Fraction Word Problems: Estimate, Solve, and Check

Start with what a fraction means. A fraction tells you an amount only when you know its whole, the one full thing it is a part of. Half of a small jug of water and half of a big jug of water are both written $\frac{1}{2}$, but they are different amounts.

So here is the rule for every problem in this lesson: compare, add or subtract fractions as amounts only when they refer to the same whole, or to wholes of equal size.

In a word problem, look at the unit, such as liter, hour or yard. In Nia's problem, both amounts are parts of 1 liter. They share a whole, so you can add them.

Benchmark fractions are easy amounts that you compare other fractions with. Your benchmarks in this lesson are $0$, $\frac{1}{2}$ and $1$ of the same whole. Every fraction you estimate in this lesson is from $0$ to $1$.

To find the nearest benchmark for $\frac{5}{8}$, think of a whole cut into 8 equal pieces. The denominator (bottom number) is 8, the number of pieces in the whole. The numerator (top number) is 5, the number of pieces you have.

• No pieces is $0$, which is $\frac{0}{8}$.
• Half of the pieces is $\frac{1}{2}$, which is $\frac{4}{8}$.
• All of the pieces is $1$, which is $\frac{8}{8}$.

So the benchmark numerators are 0, half of the denominator, and the denominator itself. Here they are 0, 4 and 8. Which one is closest to your numerator, 5? It is 1 away from 4, 3 away from 8 and 5 away from 0. So $\frac{5}{8}$ is nearest $\frac{1}{2}$.

You can write $\frac{5}{8} \approx \frac{1}{2}$. The sign $\approx$ means is about. The table shows the same steps for more fractions.

An estimate can be $0$. Suppose one piece of string is $\frac{5}{8}$ of a foot long and another is $\frac{5}{12}$ of a foot long. Both lengths are parts of 1 foot, the same whole, so you can compare them. About how much longer is the first piece?

You already found that $\frac{5}{8}$ and $\frac{5}{12}$ are each nearest $\frac{1}{2}$. So the estimate is $\frac{1}{2} - \frac{1}{2} = 0$. This does not say the pieces are the same length. An estimate is a rough guide, and this one tells you the exact difference is not far from $0$. Check it. Each eighth is 3 twenty-fourths, so $\frac{5}{8} = \frac{15}{24}$. Each twelfth is 2 twenty-fourths, so $\frac{5}{12} = \frac{10}{24}$. The exact difference is $\frac{15}{24} - \frac{10}{24} = \frac{5}{24}$ of a foot.

An exact answer can be $0$ as well. For example, $\frac{3}{6} - \frac{1}{2} = 0$, because $\frac{3}{6}$ and $\frac{1}{2}$ of the same whole are the same amount.

How close should your answer be to your estimate?

Neighboring benchmarks are $\frac{1}{2}$ apart. A fraction exactly in the middle of two neighbors is $\frac{1}{4}$ from each, and no fraction from $0$ to $1$ is farther than that from its nearest benchmark. So swapping a fraction for its nearest benchmark changes it by $\frac{1}{4}$ at most. With two fractions, the two changes add up to $\frac{1}{2}$ at most.

This gives you a check. Suppose you estimate with two fractions of the same whole, each from $0$ to $1$, and you use the nearest benchmark for each. Then a correct answer is never more than $\frac{1}{2}$ away from your estimate. If your answer is farther away, look for a mistake.

Try it on Omar's total time. Each sixth is 2 twelfths, so $\frac{5}{6} = \frac{10}{12}$. The exact sum is $\frac{10}{12} + \frac{5}{12} = \frac{15}{12} = 1\frac{3}{12} = 1\frac{1}{4}$ hours. His estimate was $1\frac{1}{2}$ hours. The two are $\frac{1}{4}$ hour apart, so the answer passes.

Passing means the answer is reasonable. It does not prove the answer is exactly right, so still compute with care.

The big idea: estimate before you solve, and compare after you solve. Replace each fraction with its nearest benchmark, $0$, $\frac{1}{2}$ or $1$, to get a quick estimate. Find the exact answer with a common denominator. Then ask whether the two are close. This works when the fractions refer to the same whole, or to wholes of equal size.

Next, you will solve word problems that take more than one step, with fractions and mixed numbers. You will name each step and the whole each fraction refers to. You will also find and fix mistakes in solved problems.