Addition of Fractions

In order to add fractions, you should have the same denominator.

Let’s try these in different cases:

Adding the fractions Having the same Denominator:

  • If the denominator of both the numbers is the same, we leave it as it is and add the numerators.

Let’s understand it with an example:

\(\frac{7}{2}\) +\(\frac{4}{2}\) = \(\frac{11}{2}\)

In the above Example, if we add \(\frac{7}{2}\) and \(\frac{4}{2}\), we leave the denominator 2 as it is and add the numerators 7 and 4, i.e 7 + 4 = 11. The result of the fraction is \(\frac{11}{2}\).

Add Fractions with Coprime Denominators

  • What are Co-prime Factors?

Coprime denominators: The denominators which do not have common factors.

Let’s study how to add co-prime factors?

Steps:

  • You need to multiply the denominator to find the denominator of the resulting fraction.
  • Multiply the numerator of one of the fractions by the denominator of the other or vice versa to get the numerator.
  • Add the result now.

Examples of addition of Fractions:

Let’s understand it with an example:

\(\frac{10}{3}\) +\(\frac{2}{4}\) =\(\frac{10 X 4}{3 x 4}\) +\(\frac{2 X 3}{4 X 3}\) = \(\frac{40}{12}\) +\(\frac{6}{12}\) = \(\frac{46}{12}\)

The denominators are 3 and 4, which are different and have no common factors,

so to calculate the numerator, you need to multiply 10 X 4 = 40 and 2 x 3 = 6, and add the results, 40 + 6 = 46, which would be the numerator of the answer.

Hence, the final answer to the addition is \(\frac{46}{12}\)

Add Fractions with a Denominator that is the Divisor of the Other:

Let’s understand it with an example:

\(\frac{10}{20}\) +\(\frac{3}{4}\) = \(\frac{10}{20}\) +\(\frac{3 X 5}{4 X 5}\) = \(\frac{10}{20}\) +\(\frac{15}{20}\) = \(\frac{10 X 15}{20}\) = \(\frac{150}{20}\) = \(\frac{15}{2}\)
  • As the denominator of these fractions are different, but as 4 is a factor of 20, which

is a multiple of 5.

  • Multiply both the numerator and the denominator of \(\frac{3}{4}\) by 5.
  • You will get \(\frac{15}{20}\).
  • Now find the sum of both fractions.
  • It will give the answer as \(\frac{150}{20}\) which can be simplified further

to \(\frac{15}{2}\).

Add Fractions with the Least Common Multiple:

Let’s understand it with an example:

Let’s add:

\(\frac{3}{12}\) + \(\frac{10}{8}\)

Steps:

  • The denominators of the added fractions are 12 and 8.
  • Both of these are different with common factors.
  • To determine the least common multiple, we need to factor the numbers.

12 = 22 X 3

8 = 23

  • Find the highest power of any common factors to find the LCM – 24 = 23 x 3
  • So, 24 is the common denominator.

To calculate the numerator of the fractions for adding or subtracting , divide the calculated LCM into the denominator.

  • Next, Multiply the result with the numerator, \(\frac{24}{12}\) = 2 and \(\frac{6}{2}\) = 3
  • \(\frac{24}{8}\) = 3 and \(\frac{30}{3}\) = 10
  • The new fractions will be \(\frac{3}{12}\) X \(\frac{2}{2}\) and \(\frac{10}{8}\) X \(\frac{3}{3}\).
  • Resulting Fraction – \(\frac{30}{24}\).
  • Now add the numerators.
  • The sum of the two fractions is \(\frac{36}{24}\)
  • Simplify it by dividing the numerator and denominator by 12.

The answer is \(\frac{3}{2}\)

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Practise This Question

There are three figures of areas x2,12x and z respectively, as shown.Figures A and C are squares while B is a rectangle, with length of one side is x. What should be the value of z so that all the 3 figures can be arranged to form a square of side (x +6)?