Question

How do you simplify $\dfrac{3}{4}n + 3m - \dfrac{1}{3}n + \dfrac{1}{4}m$ ?

Answer 1

Hint: In this question, we are given an algebraic expression in terms of “n” and “m” and we have to simplify it. We will first place the like terms together and then simplify the given algebraic expression by using the distributive property. The distributive property states that the product of a number with the sum of two other numbers is equal to the product of that number with the first number plus the product of that number with the second number and vice versa, that is, $a(b + c) = ab + ac\,or\,ab + ac = a(b + c)$ .

Complete step-by-step solution:
We have to simplify $\dfrac{3}{4}n + 3m - \dfrac{1}{3}n + \dfrac{1}{4}m$
On grouping the like terms, we get –
$\Rightarrow \dfrac{3}{4}n + 3m - \dfrac{1}{3}n + \dfrac{1}{4}m = \dfrac{3}{4}n - \dfrac{1}{3}n + 3m + \dfrac{1}{4}m$
Now applying the distributive property, we get –
$\Rightarrow \dfrac{3}{4}n + 3m - \dfrac{1}{3}n + \dfrac{1}{4}m = n(\dfrac{3}{4} - \dfrac{1}{3}) + m(3 + \dfrac{1}{4})$
In the bracket $(\dfrac{3}{4} - \dfrac{1}{3})$ , the LCM of 4 and 3 is $4 \times 3 = 12$ .
The quotient obtained on dividing 12 by 4 is 3 and the quotient obtained on dividing 12 by 3 is 4. So, we multiply the numerator of 4 by 3 and the numerator of 3 by 4
And in the bracket $(\dfrac{3}{1} + \dfrac{1}{4})$ the LCM of 4 and 1 is $4 \times 1 = 4$
The quotient obtained on dividing 4 by 1 is 4 and the quotient obtained on dividing 4 by 4 is 1. So, we multiply the numerator of 1 by 4 and the numerator of 4 by 1 –
$
   \Rightarrow \dfrac{3}{4}n + 3m - \dfrac{1}{3}n + \dfrac{1}{4}m = n(\dfrac{{9 - 4}}{{12}}) + m(\dfrac{{12 + 1}}{4}) \\
   \Rightarrow \dfrac{3}{4}n + 3m - \dfrac{1}{3}n + \dfrac{1}{4}m = \dfrac{5}{{12}}n + \dfrac{{13}}{4}m \\
 $
Hence the simplified form of$\dfrac{3}{4}n + 3m - \dfrac{1}{3}n + \dfrac{1}{4}m$ is $\dfrac{5}{{12}}n + \dfrac{{13}}{4}m$ .

Note: After applying the distributive property, we obtained two fractions that are in subtraction in one bracket and in addition in the other bracket, and we know that when two fractions are linked via some arithmetic operation like addition or subtraction, then we have to first make their denominator equal. For that, we find the least common multiple (LCM) of the denominators and then we multiply the numerator of each fraction with the quotient obtained on dividing the LCM by their denominators.