288 pennies
Given:
Question:
How many pennies did they lose altogether?
The Process:
Let k = Katya's pennies and m = Mimi's pennies.
Step-1: Finding out the amount of money in the beginning
At first, Katya and Mimi have 480 pennies in their piggy banks.
Equation-1: [tex]\boxed{ \ k + m = 480 \ }[/tex]
After Katya lost ½ of her pennies and Mimi lost ²/₃ of her pennies, they both had an equal number of pennies left. That is, the number of pennies remaining from Katya is [tex]\boxed{k - \frac{1}{2}k = \frac{1}{2}k}[/tex], while Mimi is [tex]\boxed{m - \frac{2}{3}m = \frac{1}{3}m}[/tex].
Equation-2: [tex]\boxed{ \ \frac{1}{2}k = \frac{1}{3}m \ }[/tex]
From Equation-2, we get [tex]\boxed{ \ k = \frac{2}{3}m \ }[/tex] and substitute into Equation-1.
[tex]\boxed{ \ \frac{2}{3}m + m = 480 \ }[/tex]
[tex]\boxed{ \ \frac{2}{3}m + \frac{3}{3}m = 480 \ }[/tex]
[tex]\boxed{ \ \frac{5}{3}m = 480 \ }[/tex]
[tex]\boxed{ \ m = 480 \times \frac{3}{5} \ } \rightarrow \boxed{\boxed{ \ m = 288 \ }}[/tex]
Substitute the value of m into [tex]\boxed{ \ k = \frac{2}{3}m \ }.[/tex]
[tex]\boxed{ \ k = \frac{2}{3} \times 288 \ } \rightarrow \boxed{\boxed{ \ k = 192 \ }}[/tex]
Therefore, we got Katya's money of 192 pennies and Mimi's money of 288 pennies at first.
Step-2: Finding out how many pennies did they lose altogether
Katya lost ½ of her pennies and Mimi lost ²/₃ of her pennies.
[tex]\boxed{ \ = \Big(\frac{1}{2} \times 192 \Big) + \Big(\frac{2}{3} \times 288 \Big) \ }[/tex]
[tex]\boxed{ \ = 96 + 192 \ }[/tex]
[tex]\boxed{ \ = 288 \ }[/tex]
Thus, they did lose 288 pennies altogether.
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Notes
From [tex]\boxed{ \ k = \frac{2}{3}m \ }[/tex], we can find out their initial money ratio is Katya's: Mimi's = 2: 3.
Keywords: together, Katya and Mimi, 480 pennies, piggy banks, lost, an equal number, left, lose
