What is a Unit Fraction?
A unit fraction is a fraction with one as the numerator and a positive integer as the denominator. As a result, a unit fraction, \(\frac{1}{n}\) is the reciprocal of a positive integer n. \(\frac{1}{1},\frac{1}{2}, \frac{1}{3},\frac{1}{4},\frac{1}{5}\), and so on are examples of unit fractions.
As a unit fraction has a numerator of 1, it represents one part of the whole, where the whole is divided into equal parts.
Multiply Whole Numbers by Fractions
We have two steps to when multiplying whole numbers by fractions and they are as follows:
- Repeated Addition Method
Repeated addition is the process of adding equal groups and multiplication is another name for it. If the same number appears again and again, we can write it as multiplication.
\(= 2+2+2+2+2+2+2=2\times 7 = 14\)
Here 2 is repeated 7 times, we can write this addition as \(7\times 2\).
Example 1: Find the answer of \(2\times \frac{1}{3}\) using the repeated addition method.
Solution:
We have to add \(\frac{1}{3}\), 2 times to get the answer.
\(2\times \frac{1}{3}=\frac{1}{3}+\frac{1}{3}\)
Now, both fractions have a common denominator, 3, so we can simply add up the numerators.
\(=\frac{1+1}{3}=\frac{2}{3}\)
To multiply a fraction by a whole number, we multiply the numerator with the whole number. The result is written over the same denominator of the fraction. The fraction we get as the result is the product.
- Direct multiplication method
Example 2: Find the answer to the given multiplication of whole numbers by fraction \(2\times \frac{3}{5}\)
Solution:
We have to solve \(2\times \frac{3}{5}\)
First, we multiply the numerator and the whole number
\(2\times 3=6\)
This product is written over the denominator of the fraction and the resulting fraction is the product.
Therefore, \(2\times \frac{3}{5}=\frac{2\times 3}{5}=\frac{6}{5}\)
Let’s understand this using a diagram
In the diagram below each part is \(\frac{1}{5}\). The product is \(\frac{6}{5}\).
If we write this as a multiple of a unit fraction, it is \(6\times \frac{1}{5}\). That is, we need to shade 6 parts to represent \(\frac{6}{5}\).
- Unit fraction method
We have to rewrite the fraction into a unit fraction and then perform the multiplication.
Example 3: Find the answer of \(7\times \frac{5}{3}\) using the unit fraction method.
Solution:
Firstly, write the fraction as a multiple of a unit fraction.
\(\frac{5}{3}=5\times \frac{1}{3}\) [Here, \(\frac{1}{3}\) is the unit fraction]
\(7\times \frac{5}{3}=7\times (5\times \frac{1}{3})\)
\(=(7\times 5)\times \frac{1}{3}\) [Using the associative property of multiplication]
\(=35\times \frac{1}{3}\) [Simplify]
\(=\frac{35}{3}\)
Comparison between Factors and Products
When a number is multiplied by a improper fraction, the result is greater than the original number. When a number is multiplied by a proper fraction, the result is less than the original number.
Example 4: Determine whether the product of 2 and \(\frac{5}{3}\) is less than, greater than, or equal to its factors without actual calculation.
Solution:
In the fraction shown above, the numerator is greater than the denominator. So, the fraction is greater than 1.
Hence the product of 2 and \(\frac{5}{3}\) is greater than each of both 2 and \(\frac{5}{3}\).
Real Life Examples
Example 5: Every day, a man walks \(\frac{6}{7}\) miles twice. In a week, how far does he walk?
Solution:
The man walks \(\frac{6}{7}\) miles twice a day. So, total distance travelled by the man in a day is \(\frac{6}{7}\times 2\)
The distance covered by the man in a week \(\frac{6}{7}\times 2\times 7\)
\(=\frac{6}{7}\times 14\) Multiplication of whole numbers
Now, write \(\frac{6}{7}\) as a multiple of a unit fraction
\(\frac{6}{7}=\frac{1}{7}\times 6\)
Therefore, \(\frac{6}{7}\times 14=(\frac{1}{7}\times 6)\times 14\)
\(=\frac{1}{7}\times (6\times 14)\) [Using the associative property of multiplication]
\(=\frac{1}{7}\times 84\) [Simplify]
\( =12 \)
Therefore, the man covers 12 miles every week.
Example 6: In the United States of America, four-fifths out of fifty people eat fast food on a daily basis. How many people eat fast food on a daily basis?
Solution:
We have to multiply \(\frac{4}{5}\) by 50 to get the number of people who eat fast food every day.
\(=\frac{4}{5}\times 50\)
\(=\frac{4\times 50}{5}\) [Multiply the numerator by the whole number]
\(=\frac{200}{5}\) [Divide the product by the same denominator]
\( =40 \)
Hence, every 40 out of 50 people in the United States of America eat fast food daily.
Any positive number without a fractional or decimal part is referred to as a whole number. This means that all numbers, such as 0, 1, 2, 3, 4, 5, 6, and 7 are whole numbers.
Whole numbers are numbers such as 0, 1, 2, …. Each of these numbers can be expressed as a fraction with denominator 1 that is, \(\frac{0}{1}, \frac{1}{1}, \frac{2}{1}\) … Therefore, all whole numbers are indeed fractions.
Fractions are numbers that are actually a part of a whole, that is \(\frac{1}{2}, \frac{3}{4}, \frac{9}{4}\) …. Fractions are also numbers such as \(\frac{0}{1}, \frac{1}{1}, \frac{2}{1}\)….A number divided by 1 is the number itself hence these numbers are 0, 1, 2, … which are whole numbers. So some fractions can be expressed as whole numbers. But the entire family of fractions that consist of fractions such as \(\frac{\sqrt 3}{4}, \frac{-9}{5}, \frac{2^3}{3}\)… and these are not whole numbers. So all fractions are not whole numbers.