Solving the Cousin Age Word Problem

This question asks us to determine the oldest person among five cousins: Anitha, Mahima, Rajan, Latha, and Deepti, based on specific relationships given about their ages. To solve this, we need to express the age of each cousin in terms of a single variable and then compare those expressions.

Understanding the Given Age Relationships

The relationships between the cousins' ages are:

• Anitha is twice as old as Mahima. Let $A$ be Anitha's age and $M$ be Mahima's age. This means $A = 2M$.
• Rajan is half the age of Mahima. Let $R$ be Rajan's age. This means $R = 0.5M$.
• Anitha is half the age of Deepti. Let $D$ be Deepti's age. This means $A = 0.5D$.
• Rajan is twice the age of Latha. Let $L$ be Latha's age. This means $R = 2L$.

Expressing All Cousin Ages in Terms of One Person

Let's use Mahima's age ($M$) as our base variable and express everyone else's age in terms of $M$.

• Mahima's age: $M$
• From the first relationship, Anitha's age is $A = 2M$.
• From the second relationship, Rajan's age is $R = 0.5M$.
• Now, let's find Deepti's age. We know $A = 0.5D$. We also know $A = 2M$. So, we can write $2M = 0.5D$. To find $D$, we multiply both sides of the equation by 2: $2 \times (2M) = 2 \times (0.5D)$, which gives $4M = D$. So, Deepti's age is $4M$.
• Finally, let's find Latha's age. We know $R = 2L$. We also know $R = 0.5M$. So, we can write $0.5M = 2L$. To find $L$, we divide both sides of the equation by 2: $\frac{0.5M}{2} = \frac{2L}{2}$, which gives $0.25M = L$. So, Latha's age is $0.25M$.

Comparing the Ages of the Five Cousins

Now we have the ages of all five cousins expressed in terms of $M$:

Since age must be a positive value, $M$ is a positive number. To find the oldest cousin, we need to compare the coefficients of $M$ for each person:

• Deepti: 4
• Anitha: 2
• Mahima: 1
• Rajan: 0.5
• Latha: 0.25

The largest coefficient determines the oldest person. Comparing the values (4, 2, 1, 0.5, and 0.25), we see that 4 is the largest value.

Identifying the Oldest Cousin

The age with the largest coefficient (4) is Deepti's age ($4M$). This means Deepti is the oldest cousin.

We can also list the cousins in order from youngest to oldest based on their ages relative to $M$:

$0.25M \text{ (Latha)} < 0.5M \text{ (Rajan)} < M \text{ (Mahima)} < 2M \text{ (Anitha)} < 4M \text{ (Deepti)}$

Conclusion

By analyzing the age relationships and expressing all ages in terms of Mahima's age, we found that Deepti's age ($4M$) has the largest multiple of $M$. Therefore, Deepti is the oldest among the five cousins.