Solving Poojitha's Age Word Problem

This problem asks us to find Poojitha's current age based on a relationship involving her age at different points in time. We can solve this by setting up a linear equation.

Define the Variable

Let Poojitha's current age be represented by the variable \(x\) years.

Express Ages at Different Times

• Poojitha's age three years ago was \(x - 3\) years.
• Poojitha's age two years from now will be \(x + 2\) years.

Translate the Problem into an Equation

The problem states that "one-third of Poojitha's age three years ago plus one-half of her age two years from now is twenty years". We can write this as an equation:

$$\frac{1}{3} \times (x - 3) + \frac{1}{2} \times (x + 2) = 20$$

This simplifies to:

$$\frac{x - 3}{3} + \frac{x + 2}{2} = 20$$

Solve the Linear Equation

To solve for \(x\), we first need to eliminate the fractions. The least common multiple (LCM) of the denominators (3 and 2) is 6. Multiply every term in the equation by 6:

$$6 \times \left(\frac{x - 3}{3}\right) + 6 \times \left(\frac{x + 2}{2}\right) = 6 \times 20$$

Simplify the equation:

$$2(x - 3) + 3(x + 2) = 120$$

Now, distribute the numbers outside the parentheses:

$$2x - 6 + 3x + 6 = 120$$

Combine like terms (\(2x\) and \(3x\), and \(-6\) and \(+6\)):

$$(2x + 3x) + (-6 + 6) = 120$$

$$5x + 0 = 120$$

$$5x = 120$$

Finally, isolate \(x\) by dividing both sides by 5:

$$x = \frac{120}{5}$$

$$x = 24$$

Verification

Let's check if Poojitha's age of 24 years satisfies the original condition:

• Age three years ago: \(24 - 3 = 21\)
• One-third of age three years ago: \(\frac{1}{3} \times 21 = 7\)
• Age two years from now: \(24 + 2 = 26\)
• One-half of age two years from now: \(\frac{1}{2} \times 26 = 13\)

Sum of these parts: \(7 + 13 = 20\).

The sum is 20, which matches the condition given in the problem. So, the calculated age is correct.

Conclusion

Poojitha's current age is 24 years.

Revision Table: Key Concepts

Additional Information: Solving Age Problems

Age problems are common types of word problems in algebra. They often involve relationships between a person's current age, their age in the past, or their age in the future. Here are some tips for solving age problems:

• Always define your variable clearly, usually representing the current age.
• Express past or future ages in terms of the current age variable (\(x - \text{years ago}\) or \(x + \text{years from now}\)).
• Carefully read the problem to set up the correct equation based on the relationships given.
• Solve the resulting linear equation step-by-step.
• Always verify your answer by substituting it back into the original problem statement to ensure it makes sense and satisfies all conditions.
• Pay attention to phrases like "is", "was", "will be" which usually indicate equality (=).
• Phrases like "sum", "difference", "product", "quotient", "one-third of", "twice" indicate specific mathematical operations.