Thirteen years ago, Manorama's age was \(2\over5\) of that of her sister. The ratio of Manorama's and her sister's present ages is 3 : 4. What is the sum of their present ages?
39 years
Manorama's Age Problem Setup
This problem involves finding the sum of the present ages of Manorama and her sister. We are given information about their ages 13 years ago and the current ratio of their ages. Let's break down the information and solve it step-by-step.
- Let Manorama's present age be \(M\) years.
- Let her sister's present age be \(S\) years.
Deriving Equations from Age Ratios
We are given two key pieces of information:
- Present Age Ratio: The ratio of Manorama's present age to her sister's present age is 3 : 4. This can be written as: \( \frac{M}{S} = \frac{3}{4} \) From this, we can express one age in terms of the other, or use a common multiplier. Let's say \( M = 3x \) and \( S = 4x \) for some positive value \(x\).
- Past Age Condition: Thirteen years ago, Manorama's age was \(M - 13\) and her sister's age was \(S - 13\). The problem states that Manorama's age then was \( \frac{2}{5} \) of her sister's age. This gives us the equation: \( M - 13 = \frac{2}{5} (S - 13) \)
Solving the System of Equations
Now we substitute the expressions for \(M\) and \(S\) from the present ratio (\(M = 3x\), \(S = 4x\)) into the past age condition equation:
Substitute \( M = 3x \) and \( S = 4x \) into \( M - 13 = \frac{2}{5} (S - 13) \):
\( 3x - 13 = \frac{2}{5} (4x - 13) \)To eliminate the fraction, multiply both sides of the equation by 5:
\( 5 \times (3x - 13) = 5 \times \frac{2}{5} (4x - 13) \) \( 15x - 65 = 2 (4x - 13) \)Distribute the 2 on the right side:
\( 15x - 65 = 8x - 26 \)Now, gather the \(x\) terms on one side and the constant terms on the other. Subtract \(8x\) from both sides:
\( 15x - 8x - 65 = -26 \) \( 7x - 65 = -26 \)Add 65 to both sides:
\( 7x = -26 + 65 \) \( 7x = 39 \)Solve for \(x\):
\( x = \frac{39}{7} \)Calculating the Sum of Present Ages
The question asks for the sum of their present ages. We know that Manorama's age is \(M = 3x\) and her sister's age is \(S = 4x\). The sum of their present ages is \(M + S\).
Sum = \( M + S = 3x + 4x = 7x \)
Now substitute the value of \(x\) we found:
\( \text{Sum} = 7 \times x = 7 \times \frac{39}{7} \)The 7s cancel out:
\( \text{Sum} = 39 \)So, the sum of their present ages is 39 years.
Important Questions from Age
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?
In 8 years, Subhash will be 3 times as old as he is now. After how many years will Subhash be 5 times as old as he is now?
The average ages of parents and two children are 30 years and 8 years respectively. The average age of the family is
A. 16 years
B. 19 years
C. 18 years
D. 17 years
A father is presently 3 times his daughter’s age. After 10 years he will be twice as old as her. Find the daughter’s present age.
