Like fractions are a collection of two or more fractions that have the same denominator. Like fractions are fractions with the same integers in the denominators. 1/5, 2/5 6/5, and 6/5 are all fractions with denominators of 5.
We can easily perform the addition and subtraction of like fractions. As the denominator is the same, all we need to do is to add or subtract the numerator.
Example: \(\frac{1}{5} + \frac{3}{5}\)= \(\frac{1+3}{5}\) = \(\frac{4}{5}\).
Here are some distinct characteristics of Like fractions:
- Like fractions represent the same amount of distance or points on a number line to one another.
- By reducing both the numerator and denominator by their greatest common factor, all like fractions are reduced to the same fractions in their simplest forms.
Properties of Like Fractions:
1. Commutative Property of Like Fractions:
The commutative property is a math rule that says that the order in which we add, multiply, subtract or divide the numbers does not change the product.
(a) Commutative Property of Addition: Addition of two like fractions is commutative in nature. If a/b and c/b are any two like fractions, then
(a/b) + (c/b) = (c/b) + (a/b)
Example: 2/7 + 4/7 = 6/7
4/7 + 2/7 = 6/7
So,
2/7 + 4/7 = 4/7 + 2/7
(b) Commutative Property of Subtraction: Subtraction of two like fractions is not commutative in nature. If a/b and c/b are any two like fractions, then
(a/b) – (c/b) ≠ (c/b) – (a/b)
Example: 2/7 – 4/7 = -2/7
4/7 – 2/7 = 2/7
So,
2/7 + 4/7 ≠ 4/7 + 2/7
(c) Commutative Property of Multiplication: Multiplication of like fractions is commutative. If a/b and c/b are any two like fractions, then
(a/b) x (c/b) = (c/b) x (a/b)
Example: 5/9 x 7/9 = 35/81
7/9 x 5/9 = 35/81
Hence, 5/9 x 7/9 = 7/9 x 5/9.
Therefore, commutative property is true for multiplication.
(d) Commutative Property of Division: Division of like fractions is not commutative. If a/b and c/b are two like fractions, then
a/b ÷ c/b ≠ c/b ÷ a/b
Example: 5/3 ÷ 7/3 = 5/3 x 3/7 = 5/7
7/3 ÷ 5/3 = 7/3 x 3/5 = 7/5
Therefore, Commutative property is not true for division.
2. Associative Property of Like Fractions
This law simply states that with the addition and multiplication of numbers, you can change the grouping of the numbers in the problem and it will not affect the answer.
(a) Associative Property of Addition: Addition of like fractions is associative in nature. If a/b, c/b and e/b are any three like fractions, then
a/b + (c/b + e/b) = (a/b + c/b) + e/b
Example: 2/3 + (4/3 + 1/3) = 2/3 + 5/3 = 7/3
(2/3 + 4/3) + 1/3 = 6/3 + 1/3 = 7/3
So,
2/3 + (4/3 + 1/3) = (2/3 + 4/3) + 1/3
This is the associative property of like fractions with examples.
(b) Associative Property of Subtraction: Addition of like fractions is not associative in nature. If a/b, c/b and e/b are any three like fractions, then
a/b – (c/b – e/b) ≠ (a/b – c/d) – e/b
Example: 2/3 – (4/3 – 1/3) = 2/3 – 3/3 = -1/3
(2/3 – 4/3) – 1/3 = -2/3 – 1/3 = -3/3
So,
2/3 + (4/3 + 1/3) ≠ (2/3 + 4/3) + 1/3
(c) Associative Property of Multiplication: Multiplication of like fractions is associative. If a/b, c/b and e/b are any three like fractions, then
then a/b x (c/b x e/b) = (a/b x c/b) x e/b
Example: 3/2 x (5/2 x 6/3) = 3/2 x 30/6 = 90/12
(3/2 x 5/2) x 6/3 = 15/4 x 6/5 = 90/12
Hence, 3/2 x (5/2 x 6/3) = (3/2 x 5/2) x 6/3.
Therefore, the associative property is true for multiplication.
