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Question 1201529: Mr Cody bought some pizzas for a group of pupils. There was an equal number of girls and boys. The girls received 2 times as many pizzas as the boys. Each boy ate 1/6 of a pizza and the boys finished all the pizzas given to them. Each girl ate 1/18 of a pizza and the girls had 6 2/3 pizzas left. How many pizzas did Mr Cody buy?
Found 2 solutions by math_tutor2020, greenestamps:
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Answer: 12 pizzas
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Work Shown:
p = number of pizzas the boys got
2p = number of pizzas the girls got
p+2p = 3p = total number of pizzas purchased
The goal is to find p, so we can calculate 3p.
x = number of boys = number of girls
x number of boys
1 boy eats 1/6 of a pizza
x boys eat x/6 of a pizza
Example: x = 36 boys eat x/6 = 36/6 = 6 pizzas
The boys got p number of pizzas
The boys ate x/6 number of pizzas.
They ate all of those pizzas, so p and x/6 must be the same.
p = x/6
6p = x
x = 6p
The number of boys is equal to 6 times the number of pizzas.
The same can be said about the number of girls.
x number of girls
1 girl eats 1/18 of a pizza
x girls eat x/18 = 6p/18 = p/3 of a pizza
2p = number of pizzas the girls got
p/3 = number of pizzas the girls ate
2p - p/3 = 6p/3 - p/3 = 5p/3 = number of pizzas left for the girls
Set that equal to the mixed number 6 & 2/3, aka the improper fraction 20/3, and solve for p.
5p/3 = 6 & 2/3
5p/3 = 20/3
5p = 20
p = 20/5
p = 4
The boys got 4 pizzas and the girls got 4*2 = 8 pizzas.
Total = 4+8 = 12 pizzas
(or you could say 3p = 3*4 = 12)
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Check:
p = 4 leads to x = 6*p = 6*4 = 24 boys and 24 girls (48 pupils total)
1 boy eats 1/6 of a pizza, so 24 boys eat 24/6 = 4 pizzas. This matches with p = 4, meaning the boys ate all of their allotted pizza.
1 girl eats 1/18 of a pizza, so 24 girls eat 24/18 = 4/3 = 1 & 1/3 pizza (i.e. 1 whole pizza plus another 1/3 of a pizza).
The girls got 2p = 2*4 = 8 pizzas.
Subtract off the amount they ate
8 - 4/3 = 8(3/3) - 4/3
8 - 4/3 = 24/3 - 4/3
8 - 4/3 = (24 - 4)/3
8 - 4/3 = 20/3
8 - 4/3 = (18+2)/3
8 - 4/3 = (18/3)+(2/3)
8 - 4/3 = 6+(2/3)
8 - 4/3 = 6 & 2/3
There are 6 whole pizzas, plus another 2/3 of a pizza, remaining for the girls.
The answer has been confirmed.
There is probably a more efficient route, so feel free to explore alternatives.
Answer by greenestamps(13209)  (Show Source):
You can put this solution on YOUR website!
Let x be the number of boys and the number of girls.
The boys each ate 1/6 of a pizza; the number of pizzas the boys ate was x(1/6) = x/6. They ate all the pizzas they got, so the number of pizzas they got was x/6.
The girls each ate 1/18 of a pizza; the number of pizzas they ate was x(1/18) = x/18. They finished with 6 2/3 pizzas left over. The number of pizzas they got was x/18 + 6 2/3 = x/18 + 20/3.
The number of pizzas the girls got was twice the number the boys got:
x/18 + 20/3 = 2(x/6)
Multiply by 18 to clear fractions....
x+120 = 6x
5x = 120
x = 24
There were 24 boys and 24 girls in the group.
Each of the 24 boys ate 1/6 of a pizza; that makes 4 pizzas.
The girls got twice as many pizzas as the boys, so the girls got 8 pizzas.
So the total number of pizzas Mr. Cody bought was 4+8 = 12.
ANSWER: 12
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