Working with Fractions

A fraction names part of a whole: in 3/4, the denominator (4) tells you how many equal parts the whole is divided into, and the numerator (3) tells you how many parts you have.

Simplifying fractions

Divide the numerator and denominator by their greatest common factor (GCF).

Example: simplify 12/18. The GCF of 12 and 18 is 6, so 12/18 = 2/3.

A fraction is fully simplified when the only common factor of numerator and denominator is 1.

Converting between mixed numbers and improper fractions

A mixed number like 2⅓ has a whole part and a fraction part. To convert:

1. Multiply the whole number by the denominator: 2 × 3 = 6
2. Add the numerator: 6 + 1 = 7
3. Put that over the denominator: 7/3

To go back the other way, divide: 7 ÷ 3 = 2 remainder 1, so 7/3 = 2⅓.

Adding and subtracting fractions

Fractions must share a denominator before you can add or subtract.

1/4 + 1/6: the least common denominator of 4 and 6 is 12.

1/4 = 3/12 and 1/6 = 2/12, so the sum is 5/12.

Multiplying and dividing fractions

Multiplication is the easiest operation — multiply straight across:

2/3 × 3/5 = 6/15 = 2/5

To divide, flip the second fraction and multiply:

2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6

Fractions to decimals and back

To convert a fraction to a decimal, divide: 3/8 = 0.375.

To convert a decimal to a fraction, use the place value: 0.75 is 75/100, which simplifies to 3/4. Repeating decimals like 0.333… convert to 1/3.

Common pitfalls

• Adding denominators: 1/2 + 1/2 is 2/2 = 1, not 2/4.
• Comparing without a common denominator: 3/5 is larger than 4/7 even though 4 > 3.
• Simplifying only one part: whatever you divide the numerator by must also divide the denominator.