Formula
If \(\frac{a}{b}\) and \(\frac{c}{d}\) are two fractions and we are asked to multiply the two fractions. The formula to find the product looks like this:
\(\frac{a \times b}{c \times d}=\frac{a.c}{b.d}\)
When multiplying two fractions, the numerators of both the fractions are multiplied, which provides the numerator for the product. The denominators of the given fractions are also multiplied and this provides the denominator for the product.
In the following example we need to multiply \(\frac{3}{4}\) and \(\frac{6}{7}\).
Solution:
\(\frac{3}{4} \times \frac{6}{7}\)
\(=\frac{3\times 4}{6 \times 7}\) Using \(\frac{a \times b}{c \times d}=\frac{a.c}{b.d}\)
\(=\frac{3\times 4}{6 \times 7}\)
\(=\frac{18}{28}\)
Fractions are presented in the simplest form. This means that the numerator and denominator are divided by common factors.
\(=\frac{18}{28}\)
\(=\frac{18 \div 2}{28 \div 2}\)
\(=\frac{9}{14}\)
Solved Examples
Example 1: Solve: \(\frac{1}{5} \times \frac{1}{3}\)
Solution:
\(\frac{1}{5} \times \frac{1}{3}\)
\(=\frac{1\times 1}{5 \times 3}\) Using \(\frac{a}{b}\times \frac{c}{d}=\frac{a.c}{b.d}\)
\(=\frac{1}{15}\)
Example 2: Solve : \(\frac{5}{10} \times \frac{6}{7}\)
Solution:
\(\frac{5}{10} \times \frac{6}{7}\)
\(=\frac{5\times 6}{10 \times 7}\) Using \(\frac{a}{b}\times \frac{c}{d}=\frac{a.c}{b.d}\)
\(=\frac{30}{70}\)
\(=\frac{30\div 10}{70 \div 10}\)
\(=\frac{3}{7}\)
Example 3: Solve: \(2\frac{7}{8} \times 3\frac{4}{5}\)
Solution:
\(2\frac{7}{8} \times 3\frac{4}{5}\)
\(=\frac{23}{8} \times \frac{19}{5}\) As, \(2\frac{7}{8}=\frac{2.8+7}{8}=\frac{23}{8}\) and \(3\frac{4}{5}=\frac{3.5+4}{5}=\frac{19}{5}\)
\(=\frac{23\times 9}{8 \times 5}\) Using \(\frac{a}{b}\times \frac{c}{d}=\frac{a.c}{b.d}\)
\(=\frac{437}{40}\)
Example 4: A multivitamin tablet contains \(\frac{1}{6}\)th of a gram of Zinc. Sam’s doctor prescribed the same because of multiple deficiencies that he had and he was supposed to take \(4\frac{3}{4}\) tablets in a week. What is the total quantity of Zinc that Sam consumed in a week?
Solution:
To find the total quantity of zinc that Sam consumed in a week, \(\frac{1}{6}\) must be multiplied with \(4\frac{3}{4}\) tablets, which is the total quantity of Zinc Sam needs to consume.
\(=4\frac{3}{4} \times \frac{1}{6}\)
\(=\frac{19}{4} \times \frac{1}{6}\)
\(=\frac{19\times 1}{4 \times 6}\) Using \(\frac{a}{b}\times \frac{c}{d}=\frac{a.c}{b.d}\)
\(=\frac{19}{24}\)
So, the multivitamin intake for a week is \(\frac{19}{24}\)
Example 5: Sully wants to make bread and he has \(\frac{2}{3}\)rd of a bag of flour. He uses \(\frac{3}{4}\)th of the flour from the bag. How much flour from the entire bag did he use to make the dough?
Solution:
Sully used \(\frac{3}{4}\)th of \(\frac{2}{3}\)rd of a bag of flour. To calculate the quantity of the flour used, we need to multiply \(\frac{3}{4}\) and \(\frac{2}{3}\).
\(\frac{3}{4} \times \frac{2}{3}\)
\(=\frac{3\times 2}{4 \times 3}\) Using \(\frac{a}{b}\times \frac{c}{d}=\frac{a.c}{b.d}\)
\(=\frac{6}{12}\)
\(=\frac{6\div 6}{12 \div 6}\)
\(=\frac{1}{2}\)
Therefore, Sully used \(\frac{1}{2}\) of the entire bag of flour to make bread.
Example 6: You wish to add a small fountain to your garden. The dimensions of the fountain and the garden are given below. What is the remaining area of the garden that’s left?
Solution:
Length of the Garden is \(=10\frac{1}{7}\) ft
Width of the Garden is \(=7\frac{3}{4}\) ft
Length of the fountain \(=6\frac{1}{2}\) ft
Width of the fountain \(=4\frac{1}{4}\) ft
The total area of the garden can be found by using the area of rectangle formula.
Area of a rectangle \( = \) length \( \times \) width
\(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=7\frac{3}{4} \times 10\frac{1}{7}\)
\(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{31}{4} \times \frac{71}{7}\) As, \(7\frac{3}{4}=\frac{7.4+3}{4}=\frac{31}{4}\) and \(10\frac{1}{7}=\frac{10.7+1}{7}=\frac{71}{7}\)
\(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{31\times 71}{4 \times 7}\) Using \(\frac{a}{b}\times \frac{c}{d}=\frac{a.c}{b.d}\)
\(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{221}{8}\) squared feet
Similarly, using the area of the rectangle formula we can find the total area of the fountain.
Area of a rectangle \( = \) length \( \times \) width
\(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=6\frac{1}{2} \times 6\frac{1}{4}\)
\(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{13}{2} \times \frac{17}{4}\) As, \(6\frac{1}{2}=\frac{6.2+1}{2}=\frac{13}{2}\) and \(4\frac{1}{4}=\frac{4.4+1}{4}=\frac{17}{4}\)
\(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{13\times 17}{2 \times 4}\) Using \(\frac{a}{b}\times \frac{c}{d}=\frac{a.c}{b.d}\)
\(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{221}{8}\) squared feet
Remaining area \( = \) Area of the garden \( – \) Area of the fountain
\(~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{2201}{28}-\frac{221}{8}\)
\(~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{2201.2}{28.2}-\frac{221.7}{8.7}\) Using LCM to convert them into like fractions.
\(~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{4402-1547}{56}\) Subtract and simplify.
\(~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{2855}{56}\)
Therefore the remaining area of the garden is \(\frac{2855}{56}\) squared feet.
Here are a few of the real-life applications of fractions. While eating at a restaurant and splitting a bill we use fractions. Imagine you were baking a cake. Even while following the instructions, you notice that the measurements for different ingredients are written in fractions as well.
Calculating discounted pricing for an on sale. Well, the use of fractions is pretty much everywhere, even in sports. Fractions are regularly used to evaluate a player’s or a team’s performance. They can also provide vital information about our health. We use fractions to figure out our body mass index (BMI) and whether or not we are in a healthy weight range. The list goes on.
Fractions usually represent values between two whole numbers.
Mixed fractions are a combination of a whole number and a fraction.
A mixed fraction is basically the sum of a whole number and a fraction. There is a way to obtain the result as a mixed fraction, but it’s best to convert a mixed fraction to a regular fraction before multiplying the given numbers.
The denominator represents the total number of parts and the numerator denotes the number of parts from the whole that is being considered.