Question

Divide Rs. $370$into three parts such that the second part is $\dfrac{1}{4}$of the third part and the ratio between the first and the third part is $3:5.$ Find each part.
A) $120,{\text{ 50, 200}}$
B) $100,{\text{ 50, 200}}$
C) $240,{\text{ 25, 200}}$
D) $120,{\text{ 50, 100}}$

Answer 1

Hint: Identify the known and unknown ratios and set up the ratio and proportion and solve accordingly. In these ratio and proportion types of questions, take any variable as the reference number where applicable.

Complete step by step solution:
Let us assume the common factor for the given ratio be “x”
Therefore, the first and the third parts be $3x$ and $5x$
Convert the given word statements in the form of mathematical expressions –
Second part $ = \dfrac{1}{4}$of the third part $ = \dfrac{1}{4} \times 5x = \dfrac{{5x}}{4}$
Given that the number is equal $ = 370$
Therefore, $3x + \dfrac{{5x}}{4} + 5x = 370$
Take LCM (least common multiple) in the above expression –
$\Rightarrow \dfrac{{12x + 5x + 20x}}{4} = 370$
Add the numerators of the above expression –
$\Rightarrow \dfrac{{37x}}{4} = 370$
Perform cross multiply in the above expression, where the denominator of one side is multiplied with the numerator of the opposite side.
$ \Rightarrow 37x = 370 \times 4$
Term multiplicative on one side if moved to the opposite side, then it goes to the denominator.
$ \Rightarrow x = \dfrac{{370 \times 4}}{{37}}$
Common factors from the numerator and the denominator cancels each other.
$ \Rightarrow x = 40$
Place the above values and get the values for all the three parts -
Now, the first part $ = 3x = 3(40) = 120$Rs.
Second part $ = \dfrac{{5x}}{4} = \dfrac{{5(40)}}{4} = 50$Rs.
Third part $ = 5x = 5\times 40 = 200$Rs.
This is the required solution.

Thus the correct answer is option ‘A’.

Note: Always remember, denominators should be always equal or find LCM before adding the numerators in the fraction. When denominators then and then only add numerators. Frame the given word statements into mathematical expressions and check it twice.