Question

Rs. $ 535 $ is divided among A, B and C so that if Rs. $ 15, $ Rs. $ 10, $ Rs. $ 30 $ be subtracted from their respective shares, the remainders may be in the ratio $ 4:5:7, $ what was their initial shares ( in Rs.)?
  $
  A.{\text{ 135, 160, 240}} \\
  {\text{B}}{\text{. 160, 280, 318}} \\
  {\text{C}}{\text{. 150, 210, 325}} \\
  {\text{D}}{\text{. None of these}} \\
   $

Answer 1

Hint: Identify the known and unknown ratios and set up the proportion and solve accordingly. In these ratio types of questions, take any variable as the reference number. Convert the word statements in the form of mathematical expressions and simplify for the required solution.

Complete step-by-step answer:
Given that Rs. $ 535 $ is divided among A, B and C so that if Rs. $ 15, $ Rs. $ 10, $ Rs. $ 30 $ be subtracted from their respective shares, the remainders may be in the ratio $ 4:5:7, $ -
Therefore, let us suppose that “x” is the common term among A, B and C
After subtraction, remainders are in the ratio $ 4:5:7, $ (given)
So, Let A remains with amount $ = 4x $
       Let B remains with amount $ = 5x $
       Let C remains with amount $ = 7x $
Also, the sum of ratios is equal to the amount left after subtracting Rs. $ 15, $ Rs. $ 10, $ Rs. $ 30 $ from Rs. $ 535 $
  $ \therefore 4x + 5x + 7x = 535 - (15 + 10 + 30) $
Simplify the above equation –
  $
  16x = 535 - 55 \\
  16x = 480 \\
   $
When the term multiplicative in the left moved to right goes to the denominator or becomes division and vice-versa.
  $
  x = \dfrac{{480}}{{16}} \\
  x = 30 \\
   $
Now, the initial shares received by A is $ = 4x + 15 $
Place the value of “X”
Therefore, the initial shares received by A is
  $
   = 4(30) + 15 \\
   = 120 + 15 \\
   = 135\;{\text{ }}.....{\text{(1)}} \\
   $
Therefore, the initial shares received by B is
  $
   = 5x + 10 \\
   = 5(30) + 10 \\
   = 150 + 10 \\
   = 160\;{\text{ }}.....{\text{(2)}} \\
   $
Therefore, the initial shares received by C is
  $
   = 7x + 30 \\
   = 7(30) + 30 \\
   = 210 + 30 \\
   = 240\,{\text{ }}.....{\text{(3)}} \\
   $
Now, from equations - $ \left( 1 \right){\text{, }}\left( 2 \right),\;and\left( 3 \right) $ -
The required solution – the initial shares of A, B and C was $ Rs.{\text{ 135, }}{\text{Rs}}{\text{. 160, Rs}}{\text{. 240}} $
So, the correct answer is “Option A”.

Note: Always convert the given word statement in the correct mathematical form and simplify using basic mathematical operations. Ratio is the comparison between two numbers without any units. Whereas, when two ratios are set equal to each other are called proportion. Four numbers a, b, c, and d are said to be in proportion. If $ a:b = c:d $ whereas, four numbers are said to be in continued proportion if $ a:b = b:c = c:d $