Fractions: How to Add, Subtract, Multiply and Divide

To add or subtract fractions, rewrite them with a common denominator and combine the numerators: ( frac12 + frac13 = frac36 + frac26 = frac56 ). To multiply, multiply the numerators and the denominators straight across: ( frac23 times frac45 = frac{8}{15} ). To divide, multiply by the reciprocal of the second fraction: ( frac34 div frac56 = frac34 times frac65 = frac{9}{10} ). Always simplify by dividing top and bottom by their greatest common factor.

A fraction ( frac{a}{b} ) represents a parts out of b equal parts. The top number is the numerator, and the bottom number is the denominator. Fractions provide exact values where decimals recur indefinitely (( frac13 = 0.333ldots )), making them essential for high-level algebra and calculus. This review covers step-by-step rules for working with proper fractions, improper fractions, whole numbers, mixed numbers, algebraic variables, and spreadsheet software.

Fraction review: definitions and types

Before performing arithmetic operations, identify the structure of the fraction you are working with. The table below outlines the primary fraction formats.

Type	Meaning	Example
Proper	Numerator is smaller than the denominator	( frac34 )
Improper	Numerator is equal to or greater than the denominator	( frac74 )
Mixed number	Whole number combined with a proper fraction	( 1frac34 )
Equivalent	Fractions with different numbers that represent the same value	( frac12 = frac24 = frac36 )
Unit	Fraction with a numerator of 1	( frac15 )

Simplifying fractions to lowest terms

You simplify a fraction by reducing the numerator and denominator to their smallest whole numbers without changing the fraction’s value. A fraction is in its lowest terms when the only common factor between the numerator and denominator is 1.

1. Find the greatest common factor (GCF) of the numerator and denominator.
2. Divide both the top and bottom numbers by that GCF.

Example 1: Simplify ( frac{18}{24} ). The factors of 18 are 1, 2, 3, 6, 9, 18. Also, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Their GCF is 6.

Divide top and bottom by 6: ( frac{18 div 6}{24 div 6} = frac34 ).

Example 2: Simplify ( frac{42}{56} ). The GCF of 42 and 56 is 14. Divide both numbers by 14 to get ( frac34 ).

How to add, subtract, multiply, and divide fractions with different denominators

The rules for fraction arithmetic depend on whether you are combining parts (addition and subtraction) or scaling parts (multiplication and division).

Adding and subtracting fractions with different denominators

You cannot add or subtract fractions directly when the denominators are different because the parts are not the same size. You must create equivalent fractions with a least common denominator (LCD) first.

1. First, find the least common multiple (LCM) of the denominators. This number becomes your LCD.
2. Multiply the numerator and denominator of each fraction by the factor needed to convert its denominator to the LCD.
3. Add or subtract the numerators while keeping the denominator unchanged.
4. Simplify the final fraction if needed.

Example 3 (Subtraction): Calculate ( frac23 – frac14 ).

The multiples of 3 are 3, 6, 9, 12, 15. The multiples of 4 are 4, 8, 12, 16. So, the LCD is 12. Convert each fraction: ( frac{2 times 4}{3 times 4} = frac{8}{12} ) and ( frac{1 times 3}{4 times 3} = frac{3}{12} ). Subtract the numerators: ( frac{8 – 3}{12} = frac{5}{12} ).

Example 4 (Addition): Calculate ( frac56 + frac{3}{8} ).

The LCD of 6 and 8 is 24. Convert the fractions: ( frac{5 times 4}{6 times 4} = frac{20}{24} ) and ( frac{3 times 3}{8 times 3} = frac{9}{24} ). Next, add the numerators: ( frac{20 + 9}{24} = frac{29}{24} ). Finally, convert the improper fraction to a mixed number: ( 1frac{5}{24} ).

Multiplying fractions

To multiply fractions, you do not need a common denominator. Instead, you multiply straight across.

1. Multiply numerator by numerator to get the new numerator.
2. Multiply denominator by denominator to get the new denominator.
3. Simplify the resulting fraction.

$$frac{a}{b} times frac{c}{d} = frac{ac}{bd}$$

Example 5: Calculate ( frac35 times frac{10}{9} ).

Multiplying directly gives ( frac{30}{45} ). Then, divide top and bottom by 15 to get ( frac23 ).

Cross-canceling technique: You simplify faster by canceling common factors before multiplying. In ( frac35 times frac{10}{9} ), divide 3 and 9 by 3, and divide 5 and 10 by 5: ( frac{1}{1} times frac{2}{3} = frac23 ).

Finding a fraction of a number: The word “of” means multiplication in mathematics. For example, to find ( frac34 ) of 20, write ( frac34 times 20 = frac{60}{4} = 15 ).

Dividing fractions

To divide by a fraction, multiply by its reciprocal. You obtain a reciprocal by flipping the numerator and denominator.

