Age Calculation: When Will Sara Be Exactly Twice as Old as Her Brother?

Both Sara and Abebe start as young siblings, with ages 12 and 2, respectively. This article explores the concept of age calculations and algebraic equations to determine how many years it will take for one sibling to be twice the age of the other. We will apply the same method to both cases and provide a comprehensive explanation for each.

Case Study: Sara and Her Younger Brother

Sara is 12 years old, and her brother is 2 years old. Let us denote the number of years from now as x.

Currently, Sara is 12 years old, and her brother is 2 years old. In x years, their ages will be:

Sara's age: 12 x Brother's age: 2 x

We want to find x when Sara will be exactly twice her brother's age:

12 x 2(2 x)

Now, let's solve the equation step by step:

Expand the equation: 12 x 4 2x Rearrange the equation to isolate x: 12 - 4 2x - x This simplifies to: 8 x

Therefore, in 8 years Sara will be exactly twice as old as her brother. At that time, Sara will be 12 8 20 years old, and her brother will be 2 8 10 years old.

Verification

To verify our solution, let's check the ages in 8 years:

Sara's age in 8 years: 12 8 20 Brother's age in 8 years: 2 8 10

Sara (20 years old) is indeed twice the age of her brother (10 years old).

General Case: Abebe and Aster

Abebe is 12 years old, and Aster is 2 years old. We want to find the number of years, B, when Abebe will be twice as old as Aster. Let's denote this as:

Abebe's age in B years: 12 B

Aster's age in B years: 2 B

We want Abebe to be twice as old as Aster:

12 B 2(2 B)

Now, let's solve this equation:

Expand the equation: 12 B 4 2B Rearrange the equation to isolate B: 12 - 4 2B - B This simplifies to: 8 B

Therefore, in 8 years, Abebe will be exactly twice as old as Aster. At that time, Abebe will be 12 8 20 years old, and Aster will be 2 8 10 years old.

Verification

To verify our solution, let's check the ages in 8 years:

Abebe's age in 8 years: 12 8 20 Aster's age in 8 years: 2 8 10

Abebe (20 years old) is indeed twice the age of Aster (10 years old).

Calculation Using Algebraic Expressions

Let A be Abebe's current age (12 years old), and let a be Aster's current age (2 years old). Let x be the number of years until Abebe will be twice as old as Aster. We can express this in an equation as follows:

A x 2(a x)

Now, let's solve this equation step by step:

Expand the equation: A x 2a 2x Rearrange the equation to isolate x: A - 2a 2x - x This simplifies to: A - 2a x Substitute A 12 and a 2: 12 - 2 10 x

Therefore, in 8 years, Abebe will be exactly twice as old as Aster.

Conclusion

We have shown that in both cases, the number of years required for one sibling to be twice the age of the other can be calculated using similar algebraic methods. This problem demonstrates the application of algebraic equations in solving real-life age-related problems.