Question

Sumit and Anu invested money in the ratio of $ 8:12, $ find for how much time anu invested the money if Sumit invested money for $ 9 $ months and he got Rs $ 1000 $ from a total profit of Rs $ 3000? $

Answer 1

Hint: First of all we will frame the given word statements in the form of mathematical expression and also assume the given unknown term is the period of months investment done by Anu and then will find the correlation between the two and find the required value.

Complete step by step solution:
Given that the ratio of Sumit and Anu’s investment is $ 8:12, $
Also given that Sumit invested money for $ 9 $ months and let us assume that Anu invested for “x” months.
Frame the mathematical expression using the given profit ratio –
\[ \Rightarrow 9 \times 8:12 \times x\]
Simplify the above expression finding the product of the terms –
\[ \Rightarrow 72:12x\] ….. (A)
We are given the total profit earned by both Sumit and Anu $ = 3000 $ Rs.
Also, given that profit earned by Sumit $ = 1000 $ Rs.
So, the profit earned by Anu $ = 3000 - 1000 = 2000 $ Rs.
Profit ratio for Sumit and Anu can be given by –
  $ \Rightarrow 1000:2000 $
Frame as in the form of the fraction –
  $ \Rightarrow \dfrac{{1000}}{{2000}} $
Common factors from the numerator and the denominator cancel each other.
   $ \Rightarrow \dfrac{1}{2} $ …. (B)
By using the equations (A) and (B)
Both the ratios are equal –
  $ \Rightarrow \dfrac{{72}}{{12x}} = \dfrac{1}{2} $
Perform cross multiplication, where the denominator of one side is multiplied with the numerator of the opposite side and vice-versa.
  $ \Rightarrow 72(2) = 12x(1) $
Make the required term “X” the subject and also the term multiplicative on one side if moved to the opposite side then it goes to the denominator.
  $ \Rightarrow \dfrac{{72 \times 2}}{{12}} = x $
Find the factors of the term in the numerator –
  $ \Rightarrow \dfrac{{12 \times 6 \times 2}}{{12}} = x $
Common factors from the numerator and the denominator cancels each other and therefore remove
  $ \Rightarrow x = 2 \times 6 $
Find the product of the terms in the above expression –
  $ \Rightarrow x = 12 $
Hence, Anu invested for $ 12 $ month.
So, the correct answer is “12 month ”.

Note: Frame the ratios perfectly since it is the base of the correct solution. Ratios are defined as 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 stated to be in the proportion, If $ a:b = c:d $ whereas, four numbers are stated to be in continued proportion if the terms \[\] $ a:b = b:c = c:d $