1.3.8 Multiplying Mixed Numbers

There are two major tricks involving the multiplication of mixed numbers. The first isn’t really a trick at all as it is only using technique of FOILing. Let’s illustrate with an example:

8 \frac{1}{8} \times 24 \frac{1}{8} = (8 + \frac{1}{8}) \times (24 + \frac{1}{8})

= 8 \cdot 24 + (8 + 24) \cdot \frac{1}{8} + \frac{1}{8} \cdot \frac{1}{8}

= 192 + 4 + \frac{1}{64} = 196 \frac{1}{64}

For the most part, both of the whole numbers in the mixed numbers are usually divisible by the fraction you are multiplying by (in our example both 8 and 24 are divisible by 8), which means you can just write down the fractional part of the answer immediately and then continue with the problem.

The other trick for mixed numbers occur when the sum of the fractional part is 1 and the two whole numbers are the same. For example:

9 \frac{1}{3} \times 9 \frac{2}{3} = (9 + \frac{1}{3}) \times (9 + \frac{2}{3})

= 9^2 + (9 \cdot 2 + 9) \cdot \frac{1}{3} + \frac{1}{3} \cdot \frac{2}{3}

= 9^2 + 9 + \frac{2}{9}

= 9(9 + 1) + \frac{2}{9} = 90 \frac{2}{9}

So the trick is:

The fractional part of the answer is just the two fractions multiplied together.
If the whole part in the problem is $n$ then the whole part of the answer is just $n \cdot (n + 1)$ .

Here is another example problem to show the procedure:

$7 \frac{2}{5} \times 7 \frac{3}{5} =$

Fractional Part: $\frac{2}{5} \cdot \frac{3}{5} = \frac{6}{25}$
Whole Part: $7 \cdot (7 + 1) = 56$

Answer: $56 \frac{6}{25}$

Although these tricks are great (especially FOILing the mixed numbers) sometimes FOILing is very complicated, so the best method is to convert the mixed numbers to improper fractions and see what cancels. For example, you don’t want to FOIL these mixed numbers:

4 \frac{7}{12} \times 2 \frac{2}{5} = \frac{7}{12} \cdot \frac{2}{5} + 4 \cdot \frac{2}{5} + 2 \cdot \frac{7}{12} + 4 \cdot 2

The above is really difficult to compute. Instead convert the numbers to improper fractions:

4 \frac{7}{12} \times 2 \frac{2}{5} = \frac{55}{12} \times \frac{12}{5} = 11

Usually the best method is to see if you can FOIL the numbers relatively quickly, and if you notice a stumbling block try to convert to improper fractions, then multiply.

Problem Set 1.3.8

4 \frac{1}{4} \times 8 \frac{1}{4}

8 \frac{2}{3} \times 8 \frac{1}{3}

3 \frac{4}{5} \times 3 \frac{1}{5}

4 \frac{2}{3} \times 6 \frac{1}{4}

12 \frac{1}{4} \times 8 \frac{1}{4}

15 \frac{1}{6} \times 9 \frac{1}{6}

6 \frac{1}{6} \times 12 \frac{1}{6}

11 \frac{1}{11} \times 22 \frac{1}{11}

25 \frac{2}{5} \times 5 \frac{2}{5}

5.2 \times 10.2

8 \frac{2}{3} \times 4 \frac{2}{3}

7 \frac{1}{7} \times 14 \frac{1}{7}

5 \frac{1}{5} \times 10 \frac{1}{5}

5 \frac{1}{5} \times 25 \frac{1}{5}

(5 \frac{2}{5})^2

8 \frac{1}{8} \times 16 \frac{1}{8}

10 \frac{5}{6} \times 12 \frac{4}{5}

11 \times 11 \frac{10}{11}

6 \frac{2}{3} \times 9 \frac{2}{3}

(12 \frac{2}{3})^2

7 \frac{1}{7} \times 49 \frac{1}{7}

3 \frac{3}{4} \times 2 \frac{2}{5}

4.3 \times 2.1

6 \times 6 \frac{5}{6}

(6 \frac{2}{3})^2

15.2 \times 5.2

4 \frac{3}{5} \times 4 \frac{2}{3}

3.125 \times 1.6

2.375 \times 2.4

2 \frac{2}{5} \times 5 \frac{2}{5}