SOLUTION: There are some girls and boys in a group. The average height of the girls is 135 cm and the average height of the boys is 150 cm. given that the average height of all the children

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Question 888680: There are some girls and boys in a group. The average height of the girls is 135 cm and the average height of the boys is 150 cm. given that the average height of all the children in the group is 138 cm, what fraction of the children are girls?
Answer by Theo(13342) About Me  (Show Source):
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the average height of the girls is 135.
the average height of the boys is 150.
the average height of the boys and the girls together is 138.

let x = number of girls.
let y = number of boys.
then x+y = number of girls and boys together.

the total number of girls is equal to 135 * x
the total number of boys is equal to 150 * y
the total number of boys and girls together is equal to 138 * (x + y)

you get an equation that says:

135 * x + 150 * y = 138 * (x + y)

simplify to get:

135 * x + 150 * y = 138 * x + 138 * y

subtract 135 * x from both sides of the equation and subtract 138 * y from both sides of the equation to get:

150 * y - 138 * y = 138 * x - 135 * x

combine like terms to get:

12 * y = 3 * x

divide both sides of the equation by x and divide both sides of the equation by 12 and you get:

y/x = 3/12

this equation says that the ratio of boys to girls is equal to 3/12.

this equation also says:

the ratio of boys to total = 3 / (12 + 3) = 3/15.

the ratio of girls to boys is equal to 12/3.

the ratio of girls to total is 12/ (3 + 12) = 12/15.

the fraction of girls to total number of children together is the same as the ratio of girls to total which is equal to 12/15.

is this accurate?

let's see.

assume the number of children is equal to 15.

12 are girls.
3 are boys.

total score for girls is 135 * 12
total score for boys is 150 * 3.
total score overall is 138 * 15.

if 135 * 12 + 150 * 3 = 138 * 15, then the solution is good.

we get:

1620 + 450 = 2070

simplify this to get 2070 = 2070

this confirms that the ratio of girls to total and the ratio of boys to total is good which confirms that the fraction of the children that are girls is also good and is 12/15 which can be simplified to 4/5.