2.2 Fractions: Concepts, Operations & Simplifying

What Is a Fraction?

A fraction has a numerator (top) and a denominator (bottom). The denominator tells how many equal parts the whole is cut into; the numerator tells how many of those parts you have.

• Proper fraction: numerator < denominator. Examples: 1/2, 3/4, 7/8.
• Improper fraction: numerator ≥ denominator. Examples: 5/3, 8/8, 9/2.
• Mixed number: a whole number plus a fraction. Examples: 1 1/2, 2 3/4.

The denominator can never be zero; 5/0 is undefined.

Equivalent Fractions

Two fractions are equivalent if they name the same amount. Multiply or divide both numerator and denominator by the same non-zero number and the value is unchanged: 1/2 = 2/4 = 3/6 = 50/100.

To build a fraction up to a new denominator, multiply top and bottom by the same number. To build 2/3 to sixths: 2/3 × (2/2) = 4/6.

Simplifying (Reducing) Fractions

A fraction is in simplest form when the numerator and denominator share no common factor other than 1. To simplify, divide both parts by their greatest common factor (GCF).

Worked example. Simplify 12/18.

1. Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18.
2. GCF = 6.
3. 12 ÷ 6 = 2, 18 ÷ 6 = 3.
4. 12/18 = 2/3.

Worked example. Simplify 30/45. GCF = 15. 30 ÷ 15 = 2, 45 ÷ 15 = 3. Answer: 2/3.

Converting Mixed Numbers and Improper Fractions

Mixed → improper: multiply the whole number by the denominator, add the numerator, put the result over the denominator.

• 3 1/4 → (3×4 + 1)/4 = 13/4.

Improper → mixed: divide the numerator by the denominator. The quotient is the whole number; the remainder is the new numerator over the denominator.

• 17/5 → 17 ÷ 5 = 3 remainder 2 → 3 2/5.

Adding and Subtracting Fractions

Same denominator (like fractions): add or subtract the numerators, keep the denominator.

• 3/8 + 1/8 = 4/8 = 1/2 (simplify after).
• 5/9 - 2/9 = 3/9 = 1/3.

Different denominators (unlike fractions): find a common denominator first, convert both fractions, then add or subtract the numerators.

Worked example. 2/3 + 1/4.

1. Common denominator: 12 (the LCM of 3 and 4).
2. 2/3 = 8/12, and 1/4 = 3/12.
3. 8/12 + 3/12 = 11/12.

Worked example. 5/6 - 1/4.

1. LCM of 6 and 4 is 12.
2. 5/6 = 10/12, and 1/4 = 3/12.
3. 10/12 - 3/12 = 7/12.

With mixed numbers, you can work as improper fractions or add the whole and fraction parts separately. Borrowing is sometimes needed: 3 1/4 - 1 3/4 = 2 5/4 - 1 3/4 = 1 2/4 = 1 1/2.

Multiplying Fractions

Multiply straight across — numerator times numerator, denominator times denominator — then simplify.

• 2/3 × 4/5 = 8/15.
• 3/4 × 2/3 = 6/12 = 1/2.

You may simplify before multiplying (cross-cancel): 3/4 × 2/3 → cancel the 3s → 1/4 × 2/1 = 2/4 = 1/2.

To multiply a fraction by a whole number, write the whole number as a fraction over 1: 5 × 2/3 = 5/1 × 2/3 = 10/3 = 3 1/3.

Dividing Fractions (Keep-Change-Flip)

To divide, keep the first fraction, change ÷ to ×, and flip (take the reciprocal of) the second fraction. Then multiply.

• 2/3 ÷ 1/4 = 2/3 × 4/1 = 8/3 = 2 2/3.
• 5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4 = 1 1/4.

To divide with mixed numbers, convert to improper fractions first: 2 1/4 ÷ 3/8 = 9/4 ÷ 3/8 = 9/4 × 8/3 = 72/12 = 6.

Fraction Word Problems

Nursing example. A vial holds 3/4 mL of medication. You use 1/8 mL. How much remains?

• 3/4 - 1/8. Common denominator 8: 6/8 - 1/8 = 5/8 mL.

Nursing example. A scored tablet is cut into quarters. A patient takes 1/2 tablet twice a day. Total per day?

• 1/2 × 2 = 1 tablet per day.

Example. A solution needs 3/4 cup of saline, and you make 2 1/2 batches.

• 3/4 × 2 1/2 = 3/4 × 5/2 = 15/8 = 1 7/8 cups.

Common Traps

• Never add denominators. 1/2 + 1/3 is not 2/5. Find the common denominator first.
• Always simplify the final answer. Leaving 4/8 unsimplified loses easy points.
• When dividing, flip only the second fraction, never the first.

Quick Practice

Try these before the quiz:

• Simplify 24/36. GCF is 12, so the answer is 2/3.
• 1 2/5 as an improper fraction: (1×5 + 2)/5 = 7/5.
• 3/5 + 1/2 = 6/10 + 5/10 = 11/10 = 1 1/10.
• 2/5 × 10/3 = 20/15 = 4/3 = 1 1/3.
• 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2 = 1 1/2.

If these feel automatic, you are ready for the decimals section.