Question

How can the number $39$ be divided into two parts in order that the sum of $\dfrac{2}{3}$ of one part and $\,\dfrac{3}{4}\,$ of the other part is $28$?

Answer 1

Hint: First we will start by considering the two unknown numbers as some variables. Then we will apply different conditions then take the LCM to make the denominators the same. Then finally evaluate the value of both the unknown variables.

Complete step-by-step answer:
We will first consider the unknown terms as $x\,$ and $39 - x$.
Now we will apply the first condition that $\dfrac{2}{3}$ of first part is $\dfrac{2}{3} \times x = \dfrac{{2x}}{3}$ .
Now we will apply the second condition that is $\dfrac{3}{4}$ part of other is
 \[\dfrac{3}{4} \times (39 - x) = \dfrac{3}{4} \times 39 - \dfrac{3}{4} \times x = \dfrac{{117}}{4} - \dfrac{{3x}}{4}\]
Now, here as their sum is $28$, we have
$28 = \dfrac{{2x}}{3} + \dfrac{{117}}{4} - \dfrac{{3x}}{4}$
Now, if we reduce the terms, we get
 \[\dfrac{{3x}}{4} - \dfrac{{2x}}{3} = \dfrac{{117}}{4} - 28 = \dfrac{{117}}{4} - \dfrac{{28 \times 4}}{4}\]
Now we take the LCM and make the denominators the same.
 \[\dfrac{{3x \times 3}}{{4 \times 3}} - \dfrac{{2x \times 4}}{{3 \times 4}} = \dfrac{{117}}{4} - \dfrac{{112}}{4} = \dfrac{5}{4}\]
Now reduce the terms and simplify the equation.
 \[\dfrac{{9x}}{{12}} - \dfrac{{8x}}{{12}} = \dfrac{5}{4}\]
Evaluate the value of $x$.
 \[
  \dfrac{x}{{12}} = \dfrac{5}{4} \\
  x = \dfrac{5}{4} \times 12 \\
  x = 15 \;
 \]
So, the values of the variables will be,
$
  x = 15 \\
  39 - x = 39 - 15 = 24 \;
 $
Hence, the values of the unknown variables will be $15,24$
So, the correct answer is “$15,24$”.

Note: Always be sure that all of the terms are of the same type while comparing them. When you apply the conditions, make sure you back trace. After applying the values to the unknown variables make sure you substitute the values in the conditions and those values satisfy the conditions. While reducing terms make sure you reduce by making the factors.