The salaries of Kabir and Vicky are in the ratio 5 ∶ 2 . If the increments of 20% and 12% are allowed respectively in their salaries, then what will be new ratio of their salaries?
75 ∶ 28
Understanding the Salary Ratio Problem
The problem asks us to find the new ratio of the salaries of Kabir and Vicky after they receive different percentage increments. Initially, their salaries are in the ratio 5:2. Kabir receives a 20% increment, and Vicky receives a 12% increment.
Step-by-Step Calculation of New Salaries
Let the initial salary of Kabir be \(5x\) and the initial salary of Vicky be \(2x\), where \(x\) is a common factor.
Initial Salaries
- Kabir's initial salary: \(5x\)
- Vicky's initial salary: \(2x\)
Calculating Increments
An increment is an increase in salary. A percentage increment means the increase is a certain percentage of the original salary.
- Kabir's salary increase: 20% of \(5x\)
- Vicky's salary increase: 12% of \(2x\)
New Salary of Kabir
Kabir's new salary is his initial salary plus the increment.
\(\text{New Salary of Kabir} = \text{Initial Salary} + \text{Increment}\)
\(\text{New Salary of Kabir} = 5x + 20\%\text{ of } 5x\)
\(\text{New Salary of Kabir} = 5x + \frac{20}{100} \times 5x\)
\(\text{New Salary of Kabir} = 5x + \frac{1}{5} \times 5x\)
\(\text{New Salary of Kabir} = 5x + x\)
\(\text{New Salary of Kabir} = 6x\)
Alternatively, an increase of 20% means the new salary is 100% + 20% = 120% of the original salary.
\(\text{New Salary of Kabir} = 120\%\text{ of } 5x\)
\(\text{New Salary of Kabir} = \frac{120}{100} \times 5x\)
\(\text{New Salary of Kabir} = 1.20 \times 5x\)
\(\text{New Salary of Kabir} = 6x\)
New Salary of Vicky
Vicky's new salary is her initial salary plus the increment.
\(\text{New Salary of Vicky} = \text{Initial Salary} + \text{Increment}\)
\(\text{New Salary of Vicky} = 2x + 12\%\text{ of } 2x\)
\(\text{New Salary of Vicky} = 2x + \frac{12}{100} \times 2x\)
\(\text{New Salary of Vicky} = 2x + 0.12 \times 2x\)
\(\text{New Salary of Vicky} = 2x + 0.24x\)
\(\text{New Salary of Vicky} = 2.24x\)
Alternatively, an increase of 12% means the new salary is 100% + 12% = 112% of the original salary.
\(\text{New Salary of Vicky} = 112\%\text{ of } 2x\)
\(\text{New Salary of Vicky} = \frac{112}{100} \times 2x\)
\(\text{New Salary of Vicky} = 1.12 \times 2x\)
\(\text{New Salary of Vicky} = 2.24x\)
Finding the New Ratio of Salaries
The new ratio of Kabir's salary to Vicky's salary is the ratio of their new salaries.
\(\text{New Ratio} = \frac{\text{New Salary of Kabir}}{\text{New Salary of Vicky}}\)
\(\text{New Ratio} = \frac{6x}{2.24x}\)
We can cancel out the common factor \(x\):
\(\text{New Ratio} = \frac{6}{2.24}\)
To express this as a ratio of whole numbers, we can multiply the numerator and denominator by 100 to remove the decimal:
\(\text{New Ratio} = \frac{6 \times 100}{2.24 \times 100} = \frac{600}{224}\)
Now, simplify the fraction \(\frac{600}{224}\) by dividing both the numerator and denominator by their greatest common divisor. We can do this in steps:
- Divide by 4: \(\frac{600 \div 4}{224 \div 4} = \frac{150}{56}\)
- Divide by 2: \(\frac{150 \div 2}{56 \div 2} = \frac{75}{28}\)
The simplified fraction is \(\frac{75}{28}\).
Final New Salary Ratio
The new ratio of Kabir's salary to Vicky's salary is 75 : 28.
Revision Table: Key Concepts in Salary Ratios and Percentages
| Concept | Explanation | Example |
|---|---|---|
| Ratio | A comparison of two quantities. Written as a:b or a/b. | Salaries in ratio 5:2 means for every $5 Kabir earns, Vicky earns $2. |
| Percentage Increment | An increase calculated as a percentage of the original value. | 20% increment on $100 is an increase of $20 (20/100 * 100). |
| Calculating Value after % Increase | Original Value × (1 + Percentage Increase / 100) | $100 increased by 20% is $100 × (1 + 20/100) = $100 × 1.20 = $120. |
Additional Information: Related Ratio and Percentage Problems
Problems involving ratios and percentages are common in mathematics and real-life applications. You might encounter variations such as:
- Calculating the original value before an increase or decrease.
- Finding the percentage change between two values.
- Sharing quantities in a given ratio.
- Solving problems involving consecutive percentage changes.
Understanding how to work with ratios and percentages, especially calculating values after a percentage increase or decrease, is fundamental for solving these types of questions.
Important Questions from Percentage
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The price of cooking oil increased by 25%. Find by how much percentage a family must reduce its consumption in order to maintain the same budget.
The population of a city increased by 30% in the first year and decreased by 15% in the next year. If the present population is 11,050 then population 2 years ago was:
The income of A is 30% less than the income of B and the income of B is 137.5% more than that of C. If the income of A is Rs. 28500 less than that of B, then the income (in Rs.) of C is:
