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Question
The average score of girls in class X examination in school is 67 and that of boys is 63. The average score for the whole class is 64.5. Find the percentage of girls and boys in the class.
Solution
Let the number of girls and boys be n1 and n2 respectively.
We have
`bar"X"_1` = Average score of girls = 67
`bar"X"_2` = Average score of boys = 63
`bar"X"` = Average score of the whole class = 64·5
∴ `bar"X" = ("n"_1bar"X"_1 + "n"_2bar"X"_2)/("n"_1 + "n"_2)`
⇒ 64·5 = `(67"n"_1 + 63"n"_2)/("n"_1 + "n"_2)`
⇒ 64·5n1 + 64·5n2 = 67n1 + 63n2
⇒ 2·5n1 = 1·5n2
⇒ 25n1 = 15n2
⇒ 5n1 = 3n2
Total number of students in the class = n1 + n2.
∴ Percentage of girls = `("n"_1)/("n"_1 + "n"_2) xx 100`
= `("n"_1)/("n"_1 + (5"n"_1)/(3)) xx 100`, ...[∵ 5n1 = 3n2]
= `(3"n"_1)/(3"n"_1 + 5"n"_1) xx 100`
= `(3)/(8) xx 100`
= 37·5%
and
Percentage of boys = `("n"_2)/("n"_1 + "n"_2) xx 100`
= `("n"_2)/((3"n"_2)/(5) + "n"_2) xx 100`
= `(5"n"_2)/(3"n"_2 + 5"n"_2) xx 100`
= 62·5%
Hence, there are 37·5% girls and 62·5% boys in the class.
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