This problem involves using ratios to find unknown salaries that change after a fixed increase. We are given the initial ratio of the salaries of Ravi and Sumit, and the new ratio after each person's salary is increased by the same amount. We need to find Sumit's salary after the increase.

Understanding the Initial Salary Ratio

The salaries of Ravi and Sumit are in the ratio 4 : 5. This means that for some value, let's call it \(x\), Ravi's salary is \(4x\) and Sumit's salary is \(5x\). The variable \(x\) represents a common multiplier that maintains the given ratio.

• Ravi's original salary = \(4x\)
• Sumit's original salary = \(5x\)

Applying the Salary Increase

Both Ravi's and Sumit's salaries are increased by Rs. 6,000.

• Ravi's new salary = Original salary + Increase = \(4x + 6000\)
• Sumit's new salary = Original salary + Increase = \(5x + 6000\)

Setting up the New Ratio Equation

After the increase, the new ratio of their salaries becomes 35 : 40. This ratio can be simplified by dividing both numbers by their greatest common divisor, which is 5.

\(\frac{35}{40} = \frac{35 \div 5}{40 \div 5} = \frac{7}{8}\)

So, the new ratio is 7 : 8. We can now write an equation relating the new salaries to this new ratio:

\[ \frac{\text{Ravi's new salary}}{\text{Sumit's new salary}} = \frac{7}{8} \]

Substituting the expressions for the new salaries, we get:

\[ \frac{4x + 6000}{5x + 6000} = \frac{7}{8} \]

Solving the Equation for \(x\)

To solve for \(x\), we can cross-multiply:

\[ 8 \times (4x + 6000) = 7 \times (5x + 6000) \]

Distribute the numbers on both sides:

\[ 32x + 48000 = 35x + 42000 \]

Now, we need to isolate the \(x\) terms on one side and the constant terms on the other side. Subtract \(32x\) from both sides and subtract \(42000\) from both sides:

\[ 48000 - 42000 = 35x - 32x \]

\[ 6000 = 3x \]

Divide by 3 to find the value of \(x\):

\[ x = \frac{6000}{3} \]

\[ x = 2000 \]

Calculating Original and Increased Salaries

Now that we have the value of \(x\), we can find the original salaries:

• Ravi's original salary = \(4x = 4 \times 2000 = 8000\)
• Sumit's original salary = \(5x = 5 \times 2000 = 10000\)

The question asks for Sumit's increased salary. This is his original salary plus the Rs. 6,000 increase.

Sumit's increased salary = Sumit's original salary + 6000

Sumit's increased salary = \(10000 + 6000 = 16000\)

Summary of Salary Calculation

The calculated increased salary for Sumit is Rs. 16,000.

Revision Table: Salary Ratio Problem

Additional Information on Ratios and Proportions

A ratio is a comparison of two quantities. If the ratio of quantity A to quantity B is \(a:b\), it means \(\frac{A}{B} = \frac{a}{b}\). In many problems involving ratios where quantities change, we introduce a variable (like \(x\)) to represent the actual values in terms of the ratio components (\(ax, bx\)). When the quantities are increased or decreased by a fixed amount, we modify these expressions (\(ax + \text{increase}\), \(bx + \text{increase}\)) and form a new ratio equation if the new ratio is given. Solving this equation allows us to find the value of the variable \(x\), and consequently, the actual values of the quantities.

Linear equations, like the one \((32x + 48000 = 35x + 42000)\) we solved, are fundamental in algebra. They involve variables raised to the power of 1 and can be solved by isolating the variable using inverse operations (addition/subtraction, multiplication/division) on both sides of the equation to maintain equality.