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
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:
In an examination, Ram obtained 20 % more than Ashok but 10% less than Rajesh. If the marks obtained by Ashok is 1080. Then the Percentage marks obtained by Rajesh if the full marks is 2000 ;
