Skill: Estimate fraction sums and differences

To estimate a fraction sum, replace each fraction with a nearby familiar value, such as $0$, $\frac12$, or $1$. Then add the easier values.

On a number line, $\frac12$ is halfway between $0$ and $1$. Fractions close to that point can be estimated as $\frac12$. Fractions close to $1$ can be estimated as $1$.

Example

Estimate:

$$\frac{5}{9}+\frac{7}{10}$$

Step 1: Estimate each fraction.

• $\frac{5}{9}$ is a little greater than $\frac12$, so estimate it as $\frac12$.
• $\frac{7}{10}$ is close to $\frac34$, so estimate it as $\frac34$.

You can picture each fraction as part of a bar of the same length. The first bar is about half shaded, and the second is about three-fourths shaded.

Step 2: Add the benchmark fractions.

$$\frac12+\frac34=\frac24+\frac34=\frac54=1\frac14$$

So,

$$\boxed{\frac{5}{9}+\frac{7}{10}\approx1\frac14}$$

The symbol $\approx$ means “approximately equal to.”

Step 3: Check whether the estimate makes sense.

Both original fractions are less than $1$, and together they should make a little more than $1$. The estimate $1\frac14$ fits.

For comparison, the exact sum is

$$\frac{5}{9}+\frac{7}{10}=\frac{50}{90}+\frac{63}{90}=\frac{113}{90}\approx1.26,$$

which is close to $1.25$, or $1\frac14$. Notice that for an estimate, you do not need common denominators first. Use nearby benchmark fractions to make the calculation easier.