Age Word Problem Explained

This problem requires us to find the sister's current age based on the relationship between her age and my age at two different times: 5 years ago and the present.

Defining Variables for Ages

To solve this, let's define variables for our current ages:

• Let M represent my current age (in years).
• Let S represent my sister's current age (in years).

Setting Up Age Equations

We can translate the information given in the problem into mathematical equations:

1. Relationship 5 Years Ago:

   Five years ago, my age was M - 5 and my sister's age was S - 5.

   The problem states: "my sister's age was 5 times my age". This gives us the equation:

   $$ S - 5 = 5 \times (M - 5) $$

   Let's simplify this equation:

   $$ S - 5 = 5M - 25 $$

   Rearranging to express S in terms of M:

   $$ S = 5M - 25 + 5 $$

   $$ S = 5M - 20 $$ (This is our Equation 1)

2. Relationship Now (Present):

   The problem states: "Now it is 3 times only". This means the sister's current age is 3 times my current age.

   $$ S = 3M $$ (This is our Equation 2)

Solving the System of Equations

We now have two equations for S. We can find the ages by solving this system. A good method is substitution. Since Equation 2 tells us S = 3M, we can substitute 3M for S in Equation 1:

Substitute 3M into Equation 1:

$$ 3M = 5M - 20 $$

Now, let's solve for M:

Subtract 3M from both sides:

$$ 0 = 5M - 3M - 20 $$

$$ 0 = 2M - 20 $$

Add 20 to both sides:

$$ 20 = 2M $$

Divide by 2:

$$ M = \frac{20}{2} $$

$$ M = 10 $$

This means my current age is 10 years.

Calculating the Sister's Present Age

We need to find the sister's present age, S. We can use Equation 2:

$$ S = 3M $$

Substitute the value we found for M (which is 10):

$$ S = 3 \times 10 $$

$$ S = 30 $$

So, the sister's present age is 30 years.

Verification Step

Let's check our answer using the condition from 5 years ago:

• My current age: 10 years.
• Sister's current age: 30 years.
• My age 5 years ago: 10 - 5 = 5 years.
• Sister's age 5 years ago: 30 - 5 = 25 years.
• Check: Was the sister's age (25) 5 times my age (5)? Yes, 25 = 5 \times 5.

The calculations confirm that the sister's present age is 30 years.