ 8 \ unit = \ ? \ }[/tex]
[tex]\boxed{ \ 8 \times \$ 9 = \$ 72 \ }[/tex]
Thus, the money Abbie had at first is $72.
The complete pattern of Abbie's initial money is as follows:
[tex]\boxed{\$ 9}\boxed{\$ 9}\boxed{\$ 9}\boxed{\$ 9}\boxed{\$ 9}\boxed{\$ 9}\boxed{\$ 9}\boxed{\$ 9} = \$ 72[/tex]
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Quick Steps:
Let M as Abbie's initial money.
[tex]\boxed{ \ \frac{5}{8} \times M = \$ 45 \ }[/tex]
Both sides are multiplied by ⁸/₅ so that M is the subject alone on the left side.
[tex]\boxed{ \ M = \frac{8}{5} \times \$ 45 \ }[/tex]
We crossed out 5 and 45.
[tex]\boxed{ \ M = 8 \times \$ 9 \ }[/tex]
[tex]\boxed{ \ M = \$ 72 \ }[/tex]
Thus, the money Abbie had at first is $72.
Keywords: Abbie, spent 5/8 of her money, saved the rest, $45, $72, how much money, she have at first, the pattern