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?
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Solving Age Ratio Problems
This problem involves finding the present age of a father based on given ratios of his age to his son's age at two different points in time: three years ago and four years from now.
Setting up the Age Problem Equations
Let's denote the present age of the father as \(F\) years and the present age of the son as \(S\) years.
According to the question, we have two conditions based on age ratios at different times:
- Condition 1: Three years ago
- Three years ago, the father's age was \(F-3\) years.
- Three years ago, the son's age was \(S-3\) years.
- The ratio of the father's age to the son's age three years ago was 8 ∶ 3.
- This gives us the equation:
\[ \frac{F-3}{S-3} = \frac{8}{3} \]
Cross-multiplying this equation, we get:
\[ 3(F-3) = 8(S-3) \]
\[ 3F - 9 = 8S - 24 \]
\[ 3F - 8S = -24 + 9 \]
\[ 3F - 8S = -15 \quad \text{(Equation 1)} \]
- Condition 2: After 4 years
- After 4 years, the father's age will be \(F+4\) years.
- After 4 years, the son's age will be \(S+4\) years.
- The ratio of their ages after 4 years will be 11 ∶ 5.
- This gives us the equation:
\[ \frac{F+4}{S+4} = \frac{11}{5} \]
Cross-multiplying this equation, we get:
\[ 5(F+4) = 11(S+4) \]
\[ 5F + 20 = 11S + 44 \]
\[ 5F - 11S = 44 - 20 \]
\[ 5F - 11S = 24 \quad \text{(Equation 2)} \]
Solving the System of Linear Equations
Now we have a system of two linear equations with two variables \(F\) and \(S\):
\[ 3F - 8S = -15 \quad \text{(Equation 1)} \]
\[ 5F - 11S = 24 \quad \text{(Equation 2)} \]
We can solve this system using the elimination method. To eliminate \(F\), we can multiply Equation 1 by 5 and Equation 2 by 3:
- Multiply Equation 1 by 5:
\[ 5 \times (3F - 8S) = 5 \times (-15) \]
\[ 15F - 40S = -75 \quad \text{(Equation 3)} \]
- Multiply Equation 2 by 3:
\[ 3 \times (5F - 11S) = 3 \times (24) \]
\[ 15F - 33S = 72 \quad \text{(Equation 4)} \]
Now, subtract Equation 3 from Equation 4:
\[ (15F - 33S) - (15F - 40S) = 72 - (-75) \]
\[ 15F - 33S - 15F + 40S = 72 + 75 \]
\[ 7S = 147 \]
Solving for \(S\):
\[ S = \frac{147}{7} \]
\[ S = 21 \]
The present age of the son is 21 years.
Now substitute the value of \(S\) (21) into either Equation 1 or Equation 2 to find \(F\). Let's use Equation 1:
\[ 3F - 8S = -15 \]
\[ 3F - 8(21) = -15 \]
\[ 3F - 168 = -15 \]
\[ 3F = -15 + 168 \]
\[ 3F = 153 \]
Solving for \(F\):
\[ F = \frac{153}{3} \]
\[ F = 51 \]
The present age of the father is 51 years.
Verification of the Ages
Let's verify the calculated present ages (Father = 51 years, Son = 21 years) with the given conditions:
- Three years ago:
- Father's age = 51 - 3 = 48 years
- Son's age = 21 - 3 = 18 years
- Ratio = \(\frac{48}{18}\). Dividing both by 6, we get \(\frac{8}{3}\). This matches the given ratio 8 ∶ 3.
- After 4 years:
- Father's age = 51 + 4 = 55 years
- Son's age = 21 + 4 = 25 years
- Ratio = \(\frac{55}{25}\). Dividing both by 5, we get \(\frac{11}{5}\). This matches the given ratio 11 ∶ 5.
The calculated ages satisfy both conditions.
Final Answer
The present age of the father is 51 years.
Revision Table: Age Problem Ratios
| Time Period | Father's Age | Son's Age | Ratio (Father : Son) | Equation |
|---|---|---|---|---|
| Present | \(F\) | \(S\) | - | - |
| Three years ago | \(F-3\) | \(S-3\) | 8 ∶ 3 | \(\frac{F-3}{S-3} = \frac{8}{3}\) or \(3F - 8S = -15\) |
| After 4 years | \(F+4\) | \(S+4\) | 11 ∶ 5 | \(\frac{F+4}{S+4} = \frac{11}{5}\) or \(5F - 11S = 24\) |
Additional Information: Solving System of Equations
Age problems often lead to systems of linear equations. A system of linear equations is a set of two or more linear equations involving the same variables. The solution to a system of two linear equations in two variables (\(x\) and \(y\)) is a pair of values (\(x\), \(y\)) that satisfies all equations in the system simultaneously.
Common methods to solve a system of linear equations include:
- Substitution Method: Solve one equation for one variable in terms of the other, and then substitute that expression into the other equation.
- Elimination Method: Multiply one or both equations by suitable numbers so that the coefficients of one variable become opposites. Then, add the equations to eliminate that variable and solve for the remaining one. This is the method used in the solution above.
- Graphical Method: Graph each equation on the same coordinate plane. The point where the lines intersect is the solution to the system.
- Matrix Method: Represent the system using matrices and use techniques like Cramer's rule or Gaussian elimination to solve.
Understanding these methods is crucial for solving various types of problems, including age-related word problems, which can be effectively modeled using linear equations.
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:
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?
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?
