Lesson 2.2: Fractions and Decimals

Introduction

In this lesson, we will explore the world of fractions and decimals, which forms a crucial part of your Quantitative Reasoning skill set for the GMAT. The objectives of this lesson are to:

Understand how to operate with fractions and decimals and convert between them.
Efficiently compare and order values expressed in fractions and decimals.
Choose the cleaner form (fraction vs. decimal) for a given problem.
Manipulate fractions and decimals fluently.
Select the representation that minimizes calculation efforts.

Let's dive into how fractions and decimals work, why they matter, and how you can master them in your quantitative reasoning.

Understanding Fractions

Definition of Fractions

A fraction represents a part of a whole. It consists of two numbers: the numerator (the number above the line) and the denominator (the number below the line). The fraction is defined as:

$$\text{Fraction} = \frac{\text{Numerator}}{\text{Denominator}}$$

For example, in the fraction $\frac{3}{4}$, $3$ is the numerator, and $4$ is the denominator, which means we are considering three parts out of four equal parts in total.

Operations with Fractions

Addition of Fractions

To add fractions, they must have a common denominator. If they do not, we must find one. The formula for adding fractions is:

$$\frac{a}{c} + \frac{b}{d} = \frac{ad + bc}{cd}$$

Worked Example:

Add the fractions $\frac{1}{3}$ and $\frac{1}{6}$. The common denominator of $3$ and $6$ is $6$.

$$\frac{1}{3} = \frac{2}{6} \quad \text{(multiply numerator and denominator by 2)}$$

$$\frac{1}{3} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$$

Subtraction of Fractions

Subtracting fractions follows the same rule as addition:

$$\frac{a}{c} - \frac{b}{d} = \frac{ad - bc}{cd}$$

Worked Example:

Subtract $\frac{2}{5}$ from $\frac{3}{5}$:

$$\frac{3}{5} - \frac{2}{5} = \frac{3 - 2}{5} = \frac{1}{5}$$

Multiplication of Fractions

To multiply fractions, multiply the numerators and denominators:

$$\frac{a}{c} \times \frac{b}{d} = \frac{ab}{cd}$$

Worked Example:

Multiply $\frac{3}{4}$ and $\frac{2}{5}$:

$$\frac{3}{4} \times \frac{2}{5} = \frac{6}{20} = \frac{3}{10}$$

Division of Fractions

To divide by a fraction, multiply by its reciprocal:

$$\frac{a}{c} ÷ \frac{b}{d} = \frac{a}{c} \times \frac{d}{b} = \frac{ad}{bc}$$

Worked Example:

Divide $\frac{5}{6}$ by $\frac{1}{2}$:

$$\frac{5}{6} ÷ \frac{1}{2} = \frac{5}{6} \times \frac{2}{1} = \frac{10}{6} = \frac{5}{3}$$

Common Misconceptions about Fractions

Common Denominator: A common error is thinking that fractions must always be converted to the same denominator before adding or subtracting. However, when multiplying or dividing, this is not necessary.
Improper Fractions: An improper fraction (where the numerator is greater than the denominator) can be confusing. For example, $\frac{7}{4}$ can simply be read as a number greater than 1 but can also be converted to a mixed number: $1 \frac{3}{4}$.

Understanding Decimals

Definition of Decimals

A decimal represents a fraction whose denominator is a power of ten. It is written with a decimal point. For instance, $0.75$ is equivalent to the fraction $\frac{75}{100}$.

Operations with Decimals

Addition and Subtraction of Decimals

When adding or subtracting decimals, align the decimal points:

Worked Example:

Add $0.5 + 0.75$:

   0.50
 + 0.75
 ------
   1.25

Multiplication of Decimals

To multiply decimals, multiply as if they were whole numbers, then place the decimal point in the product. To determine the placement, count the total number of decimal places in both factors:

Worked Example:

Multiply $0.2$ and $0.5$:

  2
x 5
----
 10

The product has $1$ decimal place (from $0.2$) + $1$ decimal place (from $0.5$) = $2$ total places:

$$0.2 \times 0.5 = 0.10 = 0.1$$

Division of Decimals

To divide decimals, move the decimal point in the divisor to convert it into a whole number, moving the decimal point in the dividend the same number of places:

Worked Example:

Divide $0.75$ by $0.25$:

0.75 (move decimal 2 places to the right)
---
0.25 (move decimal 2 places to the right)
 3

So, $0.75 ÷ 0.25 = 3$.

Common Misconceptions about Decimals

Decimal Points: Students sometimes forget to align decimal points when adding or subtracting their values, which can lead to errors.
Conversion Awareness: Knowing that $0.5$ is equivalent to $\frac{1}{2}$ should help students see patterns in operations that may simplify their calculations.

Converting Between Fractions and Decimals

Understanding how to convert between fractions and decimals is critical for solving quantitative problems effectively.

Converting Fractions to Decimals

To convert a fraction into a decimal, divide the numerator by the denominator:

Worked Example:

Convert $\frac{3}{8}$ to a decimal:

$$3 ÷ 8 = 0.375$$

Converting Decimals to Fractions

To convert a decimal into a fraction, count the decimal places, use that as the denominator, and simplify if necessary. For example, $0.625$ can be converted as follows:

Write as $\frac{625}{1000}$ (since there are three decimal places).
Simplify:

$$\frac{625 ÷ 125}{1000 ÷ 125} = \frac{5}{8}$$

Common Misconceptions about Conversion

Rounding Issues: Some students forget that long division can lead to repeating decimals (like $\frac{1}{3}$ being $0.333...). They should be conscious of these nuances.
Simplifying: Not simplifying fractions can lead to confusion in further calculations, especially in a timed test scenario.

Comparing and Ordering Values

When comparing fractions and decimals, understanding their values can help immensely, especially in cases where estimation is necessary or time is of the essence.

Examples of Comparison

To compare, convert values to the same form. If you are comparing $\frac{3}{4}$ and $0.8$:

Convert $0.8$ to a fraction: $0.8 = \frac{8}{10} = \frac{4}{5}$.
Now, find a common denominator ($20$ in this case):

$$\frac{3}{4} = \frac{15}{20} \quad \text{and} \quad \frac{4}{5} = \frac{16}{20}$$

Compare: $\frac{15}{20} < \frac{16}{20}$.

Efficient Comparison Techniques

Estimate: If the numbers can be quickly estimated, it may reduce complexity.
Use of Common Denominators: For fractions, this remains a crucial method to determine greater or lesser values efficiently.

Conclusion

In this lesson, we have explored fractions and decimals in a rigorous and comprehensive manner. Mastering these concepts is essential for excelling in the quantitative reasoning section of the GMAT. Always remember to practice manipulating both forms and understanding when each is appropriate to use.

Study Notes

A fraction represents a part of a whole, written as $\frac{\text{Numerator}}{\text{Denominator}}$.
Addition requires a common denominator: $\frac{a}{c} + \frac{b}{d} = \frac{ad + bc}{cd}$.
To convert a fraction to a decimal, divide the numerator by the denominator.
For operations with decimals, align decimal points for addition and subtraction, and count decimal places for multiplication and division.
Recognize and use efficient methods for comparing and ordering fractions and decimals.