Understanding the Age Ratio Problem

This problem involves understanding age relationships at different points in time, specifically relative to a couple's marriage anniversary. We are given information about Sanju's age at two specific points after his parents' marriage and the ratio of these ages. We need to find out how many years after their marriage Sanju was born.

Setting up the Age Variables

Let's denote the number of years after Akshar and Manvita's marriage that Sanju was born as \(Y\). This is the value we need to find.

Based on this definition:

• At the time of the marriage (Year 0), Sanju's age would be \(0 - Y = -Y\) years. This indicates he was not yet born.
• 18 years after the marriage, Sanju's age would be \(18 - Y\) years. This is his actual age at that point.
• 50 years after the marriage (which is their golden anniversary), Sanju's age would be \(50 - Y\) years. This is his actual age at their golden anniversary.

Formulating the Age Ratio Equation

The question states that Sanju's age 18 years after his parents' marriage and his age at their golden anniversary (50 years after marriage) are in the ratio 5 ∶ 21.

We can write this as a mathematical equation:

\(\frac{\text{Sanju's age 18 years after marriage}}{\text{Sanju's age 50 years after marriage}} = \frac{5}{21}\)

Substituting the expressions for Sanju's age in terms of \(Y\):

\(\frac{18 - Y}{50 - Y} = \frac{5}{21}\)

Solving for the Unknown Variable

Now we need to solve the equation for \(Y\). We can do this by cross-multiplication:

\(21 \times (18 - Y) = 5 \times (50 - Y)\)

Distribute the numbers on both sides:

\((21 \times 18) - (21 \times Y) = (5 \times 50) - (5 \times Y)\)

\(378 - 21Y = 250 - 5Y\)

Now, we want to isolate the variable \(Y\). Let's move the terms involving \(Y\) to one side and the constant terms to the other side. Add \(21Y\) to both sides:

\(378 = 250 - 5Y + 21Y\)

\(378 = 250 + 16Y\)

Subtract 250 from both sides:

\(378 - 250 = 16Y\)

\(128 = 16Y\)

Finally, divide by 16 to find the value of \(Y\):

\(Y = \frac{128}{16}\)

\(Y = 8\)

So, Sanju was born 8 years after his parents' marriage.

Verifying the Answer

Let's check if this answer holds true based on the given ratio:

• If Sanju was born 8 years after marriage, his age 18 years after marriage would be \(18 - 8 = 10\) years.
• His age 50 years after marriage (golden anniversary) would be \(50 - 8 = 42\) years.

The ratio of these ages is \(10 : 42\).

Let's simplify this ratio:

\(\frac{10}{42} = \frac{10 \div 2}{42 \div 2} = \frac{5}{21}\)

The calculated ratio \(5:21\) matches the ratio given in the question. This confirms our answer is correct.

Conclusion

Sanju was born 8 years after his parents' marriage.

Revision Table: Age Ratio Problems & Anniversaries

• Silver Anniversary: Marks 25 years of marriage.
• Golden Anniversary: Marks 50 years of marriage.
• Age at a Future Time: If current age is A, age after T years is A + T.
• Age Relative to an Event: If born Y years *after* an event, age T years after the event is T - Y.
• Ratio Problems: Set up a fraction representing the ratio and solve the resulting equation, often by cross-multiplication.

Additional Information: Solving Age Word Problems

Age word problems often require setting up equations based on relationships between people's ages at different points in time. It's crucial to define your variables clearly. For example, you might define a variable as a person's current age, or the number of years ago/hence an event occurred. In this specific problem, defining the variable as the time elapsed between the marriage and Sanju's birth was key. Always check your answer by plugging it back into the original problem statement to ensure it satisfies all conditions, like the given age ratio.