Solving the Age Ratio Problem: Finding the Sum of Present Ages
This problem involves finding the sum of the present ages of two individuals, A and B, given information about the ratio of their ages at two different points in time: four years ago and six years from now.
Understanding the Given Information
- Past Ratio: Four years ago, the ratio of A's age to B's age was 3:5.
- Future Ratio: Six years from the present, the ratio of A's age to B's age will be 2:3.
- Objective: Calculate the sum of A's and B's current ages.
Setting up Algebraic Equations
Let the present age of A be represented by $A$ years and the present age of B be represented by $B$ years.
- Based on the past ratio (4 years ago):
- A's age four years ago was $A - 4$.
- B's age four years ago was $B - 4$.
- The ratio is given as 3:5. So, we can write the equation: $$ \frac{A - 4}{B - 4} = \frac{3}{5} $$
- Based on the future ratio (6 years hence):
- A's age six years hence will be $A + 6$.
- B's age six years hence will be $B + 6$.
- The ratio is given as 2:3. So, we can write the equation: $$ \frac{A + 6}{B + 6} = \frac{2}{3} $$
Solving the System of Equations
Now, we need to solve these two equations to find the values of $A$ and $B$.
Equation 1 Derivation:
From $\frac{A - 4}{B - 4} = \frac{3}{5}$, cross-multiply:
$$ 5(A - 4) = 3(B - 4) $$ $$ 5A - 20 = 3B - 12 $$Rearrange to form a linear equation:
$$ 5A - 3B = 20 - 12 $$ $$ 5A - 3B = 8 \quad \textbf{(Equation 1)} $$Equation 2 Derivation:
From $\frac{A + 6}{B + 6} = \frac{2}{3}$, cross-multiply:
$$ 3(A + 6) = 2(B + 6) $$ $$ 3A + 18 = 2B + 12 $$Rearrange to form a linear equation:
$$ 3A - 2B = 12 - 18 $$ $$ 3A - 2B = -6 \quad \textbf{(Equation 2)} $$Finding the Values of A and B:
We can use the method of elimination or substitution. Let's use elimination.
Multiply Equation 1 by 2 and Equation 2 by 3 to make the coefficients of $B$ the same:
$$ 2 \times (5A - 3B = 8) \implies 10A - 6B = 16 $$ $$ 3 \times (3A - 2B = -6) \implies 9A - 6B = -18 $$Now, subtract the second modified equation from the first modified equation:
$$ (10A - 6B) - (9A - 6B) = 16 - (-18) $$ $$ 10A - 9A - 6B + 6B = 16 + 18 $$ $$ A = 34 $$Now substitute the value of $A$ (34) into Equation 2 to find $B$:
$$ 3A - 2B = -6 $$ $$ 3(34) - 2B = -6 $$ $$ 102 - 2B = -6 $$ $$ 102 + 6 = 2B $$ $$ 108 = 2B $$ $$ B = \frac{108}{2} $$ $$ B = 54 $$Calculating the Sum of Present Ages
We found the present age of A is 34 years and the present age of B is 54 years.
The sum of their present ages is:
$$ \text{Sum} = A + B $$ $$ \text{Sum} = 34 + 54 $$ $$ \text{Sum} = 88 $$Verification
Let's check if these ages satisfy the initial conditions:
- Four years ago: A was $34 - 4 = 30$, B was $54 - 4 = 50$. Ratio = $30:50 = 3:5$. (Correct)
- Six years hence: A will be $34 + 6 = 40$, B will be $54 + 6 = 60$. Ratio = $40:60 = 2:3$. (Correct)
Both conditions are met.
Conclusion
The sum of the present ages of A and B is 88 years.
Similar Questions
- The average age of 4 members of a family is 35 years. Their ages are in the ratio 8:7:1:12. What is the age of the oldest member of the family (in years)?
The ages of A and B are in the ratio 4: 7, respectively. After 8 years, the ratio of their ages will be 2: 3. What is the sum of the ages of A and B (in years)?
The present ages of Ramu and Ravi are in the ratio of 4: 3. After 5 years the ratio of their ages will be 5: 4. Find their present ages, respectively.
- Six years ago, the average of the ages of Ravi, his wife and their child was 32 years. Nine years ago, the average of the ages of Ravi's wife and the child was 24 years. Find the present age of Ravi?
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?
