Fractions: Operations, Rules & Practice

Fractions are simply another way of representing division, proportions, and parts of a whole. In aptitude tests, however, fractions are rarely tested only as basic arithmetic. You may need to compare them quickly, simplify complex expressions, convert between forms, or use them inside word problems.

Basic Fractions rules

A fraction such as \(\frac{5}{24}\), in which the numerator is less than the denominator, is called a proper fraction. Its value is less than 1.

A fraction such as \(\frac{30}{12}\), in which the numerator is greater than the denominator, is called an improper fraction. Its value is greater than 1.

A fraction such as \(\frac{8}{8}\), in which the numerator and denominator are equal, is also an improper fraction. However, its value is exactly 1.

Every fraction, whether proper or improper, can be expressed in decimal form (or as a whole number) by dividing the numerator by the denominator. For instance, \(\frac{3}{10}\) = 0.3, \(\frac{48}{16}\) = 3, and \(\frac{100}{8}\) = 12.5.

1. Understanding fractions

A fraction has two parts:

ab\frac{a}{b}
  • Numerator: — the number being divided.
  • Denominator: — the number doing the dividing.

The denominator cannot be zero.

For example:

35=3÷5\frac{3}{5} = 3÷5

So a fraction can always be interpreted as a division problem.

Quick Recall

A fraction can represent:

  • a part of a whole
  • a ratio
  • a division
  • a proportion

For aptitude tests, being comfortable switching between these interpretations is important.

2. Equivalent Fractions

A fraction has two parts:

ab=akbk\frac{a}{b}=\frac{ak}{bk}

For example:

34=68=1520\frac{3}{4}=\frac{6}{8}=\frac{15}{20}


This is useful when you need to create a common denominator or simplify an expression.

Example

Convert \(\frac{5}{6}\) into a fraction with denominator 24.

Since:

6×4=246\times4=24

multiply both numerator and denominator by 4:

56=2024\frac{5}{6}=\frac{20}{24}

.

Do not automatically convert everything to decimals. Creating an equivalent fraction is often faster.

3. Simplifying Fractions

A fraction is in lowest terms when the numerator and denominator have no common factor greater than 1.

For example:

1824\frac{18}{24}

Both numbers are divisible by 6:

1824=34\frac{18}{24}=\frac{3}{4}

Quick Method

Find the greatest common factor (GCF) of the numerator and denominator, then divide both by it.

Example

SImplify:

84126\frac{84}{126}

The GCF is 42

84126=23\frac{84}{126}=\frac{2}{3}

Why This Matters

Simplifying early can make later calculations significantly easier.

For example:

1528×1425\frac{15}{28}\times\frac{14}{25}

Instead of multiplying first, cancel common factors:

=32×15=\frac{3}{2}\times\frac{1}{5}

=310=\frac{3}{10}

Look for cancellation before multiplying.

4. Adding and Subtracting Fractions

Fractions can be added or subtracted directly only when they have the same denominator.

For example:

37+27=57\frac{3}{7}+\frac{2}{7}=\frac{5}{7}

When denominators are different, first find a common denominator, preferably the least common denominator (LCD).

Example

23+58\frac{2}{3}+\frac{5}{8}

The LCD of 3 and 8 is 24.

Convert both fractions:

23=1624\frac{2}{3}=\frac{16}{24}

58=1524\frac{5}{8}=\frac{15}{24}

Therefore:

1624+1524=3124\frac{16}{24}+\frac{15}{24}=\frac{31}{24}

or:

17241\frac{7}{24}

Important

Do not add the denominators:

23+58≠711\frac{2}{3}+\frac{5}{8}\ne\frac{7}{11}

The denominator represents the size of the parts, so it must remain consistent before the numerators can be combined.

5. Multiplying Fractions

To multiply fractions:

  1. Multiply the numerators.
  2. Multiply the denominators.
  3. Simplify.
ab×cd=acbd\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}

Example

49×1516\frac{4}{9}\times\frac{15}{16}

Cancel common factors first:

19×154\frac{1}{9}\times\frac{15}{4}

Then simplify further:

19×154=1536=512\frac{1}{9}\times\frac{15}{4}=\frac{15}{36}=\frac{5}{12}

Aptitude Strategy
Cancel first, multiply second.

This is particularly useful when numbers are large.

6. Dividing Fractions

Division by a fraction is equivalent to multiplication by its reciprocal.

ab÷cd=ab×dc\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}

Example

56÷109\frac{5}{6}\div\frac{10}{9}

Invert the second fraction:

56×910\frac{5}{6}\times\frac{9}{10}

Cancel:

16×92=912=34\frac{1}{6}\times\frac{9}{2}=\frac{9}{12}=\frac{3}{4}

Remember
Only the divisor is inverted.

7. Comparing Fractions

Aptitude tests often ask which fraction is larger without requiring you to calculate both completely.

Same denominator

The larger numerator gives the larger fraction. If two fractions have the same positive denominator, the fraction with the larger numerator is greater. For instance,

712>512\frac{7}{12}>\frac{5}{12}

Same numerator

The smaller positive denominator gives the larger fraction. If two fractions have the same numerator and their denominators are positive, the fraction with the smaller denominator is greater. For instance,

57>59\frac{5}{7}>\frac{5}{9}

Different numerators and denominators

You can use cross multiplication.