1. Keep the first fraction unchanged.
2. Change the division sign to a multiplication sign.
3. Flip the second fraction upside down (its reciprocal).
4. Multiply the fractions straight across and simplify.

$$frac{a}{b} div frac{c}{d} = frac{a}{b} times frac{d}{c}$$

Example 6: Calculate ( frac34 div frac56 ).

Keep ( frac34 ), change ( div ) to ( times ), and flip ( frac56 ) to ( frac65 ). This yields ( frac34 times frac65 = frac{18}{20} ). Then, divide numerator and denominator by 2 to get ( frac{9}{10} ).

Example 7 (Word problem): How many ( frac34 )-cup servings fit into 6 cups of flour? Calculate ( 6 div frac34 = 6 times frac43 = frac{24}{3} = 8 ) servings.

How to add, subtract, multiply, and divide fractions with whole numbers

To perform operations between whole numbers and fractions, convert the whole number into an improper fraction by setting its denominator to 1 (for example, ( 5 = frac51 )).

• Adding a whole number and a fraction: Convert the whole number to a fraction with a matching denominator. For ( 3 + frac25 ), rewrite 3 as ( frac{15}{5} ). Then calculate ( frac{15}{5} + frac25 = frac{17}{5} = 3frac25 ).
• Subtracting a fraction from a whole number: Convert the whole number to a fraction. For ( 4 – frac23 ), rewrite 4 as ( frac{12}{3} ). Then calculate ( frac{12}{3} – frac23 = frac{10}{3} = 3frac13 ).
• Multiplying a fraction by a whole number: Place the whole number over 1 and multiply straight across. For example, for ( 5 times frac38 ), calculate ( frac51 times frac38 = frac{15}{8} = 1frac78 ).
• Dividing a fraction by a whole number (or vice versa): Place the whole number over 1 and apply Keep-Change-Flip. For ( frac45 div 2 ), rewrite as ( frac45 div frac21 = frac45 times frac12 = frac{4}{10} = frac25 ). Similarly, for ( 6 div frac23 ), rewrite as ( frac61 times frac32 = frac{18}{2} = 9 ).

How to add, subtract, multiply, and divide mixed fractions

A mixed fraction combines a whole number with a proper fraction (like ( 2frac13 )). Convert mixed fractions into improper fractions before multiplying or dividing.

Converting between mixed numbers and improper fractions

Mixed to improper: Multiply the whole number by the denominator, add the numerator, and place the result over the original denominator. For example, for ( 2frac34 ), calculate ( frac{(2 times 4) + 3}{4} = frac{11}{4} ).

Improper to mixed: Divide the numerator by the denominator. The quotient becomes the whole number, and the remainder becomes the new numerator. For ( frac{29}{24} ), 29 divided by 24 gives 1 with a remainder of 5, resulting in ( 1frac{5}{24} ).

Operations with mixed fractions

Adding mixed numbers: You can add whole parts and fractional parts separately, or convert everything to improper fractions first.

Calculate ( 2frac13 + 1frac12 ). Convert fraction parts to sixths: ( 2frac26 + 1frac36 = 3frac56 ).

Subtracting mixed numbers: When the second fractional part is larger than the first, convert to improper fractions to avoid borrowing errors.

Calculate ( 3frac14 – 1frac56 ). First, convert to improper fractions: ( frac{13}{4} – frac{11}{6} ). Next, find common denominator 12: ( frac{39}{12} – frac{22}{12} = frac{17}{12} = 1frac{5}{12} ).

Multiplying mixed numbers: Convert both numbers to improper fractions before multiplying. Never multiply whole numbers with whole numbers and fractions with fractions.

Calculate ( 1frac12 times 2frac23 ). Convert to improper fractions: ( frac32 times frac83 = frac{24}{6} = 4 ).

Dividing mixed numbers: Convert both numbers to improper fractions, then multiply by the reciprocal of the second fraction.

Calculate ( 2frac14 div 1frac12 ). First, convert to improper fractions: ( frac94 div frac32 ). Then, apply Keep-Change-Flip: ( frac94 times frac23 = frac{18}{12} = frac32 = 1frac12 ).

How to add, subtract, multiply, and divide fractions with variables

Algebraic fractions (rational expressions) follow the exact same mathematical rules as numerical fractions. Treat variable expressions in numerators or denominators as single algebraic units.

• Adding and subtracting with variables: Find a common algebraic denominator. For ( frac1x + frac{1}{x+1} ), the LCD is ( x(x+1) ). Rewrite as ( frac{x+1}{x(x+1)} + frac{x}{x(x+1)} = frac{2x+1}{x(x+1)} ).
• Multiplying with variables: Multiply numerators and denominators straight across, then cancel matching factors. For ( frac{2x}{3} times frac{9}{x^2} ), calculate ( frac{18x}{3x^2} = frac{6}{x} ).
• Dividing with variables: Multiply by the reciprocal of the second term. For ( frac{x}{4} div frac{x^2}{8} ), rewrite as ( frac{x}{4} times frac{8}{x^2} = frac{8x}{4x^2} = frac{2}{x} ).

