Salaries of Ankur and Ankit are in the ratio 3 ∶ 2. If the salaries of each is decreased by Rs. 2500, the new ratio becomes 8 ∶ 5. What is the sum of their salaries?
Rs. 37500
Understanding the Salary Ratio Problem
The question provides information about the salaries of Ankur and Ankit, initially given as a ratio. It then describes a change: a fixed amount is decreased from both their salaries, resulting in a new ratio. Our goal is to find the sum of their original salaries.
Setting up the Initial Salaries
Let the initial salaries of Ankur and Ankit be represented using the given ratio. The ratio is 3 ∶ 2. We can introduce a variable, say 'x', to represent a common multiplier for this ratio.
- Ankur's initial salary = \(3x\)
- Ankit's initial salary = \(2x\)
The sum of their initial salaries is \(3x + 2x = 5x\). We need to find the value of \(5x\).
Applying the Salary Decrease
According to the problem, the salary of each person is decreased by Rs. 2500.
- Ankur's new salary = \(3x - 2500\)
- Ankit's new salary = \(2x - 2500\)
Forming an Equation from the New Ratio
The new ratio of their salaries is given as 8 ∶ 5. We can write this as an equation:
\(\frac{\text{Ankur's new salary}}{\text{Ankit's new salary}} = \frac{8}{5}\)
Substituting the expressions for the new salaries, we get:
\(\frac{3x - 2500}{2x - 2500} = \frac{8}{5}\)
Solving for the Variable 'x'
To solve for \(x\), we can cross-multiply the equation:
\(5 \times (3x - 2500) = 8 \times (2x - 2500)\)
Distribute the numbers on both sides:
\(15x - 5 \times 2500 = 16x - 8 \times 2500\)
\(15x - 12500 = 16x - 20000\)
Now, we need to isolate the term with \(x\). Let's move the \(15x\) term to the right side and the \(-20000\) term to the left side:
\(20000 - 12500 = 16x - 15x\)
\(7500 = x\)
So, the value of \(x\) is 7500.
Calculating the Original Salaries and Their Sum
Now that we have the value of \(x\), we can calculate the original salaries of Ankur and Ankit.
- Ankur's original salary = \(3x = 3 \times 7500 = 22500\)
- Ankit's original salary = \(2x = 2 \times 7500 = 15000\)
The sum of their original salaries is:
\(\text{Sum} = \text{Ankur's original salary} + \text{Ankit's original salary}\)
\(\text{Sum} = 22500 + 15000\)
\(\text{Sum} = 37500\)
The sum of their original salaries is Rs. 37500.
| Person | Initial Salary (Ratio) | Initial Salary (with x) | Initial Salary (Value) | New Salary (after −2500) |
|---|---|---|---|---|
| Ankur | 3 | \(3x\) | \(3 \times 7500 = 22500\) | \(22500 - 2500 = 20000\) |
| Ankit | 2 | \(2x\) | \(2 \times 7500 = 15000\) | \(15000 - 2500 = 12500\) |
Check the new ratio: \(\frac{20000}{12500} = \frac{200}{125}\). Dividing both by 25, we get \(\frac{8}{5}\), which matches the new ratio given in the problem. This confirms our calculation for \(x\) is correct.
Revision Table: Salary Ratio Concepts
| Concept | Explanation | How it applies here |
|---|---|---|
| Ratio | A comparison of two quantities. Written as a:b or a/b. | Initial ratio 3:2, New ratio 8:5 |
| Introducing a variable | Using a variable (like x) with the ratio components to represent the actual quantities (3x, 2x). | Allows us to work with actual salary values algebraically. |
| Forming an equation | Setting up a mathematical equation based on the relationship given after changes occur. | The new ratio after the salary decrease gives us the equation \(\frac{3x-2500}{2x-2500} = \frac{8}{5}\). |
| Solving linear equations | Using algebraic techniques (like cross-multiplication, combining like terms) to find the value of the variable. | We solved for x using cross-multiplication and basic algebra. |
Additional Information: Solving Ratio Problems
Ratio and proportion problems are common in quantitative aptitude tests. Understanding how to represent quantities using variables based on ratios is a fundamental step. When quantities change, remember to apply the changes to the variable expressions. This often leads to a linear equation that can be solved easily.
Key steps often involve:
- Assigning variables based on the initial ratio.
- Writing expressions for the quantities after the described change.
- Setting up an equation using the new ratio.
- Solving the equation for the variable.
- Using the variable's value to find the original or final quantities.
Always double-check your calculations and ensure your final answer corresponds to what the question asks for (e.g., sum of initial salaries, difference in new salaries, etc.).
Important Questions from Ratio and Proportion
If the ratio of three numbers A, B and C is 2 ∶ 3 ∶ 5, and the sum of the squares of these numbers is 3800, then the value of C is:
A bag contains ₹310 in the form of 5 rupee, 2 rupee and 1 rupee coins in the ratio 4 ∶ 3 ∶ 5. What is the number of 5 rupee coins?
The ratio of three numbers is 3 ∶ 5 ∶ 4 and the sum of their squares is 11250. Find the sum of the numbers.
When 'x' is subtracted from each of the numbers 22, 39, 56 and 107, then the resulting numbers, in this order, are in proportion. What is the mean proportional between (x + 3) and (3x - 7)?
The salaries of Ravi and Sumit are in the ratio 4 ∶ 5. If the salary of each is increased by Rs. 6,000 the new ratio becomes 35 ∶ 40. What will be Sumit's increased salary?
