Question

If 60 is divided into two parts in the ratio 2 : 3, then the difference between those two parts is \[\]
A.10\[\]
B.12\[\]
C.8\[\]
D.18\[\]

Answer 1

Hint: Use the property of ratios where with multiplication of integer to solve does not change the ratio. First assign the integer as an unknown variable $ x $ and multiply with 2 and 3. Take their sum as 60. Solve the question in $ x $ to find the values of $ 2x,3x. $ Then you can subtract to find the difference.

Complete step-by-step answer:
A ratio is a fraction with both numerator and denominator as positive integers expressed in standard form. The standard of a fraction is $ \dfrac{p}{q} $ where both $ p $ and $ q $ are positive integers and the highest common factor of $ p $ and $ q $ is 1. It means $ p $ and $ q $ are co-prime or relatively prime. \[\]
So the ratio between two positive integers numbers $ A $ and $ B $ is written as $ a:b $ where $ A=na,B=nb $ and $ n $ is highest common factor of $ A $ and $ B $ . \[\]
If two numbers $ A $ and $ B $ are in a ratio $ a:b $ and then for some positive integer $ k $ if multiplied, $ kA $ and $ kB $ will so have the same ratio $ a:b $ . \[\]
If two numbers $ A $ and $ B $ are in a ratio $ a:b $ and then for some positive integer $ k $ , if divided $ \dfrac{A}{k} $ and $ \dfrac{B}{k} $ will so have the same ratio $ a:b $ where k is a factor of both $ A $ and $ B $ .\[\]
If a number is divided by a ratio $ a:b $ , then the total number parts it is divided is $ a+b $ .\[\]

Method-1\[\]
 It is given that the number is 60. It needs to be divided into two numbers with a ratio 2:3. Let the two numbers be $ A $ and $ B $ . So $ A+B=60,A:B=2:3 $ Then for some positive integers $ x $ the numbers respectively can be expressed as $ 2x,3x $ . Then the total 60 can be expressed as a sum of $ 2x,3x $ . Now,\[\]
 $ \begin{align}
  & A+B=60 \\
 & 2x+3x=60 \\
 & \Rightarrow 5x=60 \\
\end{align} $ \[\]
Solving the above equation for the unknown $ x $ we get $ x=\dfrac{60}{5}=12 $ \[\]
Then the two numbers are $ A=2x=2\times 12=24\text{ and }B=3x=3\times 12=36 $ . Then their difference is $ B-A=36-24=12 $ \[\]
Method-2:\[\]
The total number of parts 60 is divided is $ 2+3=5 $ . So the number with the larger part is 3 parts out of 5. So the larger number is $ 60\times \dfrac{3}{5}=36 $ and the smaller number is $ 60\times \dfrac{2}{5}=24 $ . Their difference is $ 36-24=12 $ .

So, the correct answer is “Option B”.

Note: Ratio is also called proportion. There can be ratios with more than two numbers (called continued proportion ) $ a:b:c. $ . You can also solve problems involving two ratios, in symbols $ a:b::c:d $ which can also be written as $ \dfrac{a}{b}=\dfrac{c}{d} $ . Ratios are used to compare quantities of the same type, for example the comparison of prices within a pair of months or years, the amount of increase in population by how many times, etc. .