Reversing algebraic addition by breaking complex fractions into simpler components is called partial fraction decomposition. Expression problems with fractional exponents like ( x^{3/4} ) use exponent rules. Also, differentiating fractions of functions requires the calculus quotient rule.

How to add, subtract, multiply, and divide fractions in Excel

Spreadsheet applications like Microsoft Excel perform fraction math using standard arithmetic operators combined with proper cell formatting.

1. Select the cells where you want fractions to appear.
2. Right-click, select Format Cells, choose Fraction under the Number tab, and select your desired precision (for example, “Up to two digits”).
3. Type fractions into cells using a forward slash, preceded by a zero and space for proper fractions (e.g., type 0 1/2) or with the whole number for mixed numbers (e.g., 1 1/2).
4. Write formulas using standard mathematical symbols:
   • Addition: =A1+B1 or =1/2 + 1/3
   • Subtraction: =A1-B1 or =3/4 - 1/2
   • Multiplication: =A1*B1 or =2/3 * 4/5
   • Division: =A1/B1 or =(3/4)/(5/6)

Fraction calculator

Enter two fractions or mixed numbers (type space between whole numbers and fractions, like 1 1/2) and select an operation to inspect step-by-step working.

The dedicated fraction calculator provides further working steps and conversion options.

Comparing fractions and converting to decimals

To determine which of two fractions is larger, find a common denominator or cross-multiply across the inequality.

To compare ( frac{5}{8} ) and ( frac{3}{5} ), cross-multiply numerators with opposite denominators: ( 5 times 5 = 25 ) and ( 3 times 8 = 24 ). Because 25 is greater than 24, ( frac58 > frac35 ).

Alternatively, convert both to decimals by dividing numerator by denominator: ( frac58 = 0.625 ) and ( frac35 = 0.600 ). Convert decimals to percentages by multiplying by 100 (( 0.75 = 75% )). Review the percentage formula for details.

Common fraction errors and how to fix them

Avoid these frequent procedural mistakes when working through fraction calculations:

• Adding denominators together: Writing ( frac12 + frac13 = frac25 ) is incorrect. Never add denominators. Convert to common denominators first (( frac36 + frac26 = frac56 )).
• Forgetting to flip the second fraction in division: You must multiply by the reciprocal of the divisor. Flipping the first fraction instead produces an incorrect result.
• Canceling terms across addition or subtraction signs: In algebraic fractions, you cannot simplify ( frac{x + 2}{2} ) to ( x + 1 ). You can only cancel common factors that multiply an entire numerator and denominator.
• Multiplying mixed numbers without converting: Writing ( 2frac12 times 3frac12 = 6frac14 ) is incorrect. Convert to improper fractions (( frac52 times frac72 = frac{35}{4} = 8frac34 )).

Fraction review worksheet and practice problems

Test your knowledge with these six review problems. Work through each step before checking the answers below.

1. ( frac34 + frac16 )
2. ( frac78 – frac13 )
3. ( frac{4}{9} times frac{3}{8} )
4. ( frac25 div frac{4}{15} )
5. ( 1frac12 + 2frac23 )
6. Simplify ( frac{42}{56} ) to lowest terms.

Worked Solutions:

1. LCD of 4 and 6 is 12. Convert: ( frac{9}{12} + frac{2}{12} = frac{11}{12} ).

2. LCD of 8 and 3 is 24. Convert: ( frac{21}{24} – frac{8}{24} = frac{13}{24} ).

3. Multiply or cross-cancel: ( frac{4 times 3}{9 times 8} = frac{12}{72} = frac16 ).

4. Keep-Change-Flip: ( frac25 times frac{15}{4} = frac{30}{20} = frac32 = 1frac12 ).

5. Convert to improper fractions: ( frac32 + frac83 ). LCD is 6: ( frac{9}{6} + frac{16}{6} = frac{25}{6} = 4frac16 ).

6. GCF of 42 and 56 is 14: ( frac{42 div 14}{56 div 14} = frac34 ).

Sources: Unit fraction

Frequently asked questions

How do you add fractions with different denominators?

Find the least common denominator (LCD), convert each fraction into an equivalent fraction using that LCD, and add the numerators together while keeping the denominator the same.

How do you multiply fractions?

Multiply the numerators together to form the new numerator, multiply the denominators together to form the new denominator, and reduce the resulting fraction to lowest terms.

How do you divide fractions?

Keep the first fraction, change the division operator to multiplication, flip the second fraction upside down to get its reciprocal, and multiply straight across.

How do you simplify a fraction?

Find the greatest common factor (GCF) of the numerator and denominator, then divide both top and bottom numbers by that factor.

What is a reciprocal of a fraction?

A reciprocal is a fraction flipped upside down. The reciprocal of ( frac23 ) is ( frac32 ). Multiplying any non-zero fraction by its reciprocal yields 1.

Further reading

• Fraction (Wikipedia) — detailed mathematical background on rational numbers and arithmetic history.