Question

The positive difference between Sam and Lucy's ages is $a$, and the sum of their ages is $z$. If Lucy is older than Sam, then which of the following represents Lucy's age? A. $$\frac{z-a}{2}$$ B. $$a-\frac{z}{2}$$ C. $$2a+z$$ D. $$\frac{z+a}{2}$$ E. $$\frac{a-z}{2}$$

Short answer

The correct answer is D: $\frac{z+a}{2}$.

Step-by-step solution

- Define variables

Let Sam's age be represented by the variable $S$ and Lucy's age by $L$.

- Set up equations

Since Lucy is older than Sam, we have the equations: 1. The positive difference between their ages: $L-S=a$ 2. The sum of their ages: $L+S=z$.

- Isolate Sam's age

From the equation $L+S=z$, solve for Sam's age: $$S=z-L$$

- Substitute back

Substitute $S=z-L$ into the equation $L-S=a$: $L-(z-L)=a$.

- Simplify

Simplify the equation: $L-z+L=a$ $2L-z=a$.

- Solve for Lucy's age

Rearrange the equation to solve for $L$: $$2L=a+z$$$$L=\frac{a+z}{2}$$

Key concepts

Age Difference Problems

Age difference problems often appear in exams like the GMAT. They typically involve finding the age of individuals based on given differences and sums. Understanding how to define and manipulate the related algebraic equations is crucial.
For example:

• We are given the positive difference between Sam and Lucy's ages as $a$.
• We also know the sum of their ages is $z$.
• Lucy is older than Sam.

With these conditions, we can set up algebraic equations to find the required ages.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality between different expressions. In age-related problems, we use them to relate the ages of two or more individuals.
In this problem:
1. Let Sam's age be represented by $S$ and Lucy's age by $L$.
2. We have two equations from the problem description:
Equation 1: $L-S=a$
This states Lucy's age minus Sam's age equals the age difference.
Equation 2: $L+S=z$
This states the sum of their ages equals $z$.
Using these equations helps us solve for the variables involved.

GMAT Math Section

The GMAT Math section tests your ability to handle mathematical problems quickly and accurately. Age-related algebra problems are common. They test your understanding of setting up and solving equations based on given conditions. Key points to remember:

• Break the problem into manageable steps.
• First, define your variables.
• Set up the algebraic equations based on the problem statement.
• Isolate and solve for the required variables.

This systematic approach will help you tackle similar problems with confidence.

Solving for Variables in Algebra

Solving for variables in algebra involves isolating the variable you need to find. Here, we solve for Lucy's age $L$:
1. Start with the equations:
$L-S=a$
$L+S=z$
2. Isolate Sam's age $S$ from the sum equation: $S=z-L$
3. Substitute this into the difference equation: $L-(z-L)=a$
4. Simplify: $L-z+L=a$ or $2L-z=a$
5. Solve for $L$: $L=\frac{a+z}{2}$
This methodical process is essential in handling algebra questions efficiently.

Standardized Test Preparation

Preparing for standardized tests like the GMAT involves practicing different types of problems, including age-related algebra problems. Here are some tips:

• Understand the types of problems you’ll encounter.
• Practice regularly with a variety of questions.
• Focus on your weak areas.
• Use a step-by-step approach to solving problems.

Consistent practice with these strategies will boost your confidence and improve your problem-solving skills.