The sum of the presents age of a father and son is 52 years Four years hence, the son's age will be 1/4 that of the father. What will be the ratio of the age of the son and father, 10 years from now?
1 : 3
Solving the Father-Son Age Ratio Problem
This problem involves finding the current ages of a father and son based on given conditions and then determining the ratio of their ages in the future.
Let's break down the problem and solve it step-by-step.
Defining Variables for Ages
We start by assigning variables to represent the current ages:
- Let \(F\) be the present age of the father.
- Let \(S\) be the present age of the son.
Setting Up Equations from the Given Information
The problem provides two main pieces of information, which we can translate into mathematical equations:
- The sum of their present ages is 52 years. $$F + S = 52 \quad \text{(Equation 1)}$$
- Four years from now, the son's age will be \(\frac{1}{4}\) that of the father's age.
- Four years from now, the father's age will be \(F + 4\).
- Four years from now, the son's age will be \(S + 4\).
Solving for Present Ages
Now, we solve the system of equations to find the values of \(F\) and \(S\).
From Equation 1, we can express \(F\) in terms of \(S\):
$$F = 52 - S$$Substitute this expression for \(F\) into Equation 2:
$$S + 4 = \frac{1}{4}((52 - S) + 4)$$ $$S + 4 = \frac{1}{4}(56 - S)$$Multiply both sides by 4 to eliminate the fraction:
$$4(S + 4) = 56 - S$$ $$4S + 16 = 56 - S$$Collect terms with \(S\) on one side and constants on the other:
$$4S + S = 56 - 16$$ $$5S = 40$$Solve for \(S\):
$$S = \frac{40}{5}$$ $$S = 8$$So, the present age of the son is 8 years.
Now, substitute the value of \(S\) back into Equation 1 (\(F = 52 - S\)) to find the father's present age:
$$F = 52 - 8$$ $$F = 44$$The present age of the father is 44 years.
Calculating Ages 10 Years from Now
We need to find the ratio of their ages 10 years from now.
- Father's age 10 years from now will be \(F + 10\). $$44 + 10 = 54 \text{ years}$$
- Son's age 10 years from now will be \(S + 10\). $$8 + 10 = 18 \text{ years}$$
Determining the Ratio of Ages
The question asks for the ratio of the age of the son and father 10 years from now. This means the ratio is Son's Age : Father's Age.
Ratio = \((S + 10) : (F + 10)\)
Ratio = \(18 : 54\)
To simplify the ratio, find the greatest common divisor (GCD) of 18 and 54. The GCD is 18.
Divide both parts of the ratio by 18:
$$\frac{18}{18} : \frac{54}{18}$$ $$1 : 3$$The ratio of the age of the son and father 10 years from now is \(1 : 3\).
Summary of Ages
| Person | Present Age | Age in 4 Years | Age in 10 Years |
|---|---|---|---|
| Father | 44 | 48 | 54 |
| Son | 8 | 12 | 18 |
Check the condition for 4 years hence: Son's age (12) is \(1/4\) of Father's age (48), since \(12 = \frac{1}{4} \times 48\). This matches the problem statement.
The ratio of son's age to father's age in 10 years is \(18 : 54\), which simplifies to \(1 : 3\).
Revision Table: Key Concepts
| Concept | Explanation | Application in this Problem |
|---|---|---|
| Setting up Equations | Translate word problems into algebraic equations using variables. | Representing sum of ages and future age relationship. |
| Solving System of Equations | Finding variable values that satisfy multiple equations simultaneously (e.g., substitution method). | Finding the present ages (F and S). |
| Calculating Future/Past Ages | Add/subtract years to the present age. | Calculating ages after 4 years and 10 years. |
| Ratio | Comparing two quantities by division. A:B means A/B. | Finding the simplified ratio of son's age to father's age. |
Additional Information: Age Problems and Ratios
Age-based problems are common in algebra. They often involve setting up equations based on relationships between ages at different points in time (past, present, future). Here are some tips for solving them:
- Always define your variables clearly, usually representing present ages.
- Carefully read the problem to understand if the age relationship is for the present, past, or future.
- When dealing with past or future ages, remember to add or subtract the same number of years from everyone's age.
- Ratios can be simplified by dividing both parts by their greatest common divisor.
Understanding how to translate word problems into equations is a fundamental skill. Practice with different scenarios involving sums, differences, products, and ratios of ages.
Important Questions from Problem on Age
The ratio of present ages of A and B is 7 : 8. After 6 years from now, the ratio of their ages will be 8 : 9. If C's present age is 10 years more than the present age of A, then the present age (in years) of C is:
Three years ago, the ratio of the age of father to that of his son was 8 ∶ 3. After 4 years, their ages will be in the ratio 11 ∶ 5. What is the present age (in years) of the father?
At present, A is younger than B by 8 years. If 4 years ago, their ages were in the ratio 1 ∶ 2, then what is the present age of B (in years)?
Eight years ago, the ratio of ages of A and B was 5 ∶ 4. The ratio of their present ages is 6 ∶ 5. What will be the sum (in years) of the ages of A and B after 7 years from now?
Ratio of the present age of a mother to that of the daughter is 7 ∶ 1. After 5 years the ratio will become 4 ∶ 1. What is the difference (in years) in their present ages?
