Solving the Asma and Grandfather Age Word Problem

This problem involves setting up and solving a system of linear equations based on the given information about the ages of Asma and her grandfather.

Understanding the Problem Details

We are given two main pieces of information about their ages:

• The sum of their current ages is 80 years.
• In 10 years, Asma’s age will be one-fourth of her grandfather’s age.

Our goal is to find Asma’s current age.

Setting Up Equations for Ages

Let:

• $A$ be Asma’s current age.
• $G$ be her grandfather’s current age.

From the first statement, the sum of their current ages is 80. This gives us our first equation:

$$A + G = 80 \quad \text{(Equation 1)}$$

The second statement talks about their ages 10 years from now.

• In 10 years, Asma’s age will be $A + 10$.
• In 10 years, her grandfather’s age will be $G + 10$.

The statement says Asma’s age will be one-fourth of her grandfather’s age in 10 years. This gives us our second equation:

$$A + 10 = \frac{1}{4}(G + 10) \quad \text{(Equation 2)}$$

Solving the System of Equations

We now have a system of two linear equations with two variables:

$$A + G = 80$$

$$A + 10 = \frac{1}{4}(G + 10)$$

We can use the substitution method to solve this system. From Equation 1, we can express $G$ in terms of $A$:

$$G = 80 - A$$

Now substitute this expression for $G$ into Equation 2:

$$A + 10 = \frac{1}{4}((80 - A) + 10)$$

Simplify the expression inside the parentheses:

$$A + 10 = \frac{1}{4}(90 - A)$$

Multiply both sides by 4 to eliminate the fraction:

$$4(A + 10) = 90 - A$$

Distribute the 4 on the left side:

$$4A + 40 = 90 - A$$

Now, gather all terms with $A$ on one side and constant terms on the other side. Add $A$ to both sides:

$$4A + A + 40 = 90 - A + A$$

$$5A + 40 = 90$$

Subtract 40 from both sides:

$$5A + 40 - 40 = 90 - 40$$

$$5A = 50$$

Divide by 5 to solve for $A$:

$$A = \frac{50}{5}$$

$$A = 10$$

So, Asma’s current age is 10 years.

Verifying the Solution

Let’s check if this age satisfies both conditions:

• If Asma’s current age ($A$) is 10, her grandfather’s current age ($G$) must be $80 - 10 = 70$. Sum of current ages: $10 + 70 = 80$. This matches the first condition.
• In 10 years, Asma will be $10 + 10 = 20$ years old.
• In 10 years, her grandfather will be $70 + 10 = 80$ years old.
• Is Asma’s age in 10 years one-fourth of her grandfather’s age in 10 years? $20 = \frac{1}{4} \times 80$? Yes, $20 = 20$. This matches the second condition.

The solution is consistent with the problem statements.

Therefore, Asma’s current age is 10 years.

Revision Table: Key Concepts

Additional Information: Solving Age Problems

Age problems are common in algebra and quantitative aptitude. They typically involve setting up equations based on how ages change over time and relationships between different people's ages.

• Current Age: This is the age at present.
• Age in the Future: If a person's current age is $x$, their age $y$ years from now will be $x + y$.
• Age in the Past: If a person's current age is $x$, their age $y$ years ago was $x - y$.
• Ratios and Fractions: Phrases like "twice the age," "half the age," or "one-fourth of the age" translate directly into mathematical expressions (e.g., $2x$, $\frac{1}{2}x$, $\frac{1}{4}x$).
• Sum or Difference: Phrases like "sum of ages is..." or "difference in ages is..." translate into addition or subtraction equations.

Always define your variables clearly and make sure the equations accurately represent the conditions given in the problem for the specified time periods (current, future, or past).