Solving Mathematical Puzzles Through Logical Reasoning

Mathematical puzzles and riddles are great tools for developing logical thinking, reasoning, and problem-solving skills. Let's delve into a classic age-related problem and learn how to approach it using algebraic reasoning. This problem involves two generations and requires us to set up and solve a system of equations.

Understanding the Problem

Consider Smita and her mother. Smita's mother is 34 years old now. Two years from now, her mother's age will be four times Smita's current age. How old is Smita now?

Formulating the Equations

We can translate the problem into a system of equations. Let's denote Smita's current age as ( S ) and her mother's current age as ( M ).

( M 34 ) (Mita's mother is 34 years old now) ( M 2 4S ) (Two years from now, her mother's age will be four times Smita's current age)

Setting Up and Solving the System of Equations

Let's start by substituting the value of ( M ) into the second equation:

( 34 2 4S )

Now, let's solve this equation for ( S ):

( 36 4S )

( S frac{36}{4} )

( S 9 )

Therefore, Smita is currently 9 years old.

Verification

To verify, let's check the scenario two years from now:

Smita's age will be ( 9 2 11 ).

Her mother's age will be ( 34 2 36 ).

Indeed, 36 is four times 9, confirming our solution is correct.

Alternative Methods

There are several alternative methods to solve this problem, including the following steps:

Method 1: Let's denote Smita's age as ( S ) and her mother's age as ( M ):

We know:

( M 34 )

( 34 2 4S )

( S 9 )

Let's double-check:

After 2 years, Smita's age is ( 9 2 11 ).

After 2 years, her mother's age is ( 34 2 36 ).

36 is indeed 4 times 9, validating our solution.

Another Method:

Let's denote Smita's present age as ( S ) and her mother's present age as ( M ). We have:

( M - 2 4D - 2 ) (mother's age 2 years ago is 4 times her daughter's age at that time)

( M 2 3D 2 ) (mother's future age is three times her daughter's future age)

After simplification:

( M - 2 4(S - 2) )

( M 2 3(S 2) )

Solving the first equation:

( M 4S - 6 )

Solving the second equation:

( M 3S 4 )

Setting these two expressions for ( M ) equal to each other:

( 4S - 6 3S 4 )

( S 10 )

So, Smita is 10 years old.

Verification:

When ( S 10 ) and ( M 34 ):

2 years ago, Smita was 8 and her mother 32, and 32 is 4 times 8.

2 years in the future, Smita will be 12 and her mother 36, and 36 is 3 times 12.

Conclusion

Understanding and solving age puzzles like this one helps in developing critical thinking skills and provides a practical application of algebraic reasoning. By breaking down the problem into a series of logical steps and equations, we can confidently solve these types of questions.