Now that you can add and subtract any fractions and mixed numbers, it helps to know roughly what an answer should be before you work it out. The landmarks 0, $\frac{1}{2}$ and 1 make that quick.
Key idea
Round each fraction to the nearest benchmark, 0, $\frac{1}{2}$ or 1, and work with those. An exact answer far from the estimate needs a second look.
Why it works
Compare the numerator with the denominator. If the numerator is close to 0, the fraction is near 0. If it is close to half the denominator, the fraction is near $\frac{1}{2}$. If it is close to the denominator, the fraction is near 1. So $\frac{7}{8}$ is near 1 and $\frac{2}{5}$ is near $\frac{1}{2}$.
Worked examples
Worked example: Estimate a sum
Estimate $\frac{7}{8} + \frac{2}{5}$.
- $\frac{7}{8}$ is close to 1.
- $\frac{2}{5}$ is close to $\frac{1}{2}$ (half of 5 is 2.5).
- About $1 + \frac{1}{2} = 1\frac{1}{2}$. (Exactly: $1\frac{11}{40}$.)
Answer: about $1\frac{1}{2}$
Worked example: Catch a mistake
Kai says $\frac{2}{5} + \frac{1}{2} = \frac{3}{7}$. Is that reasonable?
- The sum must be more than $\frac{1}{2}$, since $\frac{1}{2}$ is one of the parts.
- $\frac{3}{7}$ is less than $\frac{1}{2}$, because 3 is less than half of 7.
Answer: No. The sum is $\frac{9}{10}$, close to 1.
Worked example: A story
Mia walks $\frac{2}{3}$ mile, then $\frac{3}{4}$ mile. How far does she walk?
- Estimate: about $\frac{1}{2} + 1 = 1\frac{1}{2}$ miles.
- Exactly: $\frac{8}{12} + \frac{9}{12} = \frac{17}{12} = 1\frac{5}{12}$.
- $1\frac{5}{12}$ is near $1\frac{1}{2}$, so it is reasonable.
Answer: $1\frac{5}{12}$ miles
Common mistake
Trusting an answer without checking its size. $\frac{2}{5} + \frac{1}{2} = \frac{3}{7}$ looks tidy, but it is smaller than one of the parts. A one-second benchmark check catches it.