Question

How do you simplify $7\dfrac{1}{2} - \left( {\dfrac{1}{9} + \dfrac{2}{3}} \right) \div \dfrac{3}{2}$ ?

Answer 1

Hint: The given expression contains more than one operation. So, to simplify the given expression and obtain a unique answer, it is necessary to do the operations in a fixed order. Thus, we will use the BODMAS rule to simplify the given expression.

Complete step-by-step solution:
According to the BODMAS rule, firstly we need to solve the brackets, then divide, then multiply, then addition and last subtraction. We need to follow this order strictly, to find an accurate and unique solution to the expression. Because the order of operations changed then the result will also change.
Now, the given expression is $7\dfrac{1}{2} - \left( {\dfrac{1}{9} + \dfrac{2}{3}} \right) \div \dfrac{3}{2}$
For the simplification, firstly, we need to simplify the brackets, this means the terms within the brackets should be solved first. So we add the two fractions by taking the L.C.M. of the denominators inside the brackets, then we get,
$= 7\dfrac{1}{2} - \left( {\dfrac{{1 + 3\left( 2 \right)}}{9}} \right) \div \dfrac{3}{2}$
$= 7\dfrac{1}{2} - \left( {\dfrac{{1 + 6}}{9}} \right) \div \dfrac{3}{2}$
This way after simplifying the brackets, we get
$= 7\dfrac{1}{2} - \dfrac{7}{9} \div \dfrac{3}{2}$
Now, we can see that the brackets has been simplified and now the next operation that we need to perform is division, but when dividing the two fractions then we change the sign of division to multiplication and do the reciprocal of the second fraction, we have
$= 7\dfrac{1}{2} - \dfrac{7}{9} \times \dfrac{2}{3}$
Next operation to perform is of multiplication, we multiply the two fractions, in a way that we multiply the numerator with the numerator and multiply the denominator with the denominator, then the expression becomes
$= 7\dfrac{1}{2} - \dfrac{{14}}{{27}}$
Now we simplify the mixed fraction also
$= \dfrac{{15}}{2} - \dfrac{{14}}{{27}}$ , we convert the mixed fraction into improper fraction using $\dfrac{{7 \times 2 + 1}}{2}$
Now, we perform the subtraction operation, and for subtracting the fractions we first take the L.C.M. of the denominators, L.C.M. of $2$ and $\;27$ is $2 \times 27 = 54$ .
$= \dfrac{{15(27) - 14(2)}}{{54}}$
$= \dfrac{{405 - 28}}{{54}}$ , subtracting the terms in the numerator the fraction becomes
$= \dfrac{{377}}{{54}}$ , which is an improper fraction, so we divide the numerator with denominator and convert the improper fraction into mixed fraction, which is
$= 6\dfrac{{53}}{{54}}$ , which is the required result.

Therefore $6\dfrac{{53}}{{54}}$ is the require answer.

Note: In the BODMAS rule, B stands for Brackets, O stands for Of, D stands for Division, M stands for Multiplication, A stands for Addition and S stands for subtraction. Thus, to obtain a unique solution, this rule should be strictly followed.