Understanding Income Ratios Across Years

This problem involves calculating incomes based on given ratios across two different years, 2020 and 2021. We are given the ratio of incomes of two individuals, A and B, in 2020, and how their individual incomes changed from 2020 to 2021. We also know their total combined income in 2021 and need to find B's income specifically in 2021.

Step-by-Step Solution to the Income Ratio Problem

Let's denote the income of A and B in 2020 as $A_{2020}$ and $B_{2020}$, respectively. Similarly, let their incomes in 2021 be $A_{2021}$ and $B_{2021}$.

We are given the following information:

• Ratio of incomes of A and B in 2020: $A_{2020} : B_{2020} = 5 : 4$
• Ratio of A's income in 2020 and 2021: $A_{2020} : A_{2021} = 4 : 5$
• Ratio of B's income in 2020 and 2021: $B_{2020} : B_{2021} = 2 : 3$
• Total income of A and B in 2021: $A_{2021} + B_{2021} = \text{Rs. } 7,05,600$

Expressing Incomes in Terms of a Common Variable

From the ratio of incomes in 2020, $A_{2020} : B_{2020} = 5 : 4$, we can write $A_{2020} = 5k$ and $B_{2020} = 4k$ for some constant $k$.

Finding 2021 Incomes in Terms of the Same Variable

Now, let's use the individual income ratios between 2020 and 2021:

• For A: $A_{2020} / A_{2021} = 4/5$
  So, $A_{2021} = \frac{5}{4} A_{2020}$. Substitute $A_{2020} = 5k$:
  $A_{2021} = \frac{5}{4} (5k) = \frac{25}{4}k$.
• For B: $B_{2020} / B_{2021} = 2/3$
  So, $B_{2021} = \frac{3}{2} B_{2020}$. Substitute $B_{2020} = 4k$:
  $B_{2021} = \frac{3}{2} (4k) = 6k$.

Using the Total Income in 2021

We know that the total income of A and B in 2021 is Rs. 7,05,600. We can set up an equation using the expressions for $A_{2021}$ and $B_{2021}$ in terms of $k$:

$A_{2021} + B_{2021} = 7,05,600$

$\frac{25}{4}k + 6k = 7,05,600$

Solving for the Variable k

To solve for $k$, we first combine the terms with $k$:

$\frac{25}{4}k + \frac{24}{4}k = 7,05,600$

$\frac{25+24}{4}k = 7,05,600$

$\frac{49}{4}k = 7,05,600$

Now, isolate $k$:

$k = \frac{7,05,600 \times 4}{49}$

To simplify the calculation, we can divide 7,05,600 by 49:

$705600 \div 49 = (7056 \times 100) \div 49$

$7056 \div 49 = 144$

So, $k = 144 \times 100 \times 4 = 14400 \times 4 = 57600$.

Thus, the value of $k$ is 57600.

Calculating B's Income in 2021

We need to find the income of B in 2021, which we expressed as $B_{2021} = 6k$. Now we substitute the value of $k$ we found:

$B_{2021} = 6 \times 57600$

$B_{2021} = 345600$

So, the income of B in 2021 was Rs. 3,45,600.

Verification (Optional)

Let's also calculate A's income in 2021 to verify the total:

$A_{2021} = \frac{25}{4}k = \frac{25}{4} \times 57600 = 25 \times (57600 \div 4) = 25 \times 14400$

$A_{2021} = 360000$

Total income in 2021 = $A_{2021} + B_{2021} = 360000 + 345600 = 705600$. This matches the given total income, confirming our calculations are correct.

Revision Table: Key Concepts in Ratio Problems

Additional Information: Solving Complex Ratio Problems

Ratio problems often involve multiple steps and require careful translation of the given information into mathematical equations. When dealing with incomes or other quantities changing over time or across different groups, it's helpful to use variables to represent the quantities in each specific context (like $A_{2020}$, $B_{2021}$).

Key strategies include:

• Assigning a common variable (like $k$) based on one of the initial ratios.
• Expressing all other unknown quantities in terms of this variable.
• Using any given total or difference to form an equation with the variable.
• Solving the equation to find the value of the variable.
• Substituting the variable's value back into the expressions for the specific quantity you need to find.

Fractional coefficients can arise when converting ratios across different contexts, as seen with the $\frac{25}{4}k$ for $A_{2021}$. Being comfortable with fraction arithmetic is essential for these types of problems.