Solving Age Word Problems: Ratio and Sum of Ages

This problem asks us to find the present ages of two sisters given information about the ratio of their ages one year ago and the sum of their present ages.

Understanding the Problem Statement

We are provided with two key pieces of information:

• The ratio of the ages of the two sisters one year ago was 2 : 3.
• The sum of their present ages is 12 years.

Our goal is to determine the present age of each sister.

Setting up the Equations

Let's denote the present age of the first sister as $A$ and the present age of the second sister as $B$.

From the given information, we can form two equations:

1. Sum of Present Ages: The sum of their present ages is 12.
   $$A + B = 12 \quad (Equation \; 1)$$
2. Ratio of Ages One Year Ago: One year ago, their ages were $A-1$ and $B-1$. The ratio was 2:3.
   $$\frac{A-1}{B-1} = \frac{2}{3}$$ Cross-multiplying this equation gives:
   $$3(A-1) = 2(B-1)$$ $$3A - 3 = 2B - 2 \quad (Equation \; 2)$$

Solving the System of Equations

Now we have a system of two linear equations with two variables:

We can use the substitution method to solve this system.

From Equation 1, we can express $B$ in terms of $A$:

Substitute this expression for $B$ into Equation 2:

Now, rearrange the equation to bring all terms with $A$ to one side and constants to the other side:

Divide by 5 to find the value of $A$:

So, the present age of the first sister is 5 years.

Now, substitute the value of $A$ back into Equation 1 ($B = 12 - A$) to find the value of $B$:

So, the present age of the second sister is 7 years.

Verifying the Solution

Let's check if these ages satisfy the original conditions:

• Present Ages Sum: $5 + 7 = 12$. This matches the given sum of present ages.
• Ratio One Year Ago: One year ago, the ages would be $5 - 1 = 4$ and $7 - 1 = 6$. The ratio is $\frac{4}{6}$, which simplifies to $\frac{2}{3}$. This matches the given ratio one year ago.

Since both conditions are satisfied, our calculated ages are correct.

Matching with Options

The calculated present ages are 5 and 7. Let's compare this with the provided options:

• Option 1: 5, 7
• Option 2: 7.5, 4.5
• Option 3: 9, 3
• Option 4: 8, 4

Our result (5, 7) matches Option 1.

Conclusion

The present ages of the two sisters are 5 years and 7 years.

Revision Table: Key Concepts for Age Problems

Additional Information: Strategies for Age Word Problems

Age problems are common in mathematics and often appear in competitive exams. Here are some strategies to tackle them:

• Identify the Variables: Always assign variables (like $x$ or $A$) to the unknown ages, usually the present ages.
• Read Carefully: Pay close attention to whether the information given refers to past ages (e.g., '5 years ago'), present ages ('now'), or future ages (e.g., 'in 10 years').
• Formulate Equations: Translate the word statements into mathematical equations. Each piece of information typically translates into one equation.
  • If the sum of ages is given, add the variables and set them equal to the sum.
  • If a ratio of ages is given, set up a fraction with the ages and equate it to the given ratio.
  • If one person's age is a multiple or fraction of another's age at a specific time, write that relationship as an equation.
• Solve the System: Use algebraic methods like substitution or elimination to solve the system of equations you've created.
• Verify Your Answer: Plug the calculated ages back into the original problem statement to ensure they satisfy all conditions.

Practicing different types of age problems will help you become more comfortable with setting up and solving the equations.