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?
80 years
Solving Age Ratio Problems to Find Sum of Present Ages
This problem asks us to find the sum of the present ages of two people, A and B, given information about the ratio of their ages at two different points in time: four years ago and eight years from now. We can solve this by setting up equations based on the given ratios and solving them simultaneously.
Setting up the Equations for Age Ratios
Let the present age of A be $A_p$ years and the present age of B be $B_p$ years.
Information 1: Ratio four years ago
- Four years ago, A's age was $A_p - 4$.
- Four years ago, B's age was $B_p - 4$.
- The ratio of their ages four years ago was 4 : 5.
This gives us the equation:
$\frac{A_p - 4}{B_p - 4} = \frac{4}{5}$
Cross-multiplying, we get:
$5(A_p - 4) = 4(B_p - 4)$
$5A_p - 20 = 4B_p - 16$
$5A_p - 4B_p = -16 + 20$
$5A_p - 4B_p = 4$ (Equation 1)
Information 2: Ratio eight years from now
- Eight years from now, A's age will be $A_p + 8$.
- Eight years from now, B's age will be $B_p + 8$.
- The ratio of their ages eight years from now will be 11 : 13.
This gives us the equation:
$\frac{A_p + 8}{B_p + 8} = \frac{11}{13}$
Cross-multiplying, we get:
$13(A_p + 8) = 11(B_p + 8)$
$13A_p + 104 = 11B_p + 88$
$13A_p - 11B_p = 88 - 104$
$13A_p - 11B_p = -16$ (Equation 2)
Solving the System of Linear Equations
We now have a system of two linear equations with two variables:
- Equation 1: $5A_p - 4B_p = 4$
- Equation 2: $13A_p - 11B_p = -16$
We can solve this system using methods like substitution or elimination. Let's use the elimination method. Multiply Equation 1 by 11 and Equation 2 by 4 to make the coefficient of $B_p$ the same (with opposite signs if we were adding, but here we'll make them the same and subtract):
- $11 \times (5A_p - 4B_p = 4) \Rightarrow 55A_p - 44B_p = 44$ (Equation 3)
- $4 \times (13A_p - 11B_p = -16) \Rightarrow 52A_p - 44B_p = -64$ (Equation 4)
Now, subtract Equation 4 from Equation 3:
$(55A_p - 44B_p) - (52A_p - 44B_p) = 44 - (-64)$
$55A_p - 52A_p - 44B_p + 44B_p = 44 + 64$
$3A_p = 108$
$A_p = \frac{108}{3}$
$A_p = 36$
So, A's present age is 36 years.
Now substitute the value of $A_p$ into Equation 1 to find $B_p$:
$5(36) - 4B_p = 4$
$180 - 4B_p = 4$
$180 - 4 = 4B_p$
$176 = 4B_p$
$B_p = \frac{176}{4}$
$B_p = 44$
So, B's present age is 44 years.
Calculating the Sum of Present Ages
The question asks for the sum of their present ages, which is $A_p + B_p$.
Sum of present ages = $36 + 44 = 80$ years.
To verify, let's check the ratios:
- Four years ago: A was $36-4=32$, B was $44-4=40$. Ratio $32:40 = 4:5$. Correct.
- Eight years from now: A will be $36+8=44$, B will be $44+8=52$. Ratio $44:52 = 11:13$. Correct.
The calculated ages satisfy the conditions given in the problem.
| Time Period | A's Age | B's Age | Ratio (A:B) |
|---|---|---|---|
| 4 years ago | $A_p - 4 = 32$ | $B_p - 4 = 40$ | 32 : 40 = 4 : 5 |
| Present | $A_p = 36$ | $B_p = 44$ | 36 : 44 = 9 : 11 |
| 8 years from now | $A_p + 8 = 44$ | $B_p + 8 = 52$ | 44 : 52 = 11 : 13 |
The sum of their present ages is 80 years.
Revision Table: Key Concepts in Age Problems
| Concept | Explanation | Example |
|---|---|---|
| Present Age | The age at the current time. Let it be $P$. | If present age is 30, age 5 years ago was 30-5=25. |
| Age in the Past | Age $n$ years ago: Present Age $- n$. | If present age is $P$, age $n$ years ago is $P-n$. |
| Age in the Future | Age $n$ years from now: Present Age $+ n$. | If present age is $P$, age $n$ years from now is $P+n$. |
| Ratio of Ages | Expressing the relationship between two ages as a fraction. If ratio is $a:b$, then $\frac{\text{Age 1}}{\text{Age 2}} = \frac{a}{b}$. | If A:B ages are 4:5, then $\frac{\text{Age of A}}{\text{Age of B}} = \frac{4}{5}$. |
Additional Information: Solving Systems of Equations
Age problems often lead to systems of linear equations. Two common methods for solving them are:
- Substitution Method: Solve one equation for one variable, then substitute that expression into the other equation.
- Elimination Method: Multiply equations by constants so that the coefficients of one variable are opposites, then add the equations to eliminate that variable.
In this age ratio problem, we used the elimination method to find the present ages of A and B before calculating their sum.
Important Questions from Age
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 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:
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?
