The total of the ages of Niraj, Umesh and Dhiraj is 93 years. Ten years ago, the ratio of their ages was 2 : 3 : 4 respectively. What is the present age of Dhiraj ?
38 years
Solving Age Ratio Problems: Finding Dhiraj's Present Age
This problem involves finding the present age of one person (Dhiraj) given the total present age of three people and the ratio of their ages from a past time point.
Understanding the Given Information
- The total present age of Niraj, Umesh, and Dhiraj is 93 years.
- Ten years ago, the ratio of their ages was 2 : 3 : 4.
Calculating Total Age Ten Years Ago
If the total present age of the three people is 93 years, then ten years ago, each person was 10 years younger. So, their total age ten years ago would be:
\(\text{Total age 10 years ago} = \text{Present total age} - (\text{Number of people} \times \text{Years ago})\)
\(\text{Total age 10 years ago} = 93 - (3 \times 10)\)
\(\text{Total age 10 years ago} = 93 - 30\)
\(\text{Total age 10 years ago} = 63 \text{ years}\)
Using the Ratio to Find Ages Ten Years Ago
Let the ages of Niraj, Umesh, and Dhiraj ten years ago be \(2x\), \(3x\), and \(4x\) respectively, based on the given ratio 2 : 3 : 4.
The sum of their ages ten years ago is \(2x + 3x + 4x\).
We know this total is 63 years. So, we can set up the equation:
\(2x + 3x + 4x = 63\)
\(9x = 63\)
Solving for x
To find the value of \(x\), divide the total age ten years ago by the sum of the ratio parts:
\(x = \frac{63}{9}\)
\(x = 7\)
Finding Ages Ten Years Ago
Now we can find their individual ages ten years ago using the value of \(x=7\):
- Niraj's age 10 years ago = \(2x = 2 \times 7 = 14\) years
- Umesh's age 10 years ago = \(3x = 3 \times 7 = 21\) years
- Dhiraj's age 10 years ago = \(4x = 4 \times 7 = 28\) years
Let's check: \(14 + 21 + 28 = 63\) years, which matches the total age ten years ago.
Calculating Present Ages
To find their present ages, we add 10 years to their ages from ten years ago:
- Niraj's present age = \(14 + 10 = 24\) years
- Umesh's present age = \(21 + 10 = 31\) years
- Dhiraj's present age = \(28 + 10 = 38\) years
Let's check the total present age: \(24 + 31 + 38 = 93\) years, which matches the given total present age.
Identifying Dhiraj's Present Age
Based on our calculations, Dhiraj's present age is 38 years.
Summary of Ages
| Person | Age 10 Years Ago | Present Age |
|---|---|---|
| Niraj | 14 | 24 |
| Umesh | 21 | 31 |
| Dhiraj | 28 | 38 |
The present age of Dhiraj is 38 years.
Important Questions from Problem on Age
The ratio of present ages of Neelam and Rajni is 6 : 7. After four years, their ages will be in the ratio of 8 : 9. What is the present age of Rajni?
Five years ago the age of the son was one third of that of his mother at that time. If the sum of their present ages is 70 years, then find the present age of the mother.
Jaya is 36 years old and her son Bharat is 11 years old. In how many years will Jaya be twice as Bharat's age?
The ratio of a father's age to his son's age is 3 ∶ 2 The product of the numbers representing their age is 486. The ratio of their ages after 5 years will be:
X said to Y, "At the time of your birth I was twice as old as you are at present." If the present age of X is 42 years, then consider the following statements:
1. 8 years ago, the age of X was five times the age of Y.
2. After 14 years, the age of X would be two times the age of Y.
Which of the above statements is/are correct?
