Mixed Numbers Multiplication | List of Mixed Numbers Multiplication Formulas You Should Know - BYJUS

Mixed Numbers Multiplication Formulas

Multiplication is one of the four basic operations in Math. It is also an extension of addition. Multiplication represents the repeated addition of the same number or equal groups. Therefore, when we repeatedly add a number, we may use addition to find the sum or we can also use multiplication. As adding the same number over and over again would be cumbersome, using multiplication makes the process faster and easier. Here, we will focus on the multiplication of mixed numbers. ...Read MoreRead Less

Definition of Mixed Number Multiplication

Mixed numbers are numbers that are a combination of whole numbers and proper fractions. Mixed numbers can be converted to improper fractions and multiplied with each other.

Steps to find the product of Mixed Numbers

Follow these steps to multiply mixed numbers:

Step 1: The equivalent of mixed numbers can be represented by improper fractions. Hence, convert the mixed numbers to improper fractions.

Step 2: Multiply the respective numerators and write the product as the numerator of the resulting fraction. Similarly, multiply the denominators and write the product as the denominator of the resulting fraction.

Step 3: The resulting fraction is the product. This fraction can be further converted back into a mixed number as well.

Formula to Multiply Mixed Numbers

Consider two mixed numbers $$c\frac{a}{b}$$ and $$w\frac{x}{y}$$.

Step 1: Convert to improper fractions

$$c\frac{a}{b}=\frac{c\times b+a}{b}$$

Let’s assume the value of $${c\times b +a}$$ as p. So,

$$c\frac{a}{b}=\frac{p}{b}$$

Similarly, $$w\frac{x}{y}=\frac{w\times y+x}{y}$$

Let’s assume the value of ($${w\times y +x}$$) as q. So,

$$w\frac{x}{y}=\frac{q}{y}$$

Step 2: Multiplying respective numerators and denominators

$$\frac{p}{b}\times\frac{q}{y}=\frac{p\times q}{b\times y}$$

$$\frac{p\times q}{b\times y}$$ is the resulting product.

Step 3: Expressing the resulting fraction as a mixed number

To write $$\frac{p\times q}{b\times y}$$ as a mixed number. We need to first perform the long division. The resulting quotient, remainder and divisor can be written as $$\text{Quotient}\frac{\text{Remainder}}{\text{Divisor}}$$ to obtain the required mixed number.

Steps to find the product of Mixed Numbers

Follow these steps to multiply mixed numbers:

Step 1: The equivalent of mixed numbers can be represented by improper fractions. Hence, convert the mixed numbers to improper fractions.

Step 2: Multiply the respective numerators and write the product as the numerator of the resulting fraction. Similarly, multiply the denominators and write the product as the denominator of the resulting fraction.

Step 3: The resulting fraction is the product. This fraction can be further converted back into a mixed number as well.

Solved Examples

Example 1: Find the product of the given mixed numbers: $$7\frac{5}{11}$$ and $$2\frac{8}{9}$$.

Solution:
Given are the mixed numbers $$7\frac{5}{11}$$ and $$2\frac{8}{9}$$

$$7\frac{5}{11}=\frac{7\times 11+5}{11}=\frac{82}{11}$$ and $$2\frac{8}{9}=\frac{2\times 9+8}{9}=\frac{26}{9}$$ (Convert to improper fractions)

$$\frac{82}{11}\times\frac{26}{9}=\frac{2132}{99}$$(Multiply numerators and denominators)

=$$21\frac{53}{99}$$ (Write as a mixed number)

Example 2: Find the area of a rectangle that has a length $$2\frac{1}{3}$$ inches and a width of $$3\frac{1}{2}$$ inches.

Solution:
The area of a rectangle is the value of its length times the width.

Therefore, area, $$A=2\frac{1}{3}\times3\frac{1}{2}$$
Writing as improper fractions,

$$2\frac{1}{3}=\frac{2\times3+1}{3}=\frac{7}{3}$$

$$3\frac{1}{2}=\frac{3\times2+1}{2}=\frac{7}{2}$$

$$A=\frac{7}{3}\times\frac{7}{2}=\frac{49}{6}$$ (Multiply numerators and denominators)

$$=8\frac{1}{6}$$ square inches. (Write as a mixed number)

Hence the area of the rectangle is $$8\frac{1}{6}$$ square inches.

Example 3:

Jake wants to buy a new football that costs $20. But he has only 6 dollars with him. To get some more money, he decides to set up a lemonade stand and sell lemonade. Every cup of lemonade is sold at$$$2\frac{1}{2}$$. He sells a cup every 15 minutes. He stood at the stand for 2 hours. Does he have enough money for his new football?

Solution:

Amount needed by Jake = $20 –$6 = $14 Time at the stand = 2 hours = 2 $$\times$$ 60 = 120 minutes He sells a cup every 15 minutes. Hence, number of cups sold = $$\frac{120}{15}=8$$ cups Each cups is sold for$$$2\frac{1}{2}$$

Hence, money obtained = $$$2\frac{1}{2}\times8$$ = $$\frac{5}{2}\times8$$(Convert to improper fraction) =$20

Jakes needs only $14 but he earns$20.

Therefore, he has enough to buy a new football with 6 dollars to spare.

Example 4:

Maya is writing her first book. Her publisher has asked her to finish her first draft in 2 weeks. The mystery novel that Maya is writing needs 250 pages. She writes $$10\frac{3}{4}$$ pages a day. Can she complete her draft in time?

Solution:

Total number of pages of the book = 250

Number of pages written in a day = $$10\frac{3}{4}$$

She has 14 days to complete the book.

Number of pages written in 14 days

= $$10\frac{3}{4}\times14$$

= $$\frac{43}{4}\times14$$    (Convert to improper fraction)

= $$\frac{602}{4}\times14$$         (Multiply numerators and denominators)

= $$150\frac{1}{2}$$ pages (Write as a mixed number)

This shows that in two weeks Maya writes only $$150\frac{1}{2}$$ pages, and she does not meet the total of 250 pages.

Improper fractions are fractions that have  numerators that are greater than or equal to the denominators. For example: $$\frac{23}{5},\frac{13}{11},\frac{7}{2},\frac{6}{6}$$ and so on.
Mixed numbers are numbers between two whole numbers. So mixed numbers have a whole number part and a proper fraction. For example, $$1\frac{2}{3}$$ is a mixed number. This number lies between the wholes, 1 and 2. In $$1\frac{2}{3}$$, 1 is the whole number and $$\frac{2}{3}$$ is the proper fraction.
A proper fraction has the numerator lesser than the denominator. For example, $$\frac{1}{2},\frac{3}{4},\frac{11}{13}$$ and so on.