2480.31 – Dividing A Tenner

A woman divides $10 among 12 children. Boys get 70 cents; girls get 90 cents. How many girls are there?

Solution

If you want to avoid algebra you might think along these lines. Counting by 90's, 10 girls makes 900 cents = $9. Add in two boys gets you $10.40. That's the right number of children, but too much money. Replacing one girl with a boy keeps the number of children the same and reduces the money by 20 cents. Do this twice, and you get 8 girls and 4 boys and just the right amount of money! So there are 8 girls.

Or . . . use algebra! Let gg the numbers of girls so that bb, the number of girls, is b=12−gb = 12 - g. Then,

.9g+.7(12−g)=10⇝.9g+.84−.7g=10⇝.2g=.16⇝g=8.\begin{aligned}.9 g + .7 (12- g) = 10 &\leadsto .9 g + .84 - .7 g = 10 \\&\leadsto .2 g = .16 \\&\leadsto g = 8.\end{aligned}