Lesson 5.1: Smart Numbers and Plugging In

Introduction

In this lesson, students, we will explore the concept of using Smart Numbers and the technique of Plugging In to effectively tackle variable-expression problems on the GMAT. Understanding these strategies is crucial, as they not only help save time but also reduce the risk of making errors during problem-solving. By the end of this lesson, you will learn how to choose convenient numbers, identify when plugging in is the best approach, avoid misleading number choices, and correctly apply the smart-numbers method to boost your performance.

Learning Objectives

• Choosing convenient numbers for variable-expression problems.
• Understanding when plugging in is faster and safer than algebra.
• Avoiding numbers that create false matches.
• Applying the smart-numbers method correctly.
• Selecting numbers that avoid coincidental answer matches.

Smart Numbers

Smart Numbers refer to specific, often simpler values substituted for variables in algebraic expressions or equations to make calculations easier. The goal is to use these numbers to simplify the math rather than dealing with complex algebraic manipulations, which can be time-consuming and error-prone.

Choosing Convenient Numbers

When faced with a problem containing variables, it's beneficial to choose numbers that simplify the calculations. For example, consider the expression:

$$\text{If } x + y = 10 \text{, find } 2x + 3y.$$

A common approach is to choose values for $x$ and $y$ that are easy to work with. Let's say we choose:

• $x = 2$
• $y = 8$

Now, substituting these values:

$$ 2(2) + 3(8) = 4 + 24 = 28.$$

But maybe we want to simplify even further. Instead, we could choose:

• $x = 5$
• $y = 5$

Thus, substituting again gives:

$$ 2(5) + 3(5) = 10 + 15 = 25.$$

Choosing simple numbers like $5$ helps you verify if an answer choice fits with the expression without carrying complicated arithmetic throughout. This method saves time and reduces errors.

Example 1

Let's apply the smart-numbers strategy to another expression:

$$\text{If } a^2 + b^2 = 25, \text{ determine } 2a + 3b.$$

For this, we may choose:

• $a = 3$
• $b = 4$

Now substituting:

$$ 2(3) + 3(4) = 6 + 12 = 18.$$

To verify, we check:

• $3^2 + 4^2 = 9 + 16 = 25$, which holds true. Hence, our choice of smart numbers is valid, and the result is $18$.

Plugging In

The Plugging In method involves replacing variables with specific values rather than solving them algebraically. This can be quicker and less error-prone when dealing with complex expressions or multiple variables.

When to Use Plugging In

1. When the answer choices are numbers: This technique is especially useful in multiple-choice questions where you can test the choices directly against the variable equation.
2. When the algebraic route seems cumbersome: If the expression contains multiple variables and complicated operations, creating specific cases can clear up confusion and make arithmetic manageable.

Example 2

Consider the equation:

$$\text{If } 2x + 3y = 12, \text{ which of the following values for } y \text{ makes } x \text{ an integer?}$$

The answer choices are:

• A) 1
• B) 2
• C) 3
• D) 4

We can plug in each value:

1. Testing $y = 1$:

$$2x + 3(1) = 12 \Rightarrow 2x + 3 = 12 \Rightarrow 2x = 9 \Rightarrow x = 4.5 \text{ (not an integer)}$$

1. Testing $y = 2$:

$$2x + 3(2) = 12 \Rightarrow 2x + 6 = 12 \Rightarrow 2x = 6 \Rightarrow x = 3 \text{ (integer)}$$

1. Testing $y = 3$:

$$2x + 3(3) = 12 \Rightarrow 2x + 9 = 12 \Rightarrow 2x = 3 \Rightarrow x = 1.5 \text{ (not an integer)}$$

1. Testing $y = 4$:

$$2x + 3(4) = 12 \Rightarrow 2x + 12 = 12 \Rightarrow 2x = 0 \Rightarrow x = 0 \text{ (integer)}$$

From our tests, the acceptable values for $y$ that yield integer results for $x$ are $y = 2$ and $y = 4$. Since we only need one, we can conclude that $y = 2$ is our first valid choice.

Avoiding Numbers that Create False Matches

While it's beneficial to use smart numbers and plug in, it's equally important to avoid numbers that might create misleading or coincidental results. Ensure the numbers you select are reasonable and not too specific to the possible answer choices.

Example 3

Consider a scenario where you encounter:

$$\text{If } z = 2x^2 - 4x + 2, \text{ choose } x = 2.$$

Substituting gives:

$$ z = 2(2)^2 - 4(2) + 2 = 2(4) - 8 + 2 = 8 - 8 + 2 = 2.$$

If the answer choices were very close, such as:

• A) 2
• B) 3
• C) 4
• D) 5

The selection of $x = 2$ made it coincide perfectly with one of the answers. Hence, it could lead to a false sense of confidence. Selecting another value, ensure the result genuinely arises from the provided function rather than sheer coincidence.

Conclusion

In this lesson, students, you learned how to utilize smart numbers and the plugging-in technique effectively. By leveraging convenient numbers and knowing when to replace variables, you can reduce calculation time and minimize mistakes. With practice, you will be able to quickly select appropriate values during the exam and navigate variable-expression problems with greater confidence.

Study Notes

• Smart Numbers simplify calculations in variable expressions.
• Choose numbers that are easy to work with.
• Plugging In is beneficial when answer choices are numbers.
• Avoid numbers that lead to false matches or coincidences in results.
• Always verify your selected numbers against the original equations.