The ratio of the present age of father to that of his son is 7 : 2. If after 10 years the ratio of their ages will become 9 : 4, then the present age of the father is:
35 years
Solving Father-Son Age Ratio Problem
This problem involves ratios of ages at different points in time. We are given the present ratio of the father's age to his son's age and the ratio after 10 years. We need to find the father's present age.
Setting up the Age Ratios
Let the present age of the father be \(F\) and the present age of the son be \(S\). The problem states that the ratio of their present ages is 7 : 2. We can represent their ages using a common multiple, say \(x\).
- Present age of Father = \(7x\) years
- Present age of Son = \(2x\) years
Ages After 10 Years
After 10 years, the age of the father will increase by 10 years, and the age of the son will also increase by 10 years.
- Father's age after 10 years = \(7x + 10\) years
- Son's age after 10 years = \(2x + 10\) years
The problem gives us the ratio of their ages after 10 years, which is 9 : 4.
\[ \frac{7x + 10}{2x + 10} = \frac{9}{4} \]Solving the Age Ratio Equation
Now, we need to solve this equation for \(x\). We can do this by cross-multiplying:
\[ 4 \times (7x + 10) = 9 \times (2x + 10) \]Distribute the numbers on both sides:
\[ 28x + 40 = 18x + 90 \]Now, gather the terms with \(x\) on one side and the constant terms on the other side. Subtract \(18x\) from both sides:
\[ 28x - 18x + 40 = 18x - 18x + 90 \] \[ 10x + 40 = 90 \]Subtract 40 from both sides:
\[ 10x + 40 - 40 = 90 - 40 \] \[ 10x = 50 \]Divide both sides by 10 to find the value of \(x\):
\[ x = \frac{50}{10} \] \[ x = 5 \]Calculating the Father's Present Age
We defined the father's present age as \(7x\). Now that we have the value of \(x\), we can calculate the father's present age:
\[ \text{Father's present age} = 7x = 7 \times 5 \] \[ \text{Father's present age} = 35 \text{ years} \]Verifying the Solution
Let's check if this value of \(x\) satisfies the condition for the ages after 10 years.
- Present age of Father = 35 years
- Present age of Son = \(2x = 2 \times 5 = 10\) years
Present ratio = 35 : 10, which simplifies to 7 : 2. This matches the given present ratio.
After 10 years:
- Father's age = \(35 + 10 = 45\) years
- Son's age = \(10 + 10 = 20\) years
Ratio after 10 years = 45 : 20. Divide both numbers by their greatest common divisor, which is 5:
\[ \frac{45}{5} : \frac{20}{5} = 9 : 4 \]This matches the given ratio after 10 years. So, our value for \(x\) is correct, and the father's present age is 35 years.
Revision Table: Key Steps for Age Ratio Problems
| Step | Description | Example (from this problem) |
|---|---|---|
| 1 | Represent present ages using variables and the given ratio (e.g., \(ax\) and \(bx\)). | Father's age = \(7x\), Son's age = \(2x\) |
| 2 | Write expressions for ages after/before a certain number of years. | Ages after 10 years: Father = \(7x + 10\), Son = \(2x + 10\) |
| 3 | Set up an equation using the ratio of the ages from Step 2. | \( \frac{7x + 10}{2x + 10} = \frac{9}{4} \) |
| 4 | Solve the equation for the variable (e.g., \(x\)) by cross-multiplication and algebraic manipulation. | \(10x = 50 \implies x = 5\) |
| 5 | Substitute the value of the variable back into the expressions for the present ages to find the required age(s). | Father's present age = \(7x = 7 \times 5 = 35\) |
Additional Information on Age Problems and Ratios
Age-related problems in mathematics often involve setting up equations based on given conditions about ages at different points in time. Ratios are frequently used to express the relationship between two or more ages. Here are some key points:
- Ratios: A ratio compares two quantities. If the ratio of A to B is \(a:b\), it means A can be written as \(ak\) and B as \(bk\) for some common factor \(k\). In age problems, this common factor is usually represented by a variable like \(x\).
- Changes in Age: When dealing with ages after or before a certain period, remember that everyone's age changes by the same amount. If 10 years pass, everyone's age increases by 10.
- Setting up Equations: The core of these problems is translating the word problem into algebraic equations. Look for keywords like "ratio," "after x years," "before y years," "sum of ages," "difference in ages."
- Solving Equations: Most age ratio problems lead to linear equations that can be solved using basic algebraic techniques like cross-multiplication, combining like terms, and isolating the variable.
Understanding how to represent ages using variables and how to set up and solve equations is crucial for tackling age-related questions effectively.
Important Questions from Age
The ratio of the ages of A and B, four years ago, was 4 : 5. Eight years from now, the ratio of the ages of A and B will be 11 : 13. What is the sum of their present ages?
The ratio of the present ages of A and B is 8 : 9. After 9 years, this ratio will become 19 : 21. C is 3 years younger to B. What is the present (in years) of C?
The ratio of the ages of A and B 8 years ago was 2 : 3. Four years ago, the ratio of their ages was 5 : 7. What will be the ratio of their ages 8 years from now?
The ages of two persons P and Q are in the ratio 5 : 7. Eight years ago, the ratio of P and Q was 7 : 13. The present ages of P and Q, respectively, are:
One year ago, the ratio of the ages of A and B was 4 : 3. The ratio of their ages, after 7 years from now, will be 9 : 7. What is the present age (in years) of B?
