Multiply Mixed Numbers by Fractions and Mixed Numbers | Grade 5 Math | LevelKite

You already know how to multiply a mixed number by a whole number. Now the other number will be a fraction or another mixed number.

Here are the words you need:

A mixed number is a whole number and a fraction written together, such as $2\frac{1}{2}$ . It means $2 + \frac{1}{2}$ . The whole number part is $2$ , and the fraction part is $\frac{1}{2}$ .
In a fraction, the numerator is the top number and the denominator is the bottom number.
An improper fraction is a fraction whose numerator is equal to or greater than its denominator, such as $\frac{5}{2}$ .
The product is the answer you get when you multiply.

Start with what the multiplication means. $\frac{3}{4} \times 2\frac{1}{2}$ means $\frac{3}{4}$ of $2\frac{1}{2}$ . You split $2\frac{1}{2}$ into 4 equal parts and take 3 of those parts.

That is hard to do with 2 wholes and 1 half, because the pieces are not all the same size. So first write $2\frac{1}{2}$ as an improper fraction: $\frac{5}{2}$ , or 5 halves. It is the same amount, but now every piece is a half. Pieces of the same size are easy to split and to count.

Worked example

A fraction times a mixed number

What is $\frac{3}{4} \times 2\frac{1}{2}$ ?

Write the mixed number as an improper fraction so that every piece is the same size. Each whole is 2 halves, so 2 wholes are 4 halves. Add the 1 extra half to get 5 halves.
$2\frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2}$
Multiply the numerators and multiply the denominators. Why? Cut each of the 5 halves into 4 equal pieces. These are eighths, because 2 halves with 4 pieces each make 8 pieces in a whole. One piece from each half is one fourth of the amount: 5 eighths. Three fourths is 3 times as many: 15 eighths. So 3 times 5 counts the pieces, and 4 times 2 names their size.
$\frac{3}{4} \times \frac{5}{2} = \frac{3 \times 5}{4 \times 2} = \frac{15}{8}$
Write the product as a mixed number. 8 eighths make 1 whole. 15 eighths are 8 eighths and 7 more eighths, so the product is 1 whole and 7 eighths.
$\frac{15}{8} = \frac{8}{8} + \frac{7}{8} = 1\frac{7}{8}$

Answer

The product of $\frac{3}{4}$ and $2\frac{1}{2}$ is $1\frac{7}{8}$ .

Now make both numbers mixed numbers. $1\frac{1}{2} \times 2\frac{1}{2}$ means $1\frac{1}{2}$ groups of $2\frac{1}{2}$ . That is one full group and half of another group.

The same plan works. This time, write both mixed numbers as improper fractions before you multiply. Then multiply the numerators to count the pieces, and multiply the denominators to name their size.

Worked example

Two mixed numbers

What is $1\frac{1}{2} \times 2\frac{1}{2}$ ?

Write the first mixed number as an improper fraction. 1 whole is 2 halves, and 1 more half makes 3 halves.
$1\frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{3}{2}$
Write the second mixed number as an improper fraction. 2 wholes are 4 halves, and 1 more half makes 5 halves.
$2\frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2}$
Multiply the numerators and multiply the denominators. Cut each of the 5 halves into 2 equal pieces to make fourths. One piece from each half is one half of the amount: 5 fourths. Three halves is 3 times as many: 15 fourths.
$\frac{3}{2} \times \frac{5}{2} = \frac{3 \times 5}{2 \times 2} = \frac{15}{4}$
Write the product as a mixed number. 4 fourths make 1 whole, so 12 fourths make 3 wholes. That leaves 3 fourths.
$\frac{15}{4} = \frac{12}{4} + \frac{3}{4} = 3\frac{3}{4}$

Answer

The product of $1\frac{1}{2}$ and $2\frac{1}{2}$ is $3\frac{3}{4}$ .

Key concept

Improper fractions first, then multiply

To multiply a mixed number by a fraction or by another mixed number, follow three steps. First, write every mixed number as an improper fraction. Next, multiply the numerators and multiply the denominators. Last, if the product is an improper fraction, write it as a mixed number, or as a whole number when nothing is left over.

An improper fraction holds the whole number part and the fraction part together as one number. So when you multiply, every part gets counted.

Worked example

Mixed numbers with different denominators

What is $3\frac{1}{2} \times 1\frac{2}{3}$ ?

Write both mixed numbers as improper fractions. 3 wholes are 6 halves, and 1 more half makes 7 halves. 1 whole is 3 thirds, and 2 more thirds make 5 thirds.
$3\frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{7}{2} \quad \text{and} \quad 1\frac{2}{3} = \frac{1 \times 3 + 2}{3} = \frac{5}{3}$
Multiply the numerators and multiply the denominators. Thirds cut into 2 equal pieces are sixths, and 7 times 5 counts 35 of them.
$\frac{7}{2} \times \frac{5}{3} = \frac{7 \times 5}{2 \times 3} = \frac{35}{6}$
Write the product as a mixed number. 6 sixths make 1 whole, so 30 sixths make 5 wholes. That leaves 5 sixths.
$\frac{35}{6} = \frac{30}{6} + \frac{5}{6} = 5\frac{5}{6}$

Answer

The product of $3\frac{1}{2}$ and $1\frac{2}{3}$ is $5\frac{5}{6}$ .

Here is a shortcut that looks faster but does not work. For $1\frac{1}{2} \times 2\frac{1}{2}$ , someone might multiply only the whole number parts: $1 \times 2 = 2$ . Then they multiply only the fraction parts: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ . They write $2\frac{1}{4}$ . But you already found that the product is $3\frac{3}{4}$ .

Think about what the multiplication means. $1\frac{1}{2}$ groups of $2\frac{1}{2}$ is one full group and half of another group. One full group is $2\frac{1}{2}$ all by itself. That is already more than $2\frac{1}{4}$ , so the shortcut has left something out.

To find what is missing, look inside each group. The full group of $2\frac{1}{2}$ has two parts: a $2$ and a $\frac{1}{2}$ . The half group has two parts too: half of the $2$ and half of the $\frac{1}{2}$ . That makes four small products, not two. The table lists them.

The four small products inside one and a half groups of two and a half

Parts multiplied	In words	Number sentence	Does the shortcut count it?
Whole number part times whole number part	The $2$ in the full group	$1 \times 2 = 2$	Yes
Whole number part times fraction part	The $\frac{1}{2}$ in the full group	$1 \times \frac{1}{2} = \frac{1}{2}$	No
Fraction part times whole number part	Half of the $2$	$\frac{1}{2} \times 2 = 1$	No
Fraction part times fraction part	Half of the $\frac{1}{2}$	$\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$	Yes

Now add all four small products. Add the wholes first: $2 + 1 = 3$ . Then add the fractions. Since $\frac{1}{2}$ is the same as $\frac{2}{4}$ , the fractions make $\frac{3}{4}$ . The sum is the same product you found with improper fractions.

2 + \frac{1}{2} + 1 + \frac{1}{4} = 3 + \frac{2}{4} + \frac{1}{4} = 3\frac{3}{4}

Takeaway: To multiply a mixed number by a fraction or by another mixed number, write every mixed number as an improper fraction first. Then multiply the numerators and multiply the denominators. When the product is an improper fraction, write it as a mixed number or a whole number. Improper fractions keep the whole number part and the fraction part together, so no part of the product gets left out.

Next, you will use this method to solve word problems, check each answer against an estimate, and find and fix mistakes.