Question

If \[\dfrac{n}{2} - \dfrac{{3n}}{4} + \dfrac{{5n}}{6} = 21\], then \[n = \]
A. 30
B. 42
C. 36
D. 28

Answer 1

Hint: We use the concept of LCM and take LCM of the three fractions in the left hand side of the equation. Calculate the left hand side of the equation and cross multiply the values with the right hand side of the equation, perform operations like multiplication or division on both sides of the equation as per the requirement so as to calculate the value of the variable ‘n’.
* LCM: LCM means least common multiple is the smallest number that is multiple of two or more given numbers.

Complete step-by-step solution:
We have to solve for the value of ‘n’ from the equation \[\dfrac{n}{2} - \dfrac{{3n}}{4} + \dfrac{{5n}}{6} = 21\]
Take LCM in left hand side of the equation
Since, \[2 = 2 \times 1;4 = 2 \times 2;6 = 2 \times 3\]
Take multiple from the three factorizations such that the number obtained is divisible by each 2, 4 and 6.
\[LCM(2,4,6) = 12\]as 12 is the least number that is divisible by each 2, 4 and 6 separately.
Multiply the remaining factor of denominator of each fraction to their respective numerator in LHS
\[ \Rightarrow \dfrac{{6 \times n - 3 \times 3n + 2 \times 5n}}{{12}} = 21\]
Calculate the product in numerator in LHS of the equation
\[ \Rightarrow \dfrac{{6n - 9n + 10n}}{{12}} = 21\]
Calculate the sum in numerator in LHS of the equation
\[ \Rightarrow \dfrac{{16n - 9n}}{{12}} = 21\]
Calculate the difference in numerator in LHS of the equation
\[ \Rightarrow \dfrac{{7n}}{{12}} = 21\]
We can write \[21 = 7 \times 3\]in RHS of the equation
\[ \Rightarrow \dfrac{{7n}}{{12}} = 7 \times 3\]
Cancel same factor i.e. 7 from both sides of the equation
\[ \Rightarrow \dfrac{n}{{12}} = 3\]
Multiply both sides of the equation by 12
\[ \Rightarrow \dfrac{n}{{12}} \times 12 = 3 \times 12\]
Cancel same factors from numerator and denominator in LHS of the equation and write the product of terms in RHS of the equation
\[ \Rightarrow n = 36\]
\[\therefore \]The value of ‘n’ is 36

\[\therefore \]Option C is correct

Note: Students many times get confused while calculating the LCM as they think the least common multiple means the smallest number that is common between the given numbers but that is wrong, we have to find the least number that is divisible by three numbers.
Also, many students make mistakes when shifting values from one side of the equation to another, keep in mind we always change sign from positive to negative and vice-versa when shifting values to the opposite side of the equation.