Fractions, ratios, and decimals are fundamental concepts in mathematics that often appear in GMAT problem-solving questions. Mastering these concepts is crucial for achieving a high score on the GMAT.

A fraction represents a part of a whole and is expressed as \frac{a}{b}, where a is the numerator and b is the denominator. The value of a fraction can be found by dividing the numerator by the denominator:

$\text{Value of fraction} = \frac{a}{b}$

To add or subtract fractions with different denominators, you must first rewrite them with a common denominator. Multiply the numerator and denominator of each fraction by a number that converts all denominators to the same value. Since multiplying by 1 preserves the fraction's value, choose multipliers like $\frac{2}{2}$ or $\frac{3}{3}$. For example:

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

To add $\frac{1}{2} + \frac{1}{3}$, convert them to sixths:

$\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$

The Bowtie Method

A quick way to add or subtract fractions without finding a common denominator first is the Bowtie method:

1. Multiply the denominators to get the new denominator.
2. Cross-multiply (denominator × opposite numerator) and place the results above the fractions.
3. Add or subtract these results to form the new numerator.

Example: $\frac{3}{5} + \frac{4}{7}$

$\text{New denominator} = 5 \times 7 = 35$

$\text{New numerator} = (3 \times 7) + (4 \times 5) = 21 + 20 = 41$

Result: $\frac{41}{35}$

For subtraction, ensure the correct order: $\frac{3}{5} - \frac{4}{7} = \frac{21 - 20}{35} = \frac{1}{35}$.

Multiply numerators together and denominators together:

$\frac{2}{3} \times \frac{5}{7} = \frac{2 \times 5}{3 \times 7} = \frac{10}{21}$

After adding or multiplying fractions, you'll often obtain a large fraction that can be simplified. To reduce a fraction:

1. Identify a common factor in both the numerator and denominator
2. Divide both numerator and denominator by this factor

While finding the greatest common factor (GCF) is most efficient, you can simplify in multiple steps by canceling smaller common factors. For example:

$\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$

Alternatively, you could simplify in two steps:

$\frac{12}{18} = \frac{12 \div 2}{18 \div 2} = \frac{6}{9} = \frac{6 \div 3}{9 \div 3} = \frac{2}{3}$

To divide fractions, follow this simple rule:

Method

1. Keep the first fraction unchanged
2. Flip the second fraction (find its reciprocal)
3. Multiply the two fractions

Examples

Basic division: $\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$

Alternative notation: $\frac{\frac{3}{4}}{\frac{2}{5}} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$

Reciprocals

The reciprocal of a fraction is simply its flipped version:

• Reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$
• Reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$

Key Property: When you multiply reciprocals, the result is always 1: $\frac{a}{b} \times \frac{b}{a} = 1$

Whole numbers and mixed numbers can be converted to fractions for easier calculations:

Whole Numbers to Fractions

Any integer can be written as a fraction with denominator 1:

$16 = \frac{16}{1}$

Mixed Numbers to Improper Fractions

For mixed numbers (combinations of whole numbers and fractions):

Example

Convert $3\frac{1}{2}$ to an improper fraction:

$3\frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{7}{2}$

Why Convert? Working with improper fractions often simplifies arithmetic operations, especially for multiplication and division.

When solving problems, you may need to determine which of several fractions is largest. While comparing fractions with identical denominators is straightforward, different denominators require special techniques.

Standard Comparison Method

To compare fractions accurately:

1. Find a common denominator
2. Convert all fractions to equivalent forms with this denominator
3. Compare the numerators

Efficient Shortcuts

1. The Bowtie Method

For quick comparison without full conversion:

1. Cross-multiply denominators and numerators
2. Compare the products

2. Estimation Technique

Compare fractions to benchmark values like $\frac{1}{2}$:

Comparing Multiple Fractions

When evaluating several fractions:

• Compare pairs sequentially
• Use the most efficient method for each pair
• Eliminate smaller fractions systematically

Operations with Fractions

• Addition: To add fractions, they must have a common denominator. The formula is: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$
• Subtraction: Similar to addition, fractions must have a common denominator: $\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$
• Multiplication: Multiply the numerators and the denominators: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$
• Division: Multiply by the reciprocal of the second fraction: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

A ratio compares two quantities and is expressed as a:b or \frac{a}{b}. Ratios can be simplified just like fractions.

Working with Ratios

To solve problems involving ratios, follow these steps:

1. Convert the ratio into a fraction if necessary.
2. Use cross-multiplication to solve for unknowns:
$a:b = c:d \Rightarrow ad = bc$

Decimals are another way to represent fractions. A decimal is expressed in the form $0.a_1a_2a_3...$, where $a_1, a_2, a_3$ are digits. To convert a fraction to a decimal, divide the numerator by the denominator:

$\text{Decimal value} = \frac{a}{b}$

Operations with Decimals

• Addition: Align the decimal points and add.
• Subtraction: Align the decimal points and subtract.
• Multiplication: Multiply as whole numbers and place the decimal point in the result based on the total number of decimal places in the factors.
• Division: Move the decimal point in the divisor to make it a whole number, and do the same in the dividend.

1. If $\frac{3}{4} + \frac{1}{2} = ?$, what is the answer?

2. If the ratio of boys to girls in a class is 2:3 and there are 12 boys, how many girls are there?

3. Convert $\frac{5}{8}$ to a decimal.

Understanding fractions, ratios, and decimals is essential for solving a variety of problems on the GMAT. Practice these concepts regularly to improve your problem-solving skills.