Learning Objectives
Explain equivalent fractions using equal-sized parts.
Add, subtract, multiply, and divide fractions.
Convert familiar fractions and decimals and compare their sizes.
Prerequisites: multiplication facts, division, and place value. A fraction a/b means a divided by b, with b ≠ 0. For a positive denominator, it can also describe a copies of a piece whose size is 1/b.
1. Equivalent Fractions Name the Same Amount
Multiplying a numerator and its denominator by the same nonzero number preserves the value of a fraction. For example, 1/2 = 2/4 = 3/6. Each description names the same portion, but uses a different-sized unit piece.
Worked Example: Simplifying
Simplify 18/24. Both numerator and denominator are divisible by 6.
Dividing both by 6 gives 3/4. You can check by multiplying 3 and 4 by 6 to recover 18/24.
Dividing only the numerator would change the amount rather than just its representation.
2. Add and Subtract Like-Sized Pieces
Before adding fractions with different denominators, rewrite them with a common denominator. Then add the numerators and keep that denominator. The denominator names the unit size; it is not another count to add.
Worked Example: A Common Denominator
Calculate 2/3 + 1/4. A common denominator is 12.
Rewrite 2/3 as 8/12 and 1/4 as 3/12. Then 8/12 + 3/12 = 11/12.
The answer is slightly less than 1, which makes sense because 2/3 is less than 3/4 and 3/4 + 1/4 = 1.
Subtraction uses the same preparation. For example, 5/6 − 1/4 = 10/12 − 3/12 = 7/12. A common denominator need not be the smallest possible, but using the least common denominator often keeps the numbers smaller.
3. Multiply and Divide Fractions
To multiply fractions, multiply numerators and multiply denominators. For example, (2/3)(3/5) = 6/15 = 2/5. Taking 2/3 of a quantity smaller than 1 can give a result smaller than either factor; multiplication does not always make a number larger.
To divide by a nonzero fraction, multiply by its reciprocal. The reciprocal of a/b is b/a when a ≠ 0. Thus (3/4) ÷ (2/5) = (3/4)(5/2) = 15/8 = 1 7/8.
Worked Example: How Many Portions?
You have 3/4 litre of juice. Each portion uses 1/8 litre. How many portions can you make?
Compute (3/4) ÷ (1/8) = (3/4) × 8 = 6 portions.
Check by multiplying: 6 × (1/8) = 6/8 = 3/4 litre. Dividing by a small portion size asks how many of those portions fit.
4. Connect Fractions to Decimals
A decimal is another way to write a number. In 0.35, the digits represent 35 hundredths, so 0.35 = 35/100 = 7/20. To turn a fraction into a decimal, divide its numerator by its denominator.
Fraction | Decimal |
|---|---|
1/2 | 0.5 |
1/4 | 0.25 |
3/4 | 0.75 |
1/5 | 0.2 |
1/3 | 0.333… (repeating) |
Common Mistakes
Do not add denominators: 1/2 + 1/2 = 1, not 2/4.
Invert the divisor when dividing fractions, not both fractions.
0.3 and 0.30 are equal. Compare decimals by place value, not by counting digits.
5. Practice
1. Calculate 3/5 + 1/10.
2. Calculate (5/6) ÷ (5/12).
3. Write 0.125 as a simplified fraction.
4. Which is larger: 0.6 or 5/8? Explain.
Worked Solutions
1. 3/5 = 6/10, so the sum is 7/10.
2. Multiply by the reciprocal: (5/6)(12/5) = 2. Two portions of 5/12 equal 5/6.
3. 0.125 = 125/1000 = 1/8 after dividing by 125.
4. 5/8 = 0.625, which is larger than 0.600. Therefore 5/8 is larger.
6. Summary
- Equivalent fractions preserve value while changing the unit pieces.
- Use common denominators for addition and subtraction.
- Multiply across for products and use the divisor's reciprocal for division.
- Use place value or division to connect fractions and decimals.