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)
= \(\frac{602}{4}\) (Multiply numerators and denominators)
= \(\frac{301}{2}\) (Divide numerators and denominators by 2)
= \(150\frac{1}{2}\) 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.
First, we need to convert the mixed numbers into improper fractions. Then the normal multiplication process followed for fractions can be applied. The numerators are multiplied, then the denominators, and the respective products are expressed as a fraction, which is the answer.
The mixed number is first converted to an improper fraction. Then the numerator and the whole number is multiplied, the product becomes the numerator of the required fraction and the denominator is the same as that of the improper fraction. Thus, the required fraction is obtained.
A proper fraction has the numerator lesser than the denominator. For example, \(\frac{1}{2},\frac{3}{4},\frac{11}{13}\) and so on.