Math 8 · Multiplying and dividing fractions
A part of a part, and how many fit
Multiplying by a fraction takes part of something, which is why ½ × ⅔ is smaller than both. Dividing by a fraction asks how many of it fit, which is why 3 ÷ ½ is 6. Each has a short rule. This unit is about why the rules work, so you can tell when an answer cannot be right.
- 1. The words, first
- 2. Multiplying fractions and mixed numbers
- 3. Dividing fractions and mixed numbers
- 4. Problems with fractions
- 5. What costs marks
The words, first
The idea: Reciprocal is the new word. It is what turns dividing by a fraction into multiplying.
| Word | What it means |
|---|---|
| Numerator, denominator | The top and bottom of a fraction: how many pieces you have, and how many equal pieces make a whole. |
| Mixed number, improper fraction | 2¼ is a mixed number, 2 wholes and a quarter. As an improper fraction, top bigger than bottom, it is 9/4. |
| Lowest terms | Top and bottom have no common factor but 1: ½, not 6/12. |
| “Of” | With fractions, “of” means multiply: ⅔ of 12 is ⅔ × 12 = 8. |
| Reciprocal | A fraction turned upside down. The reciprocal of 4/5 is 5/4, and of 5 is 1/5. A number times its reciprocal is 1. |
| Estimate | A rough answer from rounded numbers, used to check the real one. |
Multiplying fractions and mixed numbers
The idea: Multiply the numerators, multiply the denominators, and simplify. Mixed numbers must become improper fractions first.
⅔ × ¾ = (2 × 3)/(3 × 4) = 6/12 = ½
Why it works. Cut a square into 4 columns and shade 3: that is ¾. Now cut it into 3 rows and keep 2. The part in both is 2 × 3 = 6 small pieces, out of 3 × 4 = 12 in the whole square, so ⅔ of ¾ is 6/12, or ½. The numerators count the pieces you keep; the denominators count the pieces in the whole.
Simplify first. In 4/9 × 3/8, the 4 and 8 share a factor of 4, and the 3 and 9 share a factor of 3. Divide them out first: 1/3 × 1/2 = 1/6. Multiplying first gives 12/72, the same amount but harder to simplify.
A whole number times a fraction. 5 × ⅔ = 5/1 × 2/3 = 10/3 = 3⅓.
Worked example. A recipe needs 2¼ cups of flour, and you are making ⅔ of the recipe. How much flour?
- Mixed to improper: 2¼ = 9/4, because 2 wholes are 8 quarters, plus 1 more.
- Multiply: ⅔ × 9/4 = 18/12.
- Simplify: 18/12 = 3/2 = 1½ cups.
- Estimate: ⅔ of a bit over 2 is a bit over 1 ✓.
The mixed-number trap. 1½ × 2½ is not 1 × 2 plus ½ × ½, which is 2¼. As improper fractions it is 3/2 × 5/2 = 15/4 = 3¾. In decimals, 1.5 × 2.5 = 3.75, which agrees.
Dividing fractions and mixed numbers
The idea: Dividing asks how many of one amount fit into another. With a common denominator you can count them; the quick way is to multiply by the reciprocal.
How many fit. 3 ÷ ½ asks how many halves are in 3. Each whole holds 2, so there are 6. Dividing by a number less than 1 gives an answer bigger than you started with.
Same denominators: count the pieces. ¾ ÷ 1/8 = 6/8 ÷ 1/8. Six eighths hold six one-eighths, so the answer is 6.
The quick way: multiply by the reciprocal.
⅔ ÷ 4/5 = ⅔ × 5/4 = 10/12 = 5/6
Why it works. Dividing by 4/5 means counting the fifths, then putting them in groups of 4. Multiplying by 5 counts the fifths and dividing by 4 makes the groups, which is exactly multiplying by 5/4. Check by multiplying back: 5/6 × 4/5 = 20/30 = ⅔ ✓.
Worked example. How many ⅔-cup scoops of rice are in a 5-cup bag?
- Write the division: 5 ÷ ⅔.
- Multiply by the reciprocal: 5 × 3/2 = 15/2.
- As a mixed number: 15/2 = 7½. That is 7 full scoops, and half a scoop left over.
- Check: 7½ × ⅔ = 15/2 × 2/3 = 30/6 = 5 ✓.
Mixed numbers. Change them to improper fractions first. A ribbon 3¾ m long is cut into pieces 1¼ m long: 3¾ ÷ 1¼ = 15/4 ÷ 5/4, which is 15 quarters shared into groups of 5 quarters, so 3 pieces.
Problems with fractions
The idea: Decide the operation from the story: “of” and “for each” multiply, and “how many fit” divides. Estimate first, then check the answer against the estimate.
Worked example. A hiking trail is 7½ km long, and Ana has walked 3/5 of it. How far does she have left?
- “Of” means multiply: 3/5 × 7½ = 3/5 × 15/2 = 45/10 = 4½ km walked.
- Subtract: 7½ − 4½ = 3 km left.
- Check another way. She has 2/5 of the trail left, and 2/5 × 15/2 = 30/10 = 3 ✓.
Worked example. A bag holds 6¾ cups of trail mix, and one serving is ¾ cup. How many servings? The question is how many ¾s fit into 6¾, so divide: 6¾ ÷ ¾ = 27/4 ÷ 3/4 = 9 servings. With the common denominator, it is 27 quarters in groups of 3.
The order of operations still applies. ½ + ¾ × ⅔ means multiply first: ¾ × ⅔ = ½, and ½ + ½ = 1. Adding first gives 5/4 × ⅔ = 5/6, which answers a different question.
Is it reasonable? Multiplying by a fraction less than 1 makes a number smaller, and dividing by one makes it bigger. If ⅔ of a recipe needs more flour than the whole recipe, something went wrong.
What costs marks
The idea: An estimate catches most of these before anyone marks them.
- Multiplying the wholes and the fractions separately. 1½ × 2½ = 3¾, not 2¼. Make improper fractions first.
- Flipping the wrong fraction. In ⅔ ÷ 4/5, the one you divide by, 4/5, is the one that flips.
- Hunting for a common denominator to multiply. You need one to add or subtract, not to multiply.
- Expecting a product to be bigger. ⅔ × 9/4 is less than 9/4.
- Leaving the answer as 15/2 scoops. Say what it means: 7 full scoops, and half a scoop over.