P’s salary is 20% lower than Q’s salary which is 20% lower than R’s salary. By how much
percent is R’s salary more than P’s salary?
56.25%
Understanding Salary Relationships with Percentages
This question asks us to compare the salaries of three people, P, Q, and R, based on given percentage relationships. We are told that P's salary is 20% lower than Q's, and Q's salary is 20% lower than R's. We need to find out by what percentage R's salary is more than P's salary.
Setting Up the Salary Calculation
Let's use variables to represent the salaries:
- Let R's salary be \(S_R\).
- Q's salary is 20% lower than R's salary. This means Q's salary is \(100\% - 20\% = 80\%\) of R's salary.
So, \(S_Q = 80\%\) of \(S_R = \frac{80}{100} \times S_R = 0.80 \times S_R\). - P's salary is 20% lower than Q's salary. This means P's salary is \(100\% - 20\% = 80\%\) of Q's salary.
So, \(S_P = 80\%\) of \(S_Q = \frac{80}{100} \times S_Q = 0.80 \times S_Q\).
Calculating P's Salary in Terms of R's Salary
Now we need to find the relationship between P's salary (\(S_P\)) and R's salary (\(S_R\)). We can substitute the expression for \(S_Q\) into the equation for \(S_P\):
\(S_P = 0.80 \times S_Q\)
Since \(S_Q = 0.80 \times S_R\), we have:
\(S_P = 0.80 \times (0.80 \times S_R)\)
\(S_P = (0.80 \times 0.80) \times S_R\)
\(S_P = 0.64 \times S_R\)
This tells us that P's salary is 0.64 times, or 64%, of R's salary.
Finding the Percentage Increase from P's Salary to R's Salary
The question asks by how much percent R's salary is more than P's salary. To find this, we calculate the difference between R's salary and P's salary, and then express this difference as a percentage of P's salary.
Difference in salary: \(S_R - S_P\)
We know \(S_P = 0.64 \times S_R\), so the difference is:
\(S_R - 0.64 \times S_R = (1 - 0.64) \times S_R = 0.36 \times S_R\)
The percentage increase of R's salary compared to P's salary is calculated as:
\(\text{Percentage Increase} = \frac{\text{Difference}}{\text{P's Salary}} \times 100\%\)
\(\text{Percentage Increase} = \frac{S_R - S_P}{S_P} \times 100\%\)
Substitute the expressions in terms of \(S_R\):
\(\text{Percentage Increase} = \frac{0.36 \times S_R}{0.64 \times S_R} \times 100\%\)
The \(S_R\) terms cancel out:
\(\text{Percentage Increase} = \frac{0.36}{0.64} \times 100\%\)
To simplify the fraction \(\frac{0.36}{0.64}\), we can write it as \(\frac{36}{64}\). Both numbers are divisible by 4:
\(\frac{36 \div 4}{64 \div 4} = \frac{9}{16}\)
Now, we convert this fraction to a percentage:
\(\frac{9}{16} \times 100\%\)
We can perform the division and multiplication:
\(9 \div 16 = 0.5625\)
\(0.5625 \times 100\% = 56.25\%\)
So, R's salary is 56.25% more than P's salary.
Comparing with Options
Let's check our calculated percentage increase against the given options:
| Option | Percentage |
|---|---|
| 1 | 48.75% |
| 2 | 60.50% |
| 3 | 62.25% |
| 4 | 56.25% |
Our calculated value, 56.25%, matches Option 4.
Percentage Increase Calculation Steps
- Define variables for salaries (\(S_P, S_Q, S_R\)).
- Express \(S_Q\) based on \(S_R\): \(S_Q = 0.80 S_R\).
- Express \(S_P\) based on \(S_Q\): \(S_P = 0.80 S_Q\).
- Substitute \(S_Q\) into the equation for \(S_P\) to find \(S_P\) in terms of \(S_R\): \(S_P = 0.64 S_R\).
- Calculate the difference \(S_R - S_P\).
- Calculate the percentage increase: \(\frac{S_R - S_P}{S_P} \times 100\%\).
- Substitute the expressions and simplify: \(\frac{0.36 S_R}{0.64 S_R} \times 100\% = \frac{0.36}{0.64} \times 100\%\).
- Convert the fraction to a decimal and then a percentage: \(\frac{9}{16} \times 100\% = 0.5625 \times 100\% = 56.25\%\).
Revision Table: Key Concepts in Percentage Calculations
| Concept | Description | Formula Example |
|---|---|---|
| Percentage Decrease | Reducing a quantity by a percentage. If a quantity is decreased by x%, the new quantity is (100-x)% of the original. | New Value = Original Value \(\times (1 - \frac{x}{100})\) |
| Percentage Increase | Increasing a quantity by a percentage. If a quantity is increased by x%, the new quantity is (100+x)% of the original. | New Value = Original Value \(\times (1 + \frac{x}{100})\) |
| Finding Percentage Difference | Calculating how much one quantity is more or less than another, expressed as a percentage of the base quantity. | Percentage Difference = \(\frac{| \text{Value}_1 - \text{Value}_2 |}{\text{Base Value}} \times 100\%\) |
Additional Information on Percentage Word Problems
When solving percentage word problems, it's important to correctly identify the base value for the percentage calculation. In this problem, the initial decrease for P's salary is based on Q's salary, and the decrease for Q's salary is based on R's salary. However, the final question asks for the percentage increase of R's salary *more than* P's salary, which means the base for the final percentage calculation is P's salary.
A common mistake is to assume that consecutive percentage decreases (or increases) can be simply added or subtracted. A 20% decrease followed by another 20% decrease does not result in a 40% decrease overall from the original amount. Each percentage change is applied to the new, current value.
In this case, Q's salary is 80% of R's, and P's salary is 80% of Q's. So, P's salary is 80% of (80% of R's salary), which is \(0.80 \times 0.80 = 0.64\) or 64% of R's salary. The percentage increase from P to R is then calculated relative to P's salary.
Important Questions from Miscellaneous Topics
Which one of the following statements best reflects the critical message conveyed by the author of the passage?
With reference to the above passage, the following assumptions have been made:
I. No country needs to depend on ecosystems to boost national income.
II. Resource-rich countries need to share their resources with those of scant resources so as to prevent the degradation of ecosystems.
Which of the above assumptions is/are valid?Which one of the following statements best reflects the central idea of the passage?
With reference to the above passage, the following assumptions have been made:
I. Path-dependent green investments will eventually most likely benefit growth as well as public finances in a country like India.
II. If other green technologies follow the same pattern as that of solar energy, there will most likely be an easy green transition.
Which of the above assumptions is/are valid?Three prime numbers p, q and r, each less than 20, are such that p − q = q − r. How many distinct possible values can we get for (p + q + r)?
