Solving the Age Problem

Let's break down this age problem step-by-step to find the age of the younger sister. We are given several clues relating the ages of a father, mother, and their two sisters.

Understanding the Age Clues

The problem gives us the following information:

• The difference between the ages of the two sisters is 2 years. Since there is an elder sister and a younger sister, the elder sister is 2 years older than the younger sister.
• The father's age is 52 years.
• The father is elder by 2 years to the mother, meaning the father is 2 years older than the mother.
• The elder sister's age is half of the mother's age.

Calculating the Ages Step-by-Step

Let's use the given information to find the ages, starting with the known age and the relationships:

Step 1: Find the Mother's Age

We know the father's age is 52 and he is 2 years older than the mother.

Father's Age = Mother's Age + 2

So, Mother's Age = Father's Age - 2

Mother's Age $= 52 - 2 = 50$ years.

Using mathematical notation:

\(M = F - 2\)

\(M = 52 - 2\)

\(M = 50\)

The mother's age is 50 years.

Step 2: Find the Elder Sister's Age

We are told the elder sister's age is half of the mother's age.

Elder Sister's Age = (Mother's Age) / 2

Elder Sister's Age $= 50 / 2 = 25$ years.

Using mathematical notation:

\(E = \frac{M}{2}\)

\(E = \frac{50}{2}\)

\(E = 25\)

The elder sister's age is 25 years.

Step 3: Find the Younger Sister's Age

The difference between the ages of the two sisters is 2 years. This means the elder sister is 2 years older than the younger sister.

Elder Sister's Age - Younger Sister's Age = 2

So, Younger Sister's Age = Elder Sister's Age - 2

Younger Sister's Age $= 25 - 2 = 23$ years.

Using mathematical notation:

\(E - Y = 2\)

\(Y = E - 2\)

\(Y = 25 - 2\)

\(Y = 23\)

The younger sister's age is 23 years.

Final Age of the Younger Sister

Based on the calculations derived from the given clues, the age of the younger sister is 23 years.

Revision Table: Age Calculations Summary

Additional Information: Understanding Age Relations in Math Problems

Age problems are common in mathematics and often involve setting up equations based on the relationships given. Here are a few points to remember:

• Representing Unknowns: Use variables (like F, M, E, Y in this case) to represent the unknown ages.
• Translating Words to Equations: Pay close attention to keywords like "elder than" (addition), "younger than" (subtraction), "half of" (division by 2), "difference between" (subtraction), "sum of" (addition), "times" (multiplication).
• Relative Ages: The difference between two people's ages remains constant over time. In this problem, the 2-year difference between the sisters' ages is a fixed value, even if the father's age (the reference point in the problem statement) changes.
• Solving Systematically: Start with the information that gives a direct value or a simple relationship, and use that to find other unknown ages one by one.

Practicing these steps makes solving age-related word problems much easier.