Question

How do you simplify \[ \dfrac{1}{2} + \dfrac{4}{9}? \]

Answer 1

Hint: We need to convert the addition of two fraction terms into a single fraction term by using cross multiplication or algebraic formula. After that, we need to solve the arithmetic expressions present in the numerator and denominator by using arithmetic operations. After that, we try to simplify the fraction term by using a common divisor.

Complete step-by-step solution:
The given problem is shown below,
 \[ \dfrac{1}{2} + \dfrac{4}{9} = ? \to equation\left( 1 \right) \]
We know that,
 \[ \dfrac{a}{b} + \dfrac{c}{d} = \dfrac{{\left( {a \times d} \right) + \left( {c \times b} \right)}}{{\left( {b \times d} \right)}} \to equation\left( 2 \right) \]
By comparing the equation \[ \left( 1 \right) \] and \[ \left( 2 \right) \] , we get
 \[ equation\left( 1 \right) \to \dfrac{1}{2} + \dfrac{4}{9} = ? \]
 \[ equation\left( 2 \right) \to \dfrac{a}{b} + \dfrac{c}{d} = \dfrac{{\left( {a \times d} \right) + \left( {c \times b} \right)}}{{\left( {b \times d} \right)}} \]
So, we get
 \[ \dfrac{a}{b} = \dfrac{1}{2} \] ; So, we get the value of \[ a \] is equal to \[ 1 \] and the value of \[ b \] is equal to \[ 2 \]
 \[ \dfrac{c}{d} = \dfrac{4}{9} \] ; So, we get the value of \[ c \] is equal to \[ 4 \] and the value of \[ d \] is equal to \[ 9 \]
So, we have to find the value of \[ a,b,c \] and \[ d \] by comparing the equation \[ \left( 1 \right) \] and \[ \left( 2 \right) \] . Let’s substitute these values in the equation \[ \left( 2 \right) \] , we get
 \[ equation\left( 2 \right) \to \dfrac{a}{b} + \dfrac{c}{d} = \dfrac{{\left( {a \times d} \right) + \left( {c \times b} \right)}}{{\left( {b \times d} \right)}} \]
 \[ \dfrac{1}{2} + \dfrac{4}{9} = \dfrac{{\left( {1 \times 9} \right) + \left( {4 \times 2} \right)}}{{\left( {2 \times 9} \right)}} \to equation\left( 3 \right) \]
Let’s solve the numerator part in the above equation, we get
 \[ \left( {1 \times 9} \right) + \left( {4 \times 2} \right) = 9 + 8 = 17 \to equation\left( 4 \right) \]
Let’s solve the denominator part in the equation \[ \left( 3 \right) \] , we get
 \[ \left( {2 \times 9} \right) = 18 \to equation\left( 5 \right) \]
Let’s substitute the equation \[ \left( 4 \right) \] and \[ \left( 5 \right) \] in the equation \[ \left( 3 \right) \] to find the final answer for the given question. So, we get
 \[ equation\left( 3 \right) \to \dfrac{1}{2} + \dfrac{4}{9} = \dfrac{{\left( {1 \times 9} \right) + \left( {4 \times 2} \right)}}{{\left( {2 \times 9} \right)}} \]
 \[ \dfrac{1}{2} + \dfrac{4}{9} = \dfrac{{17}}{{18}} \]
There is no common divisor between \[ 17 \] and \[ 18 \] .
We can’t furtherly simplify the term \[ \dfrac{{17}}{{18}} \] .
So, the final answer is,
 \[ \dfrac{1}{2} + \dfrac{4}{9} = \dfrac{{17}}{{18}} \]

Note: Note that the formula \[ \dfrac{a}{b} + \dfrac{c}{d} = \dfrac{{\left( {a \times d} \right) + \left( {c \times b} \right)}}{{\left( {b \times d} \right)}} \] for making the easy calculation. Note that first, we would solve the expression inside the parenthesis. If we have two or more arithmetic operations in the fraction term we would solve the numerator and denominator part separately to avoid wrong calculation. The above problem can also be solved by converting the fraction terms into decimal numbers. By using this method we can easily find the answer.