word problem

[girlpower]

A box in a college bookstore contains books, and each book in the box is a history book, an English book or science book. If one third of these books are history books and one -sixth are English books, what fraction of the books are science books?

 

[MarkFL]

We should observe that the sum of the fractions of each type of book makes up the entire box, or 1. Let x be the fraction of science books, and we may state:

$\displaystyle \displaystyle \frac{1}{3}+\frac{1}{6}+x=1$

Let's combine the two fractions on the left, by getting the lowest common denominator of 6:

$\displaystyle \displaystyle \frac{1}{3}\cdot\frac{2}{2}+\frac{1}{6}+x=1$

$\displaystyle \displaystyle \frac{2}{6}+\frac{1}{6}+x=1$

Now we may combine the two terms over the common denominator, where the new numerator is the sum of the two previous numerators.

$\displaystyle \displaystyle \frac{2+1}{6}+x=1$

$\displaystyle \displaystyle \frac{3}{6}+x=1$

Reduce the fraction.

$\displaystyle \displaystyle \frac{1}{2}+x=1$

Now, we may subtract 1/2 from both sides to solve for x. What do you find?