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Twenty Coins
Thanks
Jean B
Twenty
people, (men, women, and children) receive a total of twenty coins.
Each man receives the value of three coins.
Each woman receives one and a half coins.
Each child receives half a coin.
Question:
How many men, women, and children are there?
Answer:
2
men, 5 women, 13 children
🧠
Detailed Explanation:
Let:
𝑥 =
number of men
𝑦
= number of women
𝑧 =
number of children
We are
given two conditions:
1.
Total number of people: 𝑥+𝑦+𝑧=20
2.
Total value of coins distributed:
Each
man gets 3 coins → total from men: 3𝑥
Each
woman gets 1.5 coins → total from women: 1.5𝑦
Each
child gets 0.5 coins → total from children: 0.5𝑧
So:
3𝑥+1.5𝑦+0.5𝑧=20
✏️
Solving the system:
We
simplify the second equation by multiplying everything by 2 to
eliminate decimals:
6𝑥+3𝑦+𝑧=40
Now we
solve the system:
Equation
1:
𝑥+𝑦+𝑧=20
Equation
2:
6𝑥+3𝑦+𝑧=40
Subtract
Equation 1 from Equation 2:
(6𝑥+3𝑦+𝑧)
−(𝑥+𝑦+𝑧) = 40−20 ⇒5𝑥+2𝑦=20
Now we
solve:
5𝑥+2𝑦=20
Try
integer values that satisfy this equation. Let's test
𝑥=2:
5(2)+2𝑦=20
⇒10+2𝑦=20
⇒𝑦=5
Now
plug
𝑥=2,
𝑦=5 into the first equation:
𝑥+𝑦+𝑧=20
⇒2+5+𝑧=20
⇒𝑧=13
✅ So
the solution is:
2 men
5 women
13
children
🔍
Final Check:
Total
people:
2+5+13=20
✅
Total
coins:
Men:
2x3=6
Women:
5x1.5=7.5
Children:
13x0.5=6.5
Total:
6+7.5+6.5=20
✅
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