Multiplying a Fraction by a Fraction
A fraction is a non-negative number in the form of \(\frac{p}{q}\) where q \(\neq\) 0. Here, p is known as the numerator, and q is known as the denominator of the fraction.
To multiply a fraction with a fraction, follow these steps:
- Multiply the numerator with the numerator
- Multiply the denominator with the denominator
- Reduce it to its simplest form
For example: Multiply \(\frac{2}{3}\) with \(\frac{6}{7}\)
\(\frac{2}{3}\) x \(\frac{6}{7}\) = \(\frac{2~\times~6}{3~\times~7}\) = \(\frac{12}{21}\) = \(\frac{4}{7}\)
Multiplying a Fraction by a Whole Number
To multiply a fraction with a whole number, follow these steps:
- First convert the whole number into a fraction by taking the denominator as 1
- Multiply the numerator with the whole number
- Multiply denominator by 1
- Reduce the product to its simplest form
Note : To multiply a fraction with a whole number, the fraction can be repeatedly added by itself in accordance with the whole number being the number of times.
For example: Multiply \(\frac{3}{5}\) with 4
\(\frac{3}{5}\) x 4 = \(\frac{3~\times~4}{5~\times~1}\) = \(\frac{12}{5}\)
Alternative method: \(\frac{3}{5}\) x 4 = \(\frac{3}{5}\) + \(\frac{3}{5}\) + \(\frac{3}{5}\) + \(\frac{3}{5}\) = \(\frac{3~+~3~+~3~+~3}{5}\) = \(\frac{12}{5}\)
Multiplying a Fraction by a Mixed Number
To multiply a fraction with a mixed number, follow these steps:
- First of all convert the mixed number into a fraction
- Multiply the numerator with the whole numerator
- Multiply the denominator with the denominator
- Reduce it into the simplest form that can either be a fraction or a mixed number
For example: Multiply \(\frac{8}{11}\) with 2\(\frac{2}{3}\)
\(\frac{8}{11}\) x 2\(\frac{2}{3}\) = \(\frac{8}{11}\) x \(\left(2~+~\frac{2}{3}\right)\) = \(\frac{8}{11}\) x \(\frac{8}{3}\) = \(\frac{64}{33}\) = 1\(\frac{31}{33}\).
Solved Examples
Example 1: John had 42 chocolates. He ate a total of \(\frac{3}{7}\) of the chocolates. How many chocolates does John have left?
Solution:
John had 42 chocolates, out of which he ate \(\frac{3}{7}\) of the chocolates.
Number of chocolates John ate = 42 x \(\frac{3}{7}\)
= \(\frac{42}{1}\) x \(\frac{3}{7}\) [Convert the whole number into a fraction]
= \(\frac{42~~\times~~3}{1~~\times~~7}\) [Multiply]
= \(\frac{126}{7}\) [Reduce into simplest form]
= 18 chocolates
So, the number of chocolates John has left is = 42 – 18 = 24 chocolates.
Example 2: Annie was playing basketball on a basketball court that measured \(\frac{82}{3}\) meter by \(\frac{29}{2}\) meter. Calculate the area of the basketball court.
Solution:
The basketball court is rectangular in shape with dimensions of \(\frac{82}{3}\) meters by \(\frac{29}{2}\) meters.
The area of the court can be calculated by using the formula for the area of a rectangular shape.
A = l x w [Formula of the area of a rectangular shape]
= \(\frac{82}{3}\) x \(\frac{29}{2}\) [Substitute \(\frac{82}{3}\) for l and \(\frac{29}{2}\) for w]
= \(\frac{82~\times~~29}{3~\times~~2}\) [Multiply the numerator and the denominator]
= \(\frac{2378}{6}\) [Simplify]
= \(\frac{1189}{3}\) or 396\(\frac{1}{3}\) [Simplest form]
Hence, the area of the basketball court is \(\frac{1189}{3}\) or 396\(\frac{1}{3}\) square meters.
Example 3: Jack reads \(\frac{1}{5}\) part of the book in one hour. How much of the book will Jack read in 3\(\frac{1}{2}\) hours?
Solution:
Jack reads \(\frac{1}{5}\) part of the book in one hour.
Part of the book will be read in 3\(\frac{1}{2}\) hours is,
\(\frac{1}{5}\) x 3\(\frac{1}{2}\) = \(\frac{1}{5}\) x \(\frac{7}{2}\) [Convert 3\(\frac{1}{2}\) into a fraction]
= \(\frac{1~\times~7}{5~\times~2}\) [Multiply the numerator and the denominator]
= \(\frac{7}{10}\) [Simplify]
Hence, Jack reads \(\frac{7}{10}\) part of the book in 3\(\frac{1}{2}\) hours.
A number that consists of a whole number and a proper fraction is known as a mixed number.
A fraction is a non-negative number in the form of p/q where ‘q is not equal to 0’.
Multiplication of fractions in simple terms is to multiply the numerators first, and then the denominators of two or fractions that are being multiplied.
A unit fraction is a fraction with a numerator of 1 and a denominator of any whole number.