Estimate and Multiply Mixed Numbers

An estimate tells you about how big a product should be before you do the exact work. To make one, round each factor: swap it for a nearby number that is easy to multiply.

• Mixed number: round to the nearest whole number. Look at the fraction part. If it is less than $\frac{1}{2}$, keep the whole number. If it is $\frac{1}{2}$ or more, go up to the next whole number.
• Fraction between 0 and 1: pick the closest benchmark. A benchmark is an easy number you compare with. Here the benchmarks are $0$, $\frac{1}{2}$ and $1$. If a fraction is exactly halfway between two benchmarks, as $\frac{3}{4}$ is, either one is a fair choice.

Then multiply the rounded numbers. They sit near the real factors, so their product gives you a rough idea of the real product. The symbol $\approx$ means is about.

Rounding is one way to predict a product. A second way is to ask whether the product will be greater or less than a number you start with.

Think about what multiplying means. Call one factor your starting number. The other factor tells you how many copies of it to take, or what part of one copy. Take the positive starting number $6$. The product $1\frac{2}{3} \times 6$ is one full copy of $6$ plus $\frac{2}{3}$ of another copy, so it is more than $6$. The product $\frac{5}{6} \times 6$ is only part of one copy of $6$, so it is less than $6$.

For a positive starting number:

• a positive factor greater than 1 makes the product greater than the starting number;
• a factor between 0 and 1 makes the product less than the starting number;
• a factor of exactly 1 leaves the starting number unchanged.

If the starting number is 0, the product stays 0. This check is for multiplying only. Dividing is different: dividing a positive number by a positive number below 1 can make it greater.

The rule you know multiplies one fraction by one fraction. A mixed number is a sum of two parts, so first rewrite it as an improper fraction. An improper fraction is a fraction whose numerator is greater than or equal to its denominator.

Take $2\frac{1}{4}$. It means $2 + \frac{1}{4}$. One whole is $\frac{4}{4}$, so 2 wholes are $\frac{8}{4}$. Add the extra $\frac{1}{4}$ and you have $\frac{9}{4}$.

Here is a quick way to get the same result. Multiply the whole number by the denominator, add the numerator, and keep the denominator: $2 \times 4 + 1 = 9$, so $2\frac{1}{4} = \frac{9}{4}$. The $2 \times 4$ counts the fourths in the 2 wholes, and the $+ 1$ adds the extra fourth.

Add all four products. Write each fraction in twelfths so you can add them: $\frac{4}{3} + \frac{1}{4} + \frac{1}{6} = \frac{16}{12} + \frac{3}{12} + \frac{2}{12} = \frac{21}{12}$. That is $1\frac{9}{12}$, or $1\frac{3}{4}$. With the $2$ from the first row, the total is $2 + 1\frac{3}{4} = 3\frac{3}{4}$.

This matches the exact product you found with improper fractions. The shortcut's answer, $2\frac{1}{6}$, falls short because it leaves out the two middle rows, $\frac{4}{3}$ and $\frac{1}{4}$. Improper fractions avoid the problem. $\frac{9}{4}$ already holds the whole part and the fraction part of $2\frac{1}{4}$ together, so nothing gets left out.

Takeaway. Estimate before you multiply. Round each mixed number to the nearest whole number and each fraction between 0 and 1 to its closest benchmark, then multiply the rounded numbers. For the exact product, rewrite each mixed number as an improper fraction, multiply the numerators, multiply the denominators, and simplify. Then compare the exact product with your estimate.

Next, you will use these skills to solve word problems and to understand scaling, which is how multiplying by a fraction affects the size of a quantity.