Question

One half of a number increased by \[16\] is four less than two thirds of the number. What is the number?

Answer 1

Hint: In this question we have to find the number which is one half of a number and is increased by $16$ and is four less than two thirds of the number. We will solve by making equations and simplify the equations by performing operations to find the number.

Complete step by step solution:
Let the number be $N$.
The number increased by $16$. So we add $16$ in the number. We get,
$N + 16$
Given that one half of the number is increased by $16$.
So, $\dfrac{1}{2}(N + 16)$
Now, two third of the number is $\dfrac{2}{3}N$
According to the question, one half of the number increased by $16$ is $4$ less than two thirds of the number.
So, we can write in equation as
$\dfrac{1}{2}(N + 16) = \dfrac{2}{3}N - 4$
Solving the left side of the equation. We get,
$\dfrac{N}{2} + 8 = \dfrac{2}{3}N - 4$
Taking all $N$ terms to the right side and constant terms to the left side. We get,
$8 + 4 = \dfrac{2}{3}N - \dfrac{1}{2}N$
Simplifying the right side of the equation by taking the least common factor (L.C.M). We get,
$8 + 4 = \dfrac{{4N - 3N}}{6}$
Simplifying the left and right side of the equation. We get,
$12 = \dfrac{N}{6}$
Shifting $6$ on the left side of the equation. We get,
$ \Rightarrow N = 12 \times 6$
$ \Rightarrow N = 72$
Hence, the number is $72$.

Note:
In this question we have simplified the equation by taking L.C.M. L.C.M is defined as the least common factor. This method is used to solve the fraction terms to equate the denominators so we can do operations on fraction. We can only add or subtract two fraction numbers if their denominators are equal so we have to equate the denominator of the fraction first.