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The ratio between two numbers is $3:5$. If each number is increased by 4, the ratio becomes $2:3$. Find the numbers.
A. $12,20$
B. $18,30$
C. $24,40$
D. $15,25$

Last updated date: 13th Jun 2024
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Hint: We first assume the constant ratio. Multiplying with the ratio we get the numbers. Then we find the numbers increased by 4 and form the linear equation with the new ratio of $2:3$. We solve the equation to get the value of x and solution of the problem.

Complete step by step answer:
The ratio between two numbers is $3:5$. We take x as the ratio constant.
So, the numbers are 3x and 5x.
Now the given condition is that if each number is increased by 4, the ratio becomes $2:3$.
After increasing every number by 4, the numbers become $\left( 3x+4 \right)$ and $\left( 5x+4 \right)$.
The ratio of those two numbers are $\dfrac{\left( 3x+4 \right)}{\left( 5x+4 \right)}$ which is equal to $2:3$.
We can make this into a linear equation by $\dfrac{\left( 3x+4 \right)}{\left( 5x+4 \right)}=\dfrac{2}{3}$.
We solve the equation by finding the value of x.
  & \dfrac{\left( 3x+4 \right)}{\left( 5x+4 \right)}=\dfrac{2}{3} \\
 & \Rightarrow 3\left( 3x+4 \right)=2\left( 5x+4 \right) \\
 & \Rightarrow 12-8=10x-9x \\
 & \Rightarrow x=4 \\
The value of x is 4 which is the constant ratio.
The numbers were 3x and 5x. So, they are $3\times 4=12$ and $5\times 4=20$.

So, the correct answer is “Option A”.

Note: The constant always has to be positive. We can also do this process in a backward way where we first the ratio constant for $2:3$. Then we get the ratio of $3:5$ by subtracting 4 from the previous numbers. We follow the same process to find the solution.