The ratio of the present age of Mahesh and Ajay is 3 : 2 respectively. After 8 years. Ratio of their age will be 11: 8. What will be the present age of Mahesh’s son if his age is half of the present age of Ajay?
12 years
Solving Age Ratio Problems: Finding Present Ages
This problem involves finding the present ages of two individuals based on their current age ratio and their age ratio after a certain number of years. We then use one of their present ages to find the age of a third person.
Understanding the Problem Setup
We are given two pieces of information regarding the ages of Mahesh and Ajay:
- Their present age ratio is 3:2.
- After 8 years, their age ratio will be 11:8.
We are also told that the present age of Mahesh's son is half the present age of Ajay.
Our goal is to determine the present age of Mahesh's son.
Step-by-Step Solution
Step 1: Represent Present Ages using the Ratio
Let the present age of Mahesh be \(3x\) years and the present age of Ajay be \(2x\) years, where \(x\) is a common multiplying factor for the ratio.
Step 2: Express Ages After 8 Years
After 8 years:
- Mahesh's age will be \(3x + 8\) years.
- Ajay's age will be \(2x + 8\) years.
Step 3: Set up an Equation based on the Future Ratio
We are given that the ratio of their ages after 8 years will be 11:8. We can write this as an equation:
\[ \frac{3x + 8}{2x + 8} = \frac{11}{8} \]Step 4: Solve the Equation for \(x\)
To solve for \(x\), we can cross-multiply:
\[ 8(3x + 8) = 11(2x + 8) \]Distribute the numbers on both sides:
\[ 24x + 64 = 22x + 88 \]Now, gather the terms with \(x\) on one side and constant terms on the other side:
\[ 24x - 22x = 88 - 64 \] \[ 2x = 24 \]Divide by 2 to find the value of \(x\):
\[ x = \frac{24}{2} \] \[ x = 12 \]Step 5: Calculate the Present Ages of Mahesh and Ajay
Using the value of \(x = 12\), we can find their present ages:
- Present age of Mahesh = \(3x = 3 \times 12 = 36\) years.
- Present age of Ajay = \(2x = 2 \times 12 = 24\) years.
Step 6: Calculate the Present Age of Mahesh’s Son
The problem states that Mahesh's son's present age is half of Ajay's present age.
Present age of Ajay is 24 years.
Present age of Mahesh's son = \(\frac{1}{2} \times\) Present age of Ajay
Present age of Mahesh's son = \(\frac{1}{2} \times 24\)
Present age of Mahesh's son = 12 years.
Summary of Present Ages
| Person | Present Age |
|---|---|
| Mahesh | 36 years |
| Ajay | 24 years |
| Mahesh’s Son | 12 years |
Therefore, the present age of Mahesh's son is 12 years.
Revision Table: Key Concepts in Age Problems
| Concept | Explanation |
|---|---|
| Representing Ages with Ratios | If a ratio is \(a:b\), ages can be represented as \(ax\) and \(bx\). |
| Ages After/Before Years | After \(y\) years: add \(y\) to current age. Before \(y\) years: subtract \(y\) from current age. |
| Setting up Equations | Use the given information about ratios at different times to form algebraic equations. |
| Solving Linear Equations | Basic algebraic techniques (cross-multiplication, combining like terms) are used to find the unknown variable (\(x\)). |
Additional Information: Ratio and Proportion
Ratio is a comparison of two quantities by division. For example, the ratio of Mahesh's age to Ajay's age is 3:2, meaning for every 3 years of Mahesh's age, Ajay is 2 years old at present.
Proportion is an equation stating that two ratios are equal. In this problem, we used the proportion \(\frac{3x + 8}{2x + 8} = \frac{11}{8}\) to relate the ratio of their ages after 8 years to the given future ratio.
Understanding how to represent unknown quantities using ratios and setting up proportions is fundamental to solving age-related word problems.
Important Questions from Problem on Age
The average age of a husband and his wife was 20 years at the time of their marriage. After 6 years, they have a 2 -year old child. Find the present average age of the family.
The ratio of the ages of A, B and C, 5 years ago, was 4 : 5 : 7. The sum of their present ages is 135 years. What will be the sum of the ages (in years) of B and C, 3 years from now?
In a school, the average age of boys and girls together is 16.8 years, the average age of boys is 15.4 years, and the average age of girls is 18.2 years. The ratio of number of boys to girls in the school is:
The difference between age of Sunita and Sheela is 12 years. If 9 years ago, elder's age was 4 times of younger's age, then what are their present age?
A. 11 and 23
B. 15 and 27
C. 13 and 25
D. 23 and 35
The ratio of alpha and beta to age is 2 ∶ 5. If the sum of their ages is 238, then find the difference between their age.
