Understanding and Solving the Ages Ratio Problem

This question involves solving an age problem where the ratio of ages at a past point in time and the sum of present ages are given. We need to find the sum of the ages of two individuals from the group at a future point in time.

Step-by-Step Solution for Age Calculation

Let the ages of A, B, and C, 5 years ago, be represented by a variable \(x\) based on the given ratio. The ratio of their ages 5 years ago was 4 : 5 : 7.

• Age of A, 5 years ago = \(4x\) years
• Age of B, 5 years ago = \(5x\) years
• Age of C, 5 years ago = \(7x\) years

To find their present ages, we add 5 years to their ages 5 years ago:

• Present age of A = \((4x + 5)\) years
• Present age of B = \((5x + 5)\) years
• Present age of C = \((7x + 5)\) years

The sum of their present ages is given as 135 years. We can set up an equation using their present ages:

\((4x + 5) + (5x + 5) + (7x + 5) = 135\)

Now, let's solve this equation to find the value of \(x\):

\(4x + 5x + 7x + 5 + 5 + 5 = 135\)

\(16x + 15 = 135\)

Subtract 15 from both sides:

\(16x = 135 - 15\)

\(16x = 120\)

Divide by 16:

\(x = \frac{120}{16}\)

Simplify the fraction:

\(x = \frac{15 \times 8}{2 \times 8} = \frac{15}{2} = 7.5\)

Now that we have the value of \(x\), we can find the present ages of A, B, and C:

• Present age of A = \(4x + 5 = 4(7.5) + 5 = 30 + 5 = 35\) years
• Present age of B = \(5x + 5 = 5(7.5) + 5 = 37.5 + 5 = 42.5\) years
• Present age of C = \(7x + 5 = 7(7.5) + 5 = 52.5 + 5 = 57.5\) years

Let's verify the sum of present ages:

\(35 + 42.5 + 57.5 = 35 + 100 = 135\) years. This matches the given information.

The question asks for the sum of the ages of B and C, 3 years from now. First, we find their ages 3 years from now:

• Age of B, 3 years from now = Present age of B + 3 = \(42.5 + 3 = 45.5\) years
• Age of C, 3 years from now = Present age of C + 3 = \(57.5 + 3 = 60.5\) years

Finally, we find the sum of the ages of B and C, 3 years from now:

Sum of ages of B and C, 3 years from now = Age of B (3 years from now) + Age of C (3 years from now)

Sum = \(45.5 + 60.5 = 106\) years.

Thus, the sum of the ages of B and C, 3 years from now, will be 106 years.

Calculation using \(x\):

• Sum of B and C's ages 3 years from now = \((5x+8) + (7x+8) = 12x + 16\)
• Substitute \(x = 7.5\): \(12(7.5) + 16 = 90 + 16 = 106\) years.

Conclusion on Ages Problem

By using the given ratio of ages from the past and the sum of present ages, we were able to determine the value of the variable representing the ratio parts. This allowed us to calculate the present ages and subsequently project the ages 3 years into the future to find the required sum.

Revision Table: Key Concepts in Ages Problems

Additional Information: Solving Ratio and Age Questions

Age problems often involve ratios, sums, and differences across different points in time (past, present, future). The key is to establish a consistent variable (like \(x\)) representing the unit of the ratio at a specific time (usually the time point mentioned in the ratio) and then express all other ages relative to that variable and time point.

• Always be careful with the time reference (e.g., "5 years ago", "3 years from now", "present").
• When moving from a past age to the present, you add years.
• When moving from a future age to the present, you subtract years.
• When comparing two ages, their difference remains constant over time, but their ratio changes.
• Set up an equation based on the information given about the sum, difference, or relationship between ages at a particular time.
• Solve the equation for the variable and then substitute it back to find the required ages.

Practicing various types of age problems with different combinations of past, present, and future references and ratios will help build confidence in solving them efficiently.