Compare:

712and59\frac{7}{12} and \frac{5}{9}

Compare:

7×9=637\times9=63

and

5×12=605\times12=60

since

63>6063>60

we know:

512>59\frac{5}{12}>\frac{5}{9}

Faster Approach: Benchmarking

Sometimes you can compare fractions against a familiar value such as:

12,1,14\frac{1}{2},1,\frac{1}{4}

For example:

49100<12\frac{49}{100}<\frac{1}{2}

while

51100>12\frac{51}{100}>\frac{1}{2}

This can be much faster than performing calculations.

8. Improper Fractions and Mixed Numbers

An improper fraction has a numerator greater than or equal to its denominator.

Example:

175\frac{17}{5}

It can be converted into a mixed number:

175=325\frac{17}{5}=3\frac{2}{5}

because

17=5(3)+217=5(3)+2

Similarly:

325=1753\frac{2}{5}=\frac{17}{5}

Quick Conversion

For:

abca\frac{b}{c}

multiply the whole number by the denominator and add the numerator:

ac+bc\frac{ac+b}{c}
9. Complex Fractions

A complex fraction is a fraction that contains one or more fractions within its numerator or denominator.

For example:

23+1654−12\frac{\frac{2}{3}+\frac{1}{6}}{\frac{5}{4}-\frac{1}{2}}

The safest approach is usually to simplify the numerator and denominator separately.

Example

23+1334\frac{\frac{2}{3}+\frac{1}{3}}{\frac{3}{4}}

Simplify the numerator:

23+13=1\frac{2}{3}+\frac{1}{3}=1

Therefore:

134=1×43=43\frac{1}{\frac{3}{4}}=1\times\frac{4}{3}=\frac{4}{3}

Aptitude Tip
When a complex fraction looks intimidating, break it into:
numerator → simplify
denominator → simplify
divide the two results

10. Fractions, Decimals and Percentages

The same quantity can often be represented in three forms:

Fraction↔Decimal↔Percentage

For example:

34=0.75=75%\frac{3}{4}=0.75=75\%

Some common conversions worth recognizing quickly:

These conversions can save substantial time in multiple-choice questions.

11. Fractions in Aptitude Problems

Fractions frequently appear inside word problems, rather than as isolated calculations.

For example:

A company spends \(\frac{3}{5}\) of its budget on operations and \(\frac{1}{4}\) on marketing. What fraction of the budget remains?

Combine the portions spent:

35+14=1220+520=1720\frac{3}{5}+\frac{1}{4}=\frac{12}{20}+\frac{5}{20}=\frac{17}{20}

Therefore, the remaining portion is:

1−1720=3201-\frac{17}{20}=\frac{3}{20}

Always ask:
“What does the fraction represent?”

A fraction might represent a portion of:

  • a quantity
  • a budget
  • time
  • distance
  • a population
  • a group
  • a probability
  • a rate

Understanding the context is often more important than the arithmetic itself.

Aptitude-Test Shortcuts

Keep these rules in mind:

  1. Simplify early.
    Reduce fractions before performing lengthy calculations.
  2. Cancel before multiplying.
    This can turn large-number calculations into very small ones.
  3. Don’t add denominators.
    Find a common denominator first.
  4. Division → reciprocal.
    Invert the divisor, then multiply.
  5. Compare intelligently.
    Use cross multiplication, benchmarks, or decimal conversion depending on which is fastest.
  6. Recognize common fractions.
    Knowing common fraction-decimal-percentage equivalents saves time.
  7. Don’t calculate unnecessarily.
    In multiple-choice aptitude tests, estimation or comparison may be enough.

Practice questions

1. Which of the following lists the fractions \(\frac{2}{3}\), \(\frac{5}{8}\), \(\frac{7}{11}\), and \(\frac{13}{20}\) in order from least to greatest?

  • \(\frac{2}{3}\), \(\frac{5}{8}\), \(\frac{7}{11}\), \(\frac{13}{20}\)
  • \(\frac{5}{8}\), \(\frac{7}{11}\), \(\frac{13}{20}\), \(\frac{2}{3}\)
  • \(\frac{5}{8}\), \(\frac{13}{20}\), \(\frac{7}{11}\), \(\frac{2}{3}\)
  • \(\frac{13}{20}\), \(\frac{7}{11}\), \(\frac{5}{8}\), \(\frac{2}{3}\)

2. Which of the following is NOT equivalent to \(\frac{15}{24}\)?

  • \(\frac{45}{72}\)
  • \(\frac{60}{96}\)
  • \(\frac{5}{8}\)
  • \(\frac{3}{5}\)

3. For any positive integer n (where n! means the product of all the integers from 1 to n), what is the value of \(\frac{6!}{8!}\)?

  • \(\frac{1}{56}\)
  • \(\frac{1}{48}\)
  • \(\frac{1}{8}\)
  • \(\frac{1}{4}\)

4. Which of the following is less than \(\frac{5}{9}\)?

  • \(\frac{5}{6}\)
  • \(\frac{21}{36}\)
  • \(\frac{25}{45}\)
  • \(\frac{55}{100}\)

5. Which of the following is (are) greater than x when \(x=\frac{9}{11}\)?

  • \(\frac{1}{x}\)
  • \(\frac{x+1}{x}\)
  • \(\frac{x+1}{x-1}\)
  • 1. I only
    2. II only
    3. I and II only
    4. I, II, and III

Click to reveal the answers
  1. B
  2. D
  3. A
  4. D
  5. C I and II only