Question

A fort is provisioned for \[42\] days; after $ 10 $ days, a reinforcement of $ 200 $ men arrives and the food will now last only for $ 24 $ days. How many men were there in the fort?
A. $ {\text{400}} $
B. $ {\text{500}} $
C. $ {\text{600}} $
D. $ {\text{160}} $

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 answer:
Let us consider that there are “x” soldiers in the fort and initially provisioned is for \[42\] days.
After $ 10 $ days, a reinforcement of $ 200 $ men arrives and provision will remain for “x” soldiers who are remaining till $ 32 $ days.
But, when the total $ (x + 200) $ soldiers the provision will last for only $ 24 $ days.
Now, using the inverse proportion, with increase in the number of soldiers, there will be decrease in the provision.
 $ \dfrac{{32}}{{24}} = \dfrac{{x + 200}}{x} $
Do cross-multiplication –
 $
  32x = 24(x + 200) \\
  32x = 24x + 4800 \;
  $
Simplify and make unknown “x” the subject –
 $
  32x - 24x = 4800 \\
  8x = 4800 \;
  $
Number in multiplicative when changes the side, goes to the division
 $
   \Rightarrow x = \dfrac{{4800}}{8} \\
   \Rightarrow x = 600 \;
  $
So, the correct answer is “Option C”.

Note: Always read the question twice and frame the word statement in the mathematical form. Segregate all the known and unknown terms and form the corresponding ratios and the proportions accordingly.
Ratio is the comparison between two numbers without any units.
Whereas, when two ratios are set equal to each other are called the proportion.
Four numbers a, b, c, and d are said to be in the proportion. If $ a:b = c:d $ whereas, four numbers are said to be in continued proportion if the terms $ a:b = b:c = c:d $