Respuesta :
288 pennies
Further explanation
Given:
- Together, Katya and Mimi have 480 pennies in their piggy banks.
- After Katya lost ½ of her pennies and Mimi lost ²/₃ of her pennies, they both had an equal number of pennies left.
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.
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Keywords: together, Katya and Mimi, 480 pennies, piggy banks, lost, an equal number, left, lose
- Katya and Mimi lost 288 pennies altogether
To start with, we are going to form equations and then solve them simultaneously.
From the question, we are told that both Katya and Mimi have 480 pennies. Let a represent Katya and b represent Mimi. This means that
- a + b = 480
In the second part, we are told that after they both lose certain portions of their pennies, they are left with the same amount of pennies.
- Katya loses half of her pennies, a - [tex]\frac{1}{2}[/tex] = [tex]\frac{1}{2}a[/tex]
- Mimi loses 2/3 of her pennies, b - [tex]\frac{2}{3}[/tex] = [tex]\frac{1}{3}b[/tex]
Again we are told that after both of them loses the said portion they both have equal number of pennies. This mean that we equate both of them to each other.
[tex]\frac{1}{2}a[/tex] = [tex]\frac{1}{3}b[/tex] cross multiplying, we have
- 3a = 2b and then,
- a = [tex]\frac{2}{3}b[/tex]
substituting this in the first equation, we have
[tex]\frac{2}{3}b[/tex] + b = 480
[tex]\frac{5}{3}b[/tex] = 480
5b = 1440
b = [tex]\frac{1440}{5}[/tex]
b = 288
if b = 288, then a = 480 - 288 = 192
This means that from the beginning, Katya had 192 pennies and Mimi had 288 pennies.
how many pennies did they lose
- Katya lost half. 192 × [tex]\frac{1}{2}[/tex] = 96
- Mimi lost two-thirds. 288 × [tex]\frac{2}{3}[/tex] = 192
they therefore lost 96 + 192 pennies altogether = 288 pennies.
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