(d) Associative Property of Division: Division of like fractions is not associative. If a/b, c/b and e/b are any three like fractions, then
a/b ÷ (c/b ÷ e/b) ≠ (a/b ÷ c/b) ÷ e/b
Example: 2/3 ÷ (4/3 ÷ 1/3) = 2/3 ÷ 4 = 1/6
(2/3 ÷ 4/3) ÷ 1/3 = 1/2 ÷ 1/3 = 3/2
Therefore, Associative property is not true for division.
3. Distributive Property of Like Fractions
Distributive Property is used to solve expressions easily by distributing a number to the numbers given in brackets.
(a) Distributive Property of Multiplication over Addition: Multiplication of like fractions is distributive over addition. If a/b, c/b and e/b are any three like fractions, then a/b x (c/b + e/b) = a/b x c/b + a/b x e/b
Example: 1/5 x (2/5 + 1/5) = 1/5 x 3/5 = 3/25
(1/5 x 2/5) + (1/5 x 1/5) = 2/25 + 1/25 = (2 + 1)/25 = 3/25
Therefore, Multiplication is distributive over addition.
(b) Distributive Property of Multiplication over Subtraction: Multiplication of like fractions is distributive over subtraction. If a/b, c/b and e/b are any three like fractions, then a/b x (c/b – e/b) = a/b x c/b – a/b x e/b
Example: 1/5 x (2/5 – 1/5) = 1/5 x 1/5 = 1/25
(1/5 x 2/5) – (1/5 x 1/5) = 2/25 – 1/25 = (2 – 1)/25 = 2/15
1/5 x (2/5 – 1/5) = 1/5 x 2/5 – 1/5 x 1/5
Therefore, Multiplication is distributive over subtraction.
Operations on Like Fractions:
There are 4 mathematical operations on Like Fractions:
1. Addition of Like Fractions
The most fundamental mathematical operation is addition. Addition combines two quantities into a single quantity, or sum, in its most basic form. Let’s imagine you have a group of two boxes and a group of three boxes. When you merge the two groups, you’ll end up with a five-box group.
In the case of like fractions, the addition operation is performed by adding the numerator terms and keeping the denominator constant.
Example: 2/9 + 4/9 = 6/9.
2. Subtraction of Like Fractions
The opposite of addition is subtraction. To find the difference between two quantities, we subtract one from the other instead of adding them together. Assume you start with a group of five boxes, like in the previous example. After removing three boxes from that group, you’re left with two.
In the case of like fractions, the subtraction operation is performed by subtracting the numerator terms and keeping the denominator constant.
Example: 4/9 – 2/9 = 2/9
3. Multiplication of Like Fractions
This is the simplest operation of all. It doesn’t require the conversion of unlike fractions to like fractions. We can easily multiply fractions by multiplying numerator with numerator and denominator with denominator. Multiplication also combines several quantities into a single quantity, which is referred to as the product. Multiplication, in fact, can be conceived of as the sum of numerous additions. The product of x and y is the result of adding x and y together y times. The product of two fractions is equal to the numerators’ product divided by the denominators’ product.
\(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)
In the case of like fractions, the multiplication operation is performed by multiplying numerator with numerator and denominator with denominator. In the case of like fractions, the denominator after multiplication becomes a perfect square.
Example: 4/9 x 2/9 = 8/81
4. Division of Like Fractions
Division is the opposite of multiplication. You split a quantity into a smaller value, called the quotient, rather than multiplying two or more numbers together to create a bigger value. Multiplying by the reciprocal of a fraction is identical to dividing by it. 1 divided by the number equals the reciprocal. A number’s multiplicative inverse is the reciprocal of that number. A number’s reciprocal and its product equals one. There is a reciprocal for all numbers except 0.
\(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}\)
In the case of like fractions, since the denominator is the same, they cancel out after taking reciprocal. So, what remains is the ratio of numerators.
Example: \(\frac{4}{9} \div \frac{3}{9} = \frac{4}{9} \times \frac{9}{2} = \frac{4\times9}{2\times 9} = \frac{4}{2} = 2\)
