The income of A is 45% more than the income of B and the income of C is 60% less than the sum of the incomes of A and B. The income of D is 20% more than that of C. If the difference between the incomes of B and D is Rs. 13200, then the income (in Rs.) of C is:

Question

Prepp Team · Accepted Answer

Solving Income Percentage Problems

This problem involves calculating the incomes of four individuals, A, B, C, and D, based on percentage relationships and a given difference in income between two of them. We need to find the income of C.

Step-by-Step Income Calculation

Let's denote the incomes of A, B, C, and D as $I_A$, $I_B$, $I_C$, and $I_D$ respectively.

We are given the following relationships:

The income of A is 45% more than the income of B.
The income of C is 60% less than the sum of the incomes of A and B.
The income of D is 20% more than that of C.
The difference between the incomes of B and D is Rs. 13200.

Expressing Incomes in Terms of One Variable

It's helpful to express all incomes in terms of a single variable. Let's use the income of B, $I_B$.

Income of A ($I_A$) in terms of $I_B$:
A's income is 45% more than B's income. $$I_A = I_B + 45\% \text{ of } I_B$$ $$I_A = I_B + 0.45 \times I_B$$ $$I_A = I_B (1 + 0.45)$$ $$I_A = 1.45 I_B$$
Sum of Incomes of A and B ($I_A + I_B$):
$$I_A + I_B = 1.45 I_B + I_B$$ $$I_A + I_B = (1.45 + 1) I_B$$ $$I_A + I_B = 2.45 I_B$$
Income of C ($I_C$) in terms of $I_B$:
C's income is 60% less than the sum of A and B's incomes. $$I_C = (I_A + I_B) - 60\% \text{ of } (I_A + I_B)$$ $$I_C = (I_A + I_B) (1 - 0.60)$$ $$I_C = 0.40 (I_A + I_B)$$ Substitute the sum $I_A + I_B = 2.45 I_B$: $$I_C = 0.40 \times 2.45 I_B$$ $$I_C = 0.98 I_B$$
Income of D ($I_D$) in terms of $I_C$ and then $I_B$:
D's income is 20% more than C's income. $$I_D = I_C + 20\% \text{ of } I_C$$ $$I_D = I_C (1 + 0.20)$$ $$I_D = 1.20 I_C$$ Substitute $I_C = 0.98 I_B$: $$I_D = 1.20 \times (0.98 I_B)$$ $$I_D = 1.176 I_B$$

Using the Income Difference to Find $I_B$

We are given that the difference between the incomes of B and D is Rs. 13200. Let's compare $I_B$ and $I_D$:

$I_B = I_B$
$I_D = 1.176 I_B$

Since $1.176 > 1$, $I_D$ is greater than $I_B$. The difference is $I_D - I_B$.

$$I_D - I_B = 13200$$ $$1.176 I_B - I_B = 13200$$ $$(1.176 - 1) I_B = 13200$$ $$0.176 I_B = 13200$$

Now, solve for $I_B$:

$$I_B = \frac{13200}{0.176}$$

To remove the decimal from the denominator, multiply both numerator and denominator by 1000:

$$I_B = \frac{13200 \times 1000}{0.176 \times 1000}$$ $$I_B = \frac{13200000}{176}$$

Let's perform the division:

$13200000 \div 176 = 75000$

So, the income of B is $I_B = 75000$ Rs.

Calculating the Income of C

The question asks for the income of C ($I_C$). We found the relationship $I_C = 0.98 I_B$.

$$I_C = 0.98 \times 75000$$ $$I_C = \frac{98}{100} \times 75000$$ $$I_C = 98 \times \frac{75000}{100}$$ $$I_C = 98 \times 750$$

Let's calculate $98 \times 750$:

$$98 \times 750 = (100 - 2) \times 750$$ $$= 100 \times 750 - 2 \times 750$$ $$= 75000 - 1500$$ $$= 73500$$

So, the income of C is $I_C = 73500$ Rs.

Summary of Incomes

Based on $I_B = 75000$:

Person	Income Calculation	Income (Rs.)
B	$I_B$	75000
A	$1.45 \times I_B = 1.45 \times 75000$	108750
C	$0.98 \times I_B = 0.98 \times 75000$	73500
D	$1.176 \times I_B = 1.176 \times 75000$	88200

Let's check the difference between D and B: $I_D - I_B = 88200 - 75000 = 13200$, which matches the given information.

The income of C is 73500 Rs.

Revision Table: Key Concepts in Income Problems

Concept	Explanation	Formula/Example
Percentage Increase	Adding a percentage of a value to the original value.	Original Value + (Percentage / 100) * Original Value or Original Value * (1 + Percentage / 100)
Percentage Decrease	Subtracting a percentage of a value from the original value.	Original Value - (Percentage / 100) * Original Value or Original Value * (1 - Percentage / 100)
Solving Equations	Using algebraic methods to find the value of an unknown variable based on given relationships.	If $ax = b$, then $x = b/a$.
Translating Word Problems	Converting written descriptions into mathematical expressions and equations.	"A is 45% more than B" translates to $A = B(1 + 0.45)$.

Additional Information: Working with Percentages

Understanding how to work with percentages is crucial for solving problems like this. A percentage is a fraction out of 100. For example, 45% means 45/100 or 0.45.

"X% more than Y" means $Y + (\frac{X}{100} \times Y) = Y(1 + \frac{X}{100})$.
"X% less than Y" means $Y - (\frac{X}{100} \times Y) = Y(1 - \frac{X}{100})$.

In this problem, we repeatedly used these concepts to set up the equations relating the incomes. It's important to be careful with calculations involving decimals or fractions.

Prepp Team · Answer

72000

Prepp Team · Answer

75000

Prepp Team · Answer

72500