Lesson 5.2: Back-Solving from Answer Choices

Introduction

In this lesson, we will explore the technique of back-solving from answer choices, a valuable strategy in quantitative reasoning. The GMAT is designed to test not only your mathematical knowledge but also your ability to apply that knowledge efficiently under time constraints. This lesson will help you develop the intuition and skills necessary to maximize your scoring potential by employing back-solving effectively.

Learning Objectives

• Understand how to test answer choices to find the correct one.
• Learn to start from the middle answer choice to minimize tests.
• Recognize which questions are suitable for back-solving.
• Use back-solving efficiently on problems that lend themselves to this approach.
• Decide quickly whether back-solving or algebra is faster based on the question context.

Understanding Back-Solving

Back-solving involves plugging the answer choices back into the original problem to find the correct answer. This method is especially useful when the answer choices are specific numbers or values. Here’s why back-solving can be advantageous:

1. Efficiency: Instead of working through the algebraic process to find a solution, you test each answer choice to see which one fits the problem.
2. Elimination: Back-solving can help eliminate wrong answers quickly, especially in problems where the calculations are complex.
3. Intuition: Sometimes, you can gauge the appropriateness of an answer based on your understanding of the problem, allowing you to eliminate choices without full calculations.

Example 1

Consider the following problem:

Question: If $x$ is a positive integer, which of the following values of $x$ makes the expression $2x + 7$ equal to $19$?

Answer Choices:

• (A) 5
• (B) 6
• (C) 7
• (D) 8
• (E) 9

Solution Steps

1. Identify the expression: We have $2x + 7$ and we want it to equal $19$.
2. Rearranging the equation: From the equation, we can express it as:

$$2x + 7 = 19$$

Subtracting $7$ from both sides gives:

$$2x = 12$$

We can solve for $x$ by dividing both sides by $2$:

$$x = 6$$

1. Check answer choices using back-solving:

• Start at the middle choice (C)
• Testing $x = 7$:

$$2(7) + 7 = 14 + 7 = 21 \quad (Too high)$$

• Testing $x = 6$:

$$2(6) + 7 = 12 + 7 = 19 \quad (Correct)$$

• Testing $x = 5$:

$$2(5) + 7 = 10 + 7 = 17 \quad (Too low)$$

Thus, the correct answer is (B) 6. Back-solving provided a quick method to confirm our computation and validate the answer.

When to Use Back-Solving

Back-solving is particularly effective in several situations:

• When answer choices are numerical values.
• When the calculations involved are straightforward to plug in.
• In problems where you can identify approximate answers and narrow down your choices quickly.
• When you are unsure how to set up an equation but can estimate likely outcomes.

Example 2

Let’s analyze another problem:

Question: A rectangular garden has a perimeter of $30$ meters. If the length of the garden is $x$ meters, what is the width in terms of $x$?

Answer Choices:

• (A) $15 - x$
• (B) $30 - 2x$
• (C) $x + 5$
• (D) $15 + x$
• (E) $x - 10$

Solution Steps

1. Understanding the perimeter formula: The perimeter $P$ of a rectangle is given by:

$$P = 2(\text{length} + \text{width})$$

Setting this equal to $30$ meters, we have:

$$30 = 2(x + \text{width})$$

1. Rearranging: Dividing both sides by $2$ gives:

$$15 = x + \text{width}$$

Thus:

$$\text{width} = 15 - x$$

1. Testing answer choices:

• The answer is clearly (A). To validate, let's check a few options quickly:
• Testing (B): $30 - 2x = 30 - 2(10) = 10 \quad (Not valid for $x=10)
• Testing (C): x + 5 = 10 + 5 = 15 \quad (Not valid given width must decrease)
• Testing (D): $15 + 10 = 25 \quad (this would increase total)
• Testing (E): $10 - 10 = 0 \quad (not valid)

By elimination, the correct choice is indeed (A) $15 - x$.

Estimation vs. Back-Solving

In many cases, you might find that estimation is faster than back-solving or vice versa. Here’s a guideline for deciding:

• Use back-solving when:
• The problem is sufficiently straightforward that plugging in values won’t take a lot of time.
• You’re confident that answer choices are all within a reasonable range relative to the question.
• Use estimation when:
• The numbers in the question are cumbersome and require complex calculations.
• You can identify approximate ranges for your answers, allowing you to narrow down more efficiently.

Example 3: Comparison Problem

Question: If $6x + 5 = 8x + 3$, what is the value of $x$?

Answer Choices:

• (A) $1$
• (B) $2$
• (C) $3$
• (D) $4$
• (E) $5$

1. Rearranging the equation:

$$6x + 5 = 8x + 3$$

Move $6x$ from left to right:

$$5 = 2x + 3$$

Then subtract $3$ from both sides:

$$2 = 2x$$

Thus, $x = 1$.

1. Start at (A) $1$:

• Testing (A): $6(1) + 5 = 11$ and 8(1) + 3 = 11 \quad (Correct)
• Testing (B): $6(2) + 5 = 17$ vs. 8(2) + 3 = 19\quad (Incorrect)
• Continuing to check isn’t needed as (A) is already correct.

Conclusion

Back-solving is a powerful tool that, when utilized correctly, can save you time and increase your accuracy on the GMAT. By understanding when and how to apply this strategy, you enhance your problem-solving efficiency. Always carefully consider the context of the question to determine whether back-solving or traditional algebraic methods will yield the best results. Remember, the key to success in quantitative reasoning is not only knowing the math but also applying it strategically.

Study Notes

• Back-solving is testing answer choices instead of calculating values directly.
• Start with the middle answer choice for efficiency.
• Effective for numerical values and straightforward problems.
• Be mindful of scenarios where estimation may be faster than back-solving.
• Use elimination strategically to narrow down options.