Question

How do you evaluate $ \dfrac{3}{4} + \dfrac{1}{3} + \dfrac{1}{2} $ ?

Answer 1

Hint: In this question we need to perform fraction addition. To solve this question we use associative property of fraction addition. To solve this question we also need to know how to perform addition, multiplication and finding L.C.M (Least Common Multiple) of two numbers. This is a very basic question on fraction addition, try once before looking at a complete solution.

Complete step-by-step solution:
Let us try to find the value of $ \dfrac{3}{4} + \dfrac{1}{3} + \dfrac{1}{2} $ . To find the value of this we will perform fraction addition. We will first evaluate $ \dfrac{3}{4} + \dfrac{1}{3} $ and its result with $ \dfrac{1}{2} $ to finally get the value of whole expression. To perform fraction we required to have knowledge of addition, multiplication division and subtraction. Also, knowledge of L.C.M (Least Common Multiple) of two numbers. L.C.M of two numbers is the smallest common multiple of both numbers. For example $ L.C.M(2,4) = 4 $ , $ 4 $ is the smallest number which is multiple of both $ 2 $ and $ 4 $ . Let’s come back to our problem.
We have to evaluate $ \dfrac{3}{4} + \dfrac{1}{3} + \dfrac{1}{2} $ , first evaluate $ \dfrac{3}{4} + \dfrac{1}{3} $
After performing fraction addition, we get
 $ \dfrac{3}{4} + \dfrac{1}{3}\, = \,\dfrac{{3 \times 3 + 4 \times 1}}{{12}} $
Because $ L.C.M(3,4) $ is equal to 12. So value of $ \dfrac{3}{4} + \dfrac{1}{3} $ , after fraction addition is
 $
 = \dfrac{3}{4} + \dfrac{1}{3}\, = \,\dfrac{{3 \times 3 + 4 \times 1}}{{12}} \\
 = \,\dfrac{{9 + 4}}{{12}} \\
 = \,\dfrac{{13}}{{12}} \\
  $
Now, we have $ \dfrac{3}{4} + \dfrac{1}{3} = \dfrac{{13}}{{12}} $ and add $ \dfrac{1}{2} $ to get value of $ \dfrac{3}{4} + \dfrac{1}{3} + \dfrac{1}{2} $ .
 $ \dfrac{{13}}{{12}} + \dfrac{1}{2} = \dfrac{{13 \times 1 + 6 \times 1}}{{12}} $
Because $ L.C.M(12,2) $ is equal to 12. So value of $ \dfrac{{13}}{{12}} + \dfrac{1}{2} $ , after fraction addition is
 $
 = \dfrac{{13}}{{12}} + \dfrac{1}{2} = \dfrac{{13 \times 1 + 6 \times 1}}{{12}} \\
 = \,\dfrac{{13 + 6}}{{12}} \\
= \dfrac{{19}}{{12}} \\
  $
Hence the expression $ \dfrac{3}{4} + \dfrac{1}{3} + \dfrac{1}{2} $ evaluate to $ \dfrac{{19}}{{12}} $ .

Note: Fraction addition is both commutative and associative. Types of fraction: proper fraction, improper fraction. Proper fractions are those fractions in which the numerator is smaller than the denominator. Improper fractions are those fractions in which the numerator is greater than the denominator.