Mohan's income is 40% more than Shyam's income. Shyam's income is what percentage less than Mohan's income?
Understanding the Income Comparison Problem
The question asks us to compare the incomes of Mohan and Shyam. We are given that Mohan's income is 40% more than Shyam's income. We need to find out by what percentage Shyam's income is less than Mohan's income.
This is a common type of percentage problem involving relative changes. If quantity A is P% more than quantity B, then quantity B is not P% less than quantity A. We need to calculate the percentage difference based on the second quantity (Mohan's income in this case).
Step-by-Step Calculation of Percentage Difference
Let's break down the problem into simple steps. A good way to approach this is to assume a base value for Shyam's income.
Step 1: Assume Shyam's Income
Let's assume Shyam's income is \(100\). This makes the calculation easier.
- Shyam's income \( = 100 \) units
Step 2: Calculate Mohan's Income
Mohan's income is 40% more than Shyam's income.
- Increase in income \( = 40\% \text{ of Shyam's income} \)
- Increase in income \( = 40\% \text{ of } 100 \)
- Increase in income \( = \frac{40}{100} \times 100 = 40 \) units
- Mohan's income \( = \text{Shyam's income} + \text{Increase} \)
- Mohan's income \( = 100 + 40 = 140 \) units
So, if Shyam's income is 100, Mohan's income is 140.
Step 3: Find the Difference in Incomes
Now, let's find the difference between Mohan's income and Shyam's income.
- Difference \( = \text{Mohan's income} - \text{Shyam's income} \)
- Difference \( = 140 - 100 = 40 \) units
The difference is 40 units.
Step 4: Calculate the Percentage Less
We need to find what percentage Shyam's income is less than Mohan's income. This means we need to calculate the difference (40) as a percentage of Mohan's income (140).
The formula for percentage less is:
\[ \text{Percentage Less} = \left( \frac{\text{Difference}}{\text{Mohan's Income}} \right) \times 100\% \]Substituting the values:
\[ \text{Percentage Less} = \left( \frac{40}{140} \right) \times 100\% \]Simplify the fraction:
\[ \text{Percentage Less} = \left( \frac{4}{14} \right) \times 100\% = \left( \frac{2}{7} \right) \times 100\% \] \[ \text{Percentage Less} = \frac{200}{7}\% \]Step 5: Convert to a Mixed Fraction
To match the options, we need to convert the improper fraction \( \frac{200}{7} \) into a mixed fraction.
Divide 200 by 7:
- \( 200 \div 7 \)
- \( 7 \times 28 = 196 \)
- Remainder \( = 200 - 196 = 4 \)
So, \( \frac{200}{7} \) as a mixed fraction is \( 28\frac{4}{7} \).
\[ \frac{200}{7}\% = 28\frac{4}{7}\% \]Therefore, Shyam's income is \( 28\frac{4}{7}\% \) less than Mohan's income.
Summary of the Income Calculation
Here is a quick summary of the steps:
| Assumption | Shyam's Income = 100 |
|---|---|
| Mohan's Income | \( 100 + 40\% \text{ of } 100 = 140 \) |
| Income Difference | \( 140 - 100 = 40 \) |
| Percentage Less | \( \left( \frac{40}{140} \right) \times 100\% \) |
| Result | \( \frac{200}{7}\% = 28\frac{4}{7}\% \) |
Revision Table: Key Concepts for Income Percentage Problems
| Concept | Explanation | Formula/Example |
|---|---|---|
| Percentage Increase | When a quantity increases by a certain percentage of the original quantity. | If A is P% more than B, \( A = B + \frac{P}{100}B = B\left(1 + \frac{P}{100}\right) \) |
| Percentage Decrease | When a quantity decreases by a certain percentage of the original quantity. | If A is P% less than B, \( A = B - \frac{P}{100}B = B\left(1 - \frac{P}{100}\right) \) |
| Percentage Change Formula | \( \text{Percentage Change} = \left( \frac{\text{Change}}{\text{Original Value}} \right) \times 100\% \) | Used to find how much a value has increased or decreased relative to a starting point. |
| Relative Percentage | Calculating percentage change relative to a different value than the original base. | If A is P% more than B, B is \( \left( \frac{\text{A-B}}{\text{A}} \right) \times 100\% \) less than A. |
Additional Information on Percentage Calculations
When dealing with percentage increase and decrease problems, it's crucial to identify the base value correctly for the calculation. In the question, "Mohan's income is 40% more than Shyam's income" uses Shyam's income as the base for the 40% increase. However, when asking "Shyam's income is what percentage less than Mohan's income", Mohan's income becomes the new base for calculating the percentage decrease.
Let's consider the relationship using fractions:
- Mohan's income is 40% more than Shyam's. \( 40\% = \frac{40}{100} = \frac{2}{5} \).
- If Shyam's income is \( 5 \) units, the increase is \( 2 \) units.
- Mohan's income is \( 5 + 2 = 7 \) units.
- Shyam's income is \( 5 \) units, and Mohan's is \( 7 \) units.
- The difference is \( 7 - 5 = 2 \) units.
- Shyam's income is less than Mohan's by this difference (2 units).
- To find the percentage less than Mohan's income, we use Mohan's income as the base: \( \left( \frac{2}{7} \right) \times 100\% = \frac{200}{7}\% = 28\frac{4}{7}\% \).
This fractional approach confirms the result obtained by assuming a base of 100 and provides another way to solve such percentage problems.
Important Questions from Percentage
A batsman scored 160 runs in a cricket match. He hit 24 fours and 8 sixes during his innings. What percentage of his runs were in boundaries?
In an examination, 25% of the candidates failed in Mathematics and 12% failed in English. If 10% of the candidates failed in both the subjects and 292 candidates passed in both the subjects, which one of the following is the number of total candidates appeared in the examination?
What is the value of 9% of 5500 + 2.4% of 1100 - 40% of 1600?
Population of a village is 7960 in which 4660 are female. If in that village 60% are literate in which 70% female are literate, then what is the number of literate male ?
The numbers of students of three classes of a school are in the ratio 4 : 5 : 6. If numbers of students in these classes increase by 25%, 20% and 25% respectively, then ratio of numbers of students will become:
