GCSE Maths: Multiplying fractions (Foundation)
Part of the GCSE Maths fractions chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Multiplying fractions
Finding “a fraction of a fraction” means multiplying. In the diagram, the blue columns show 34 and the orange rows show 23. The dark overlap is 34 × 23: 6 squares out of 12.
Key ideas
- Multiply the numerators and multiply the denominators, then simplify.
- You can cancel first: divide a numerator and a denominator by a common factor before multiplying.
- Write whole numbers as fractions over 1, for example 4 = 41.
- For mixed numbers, change to improper fractions first.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.23 × 45
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.Tops: 2, 4. Bottoms: 3, 5. Nothing cancels here.
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.2 × 4 = 8
3 × 5 = 15
815 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.8 and 15 have no common factor except 1, so 815 is the answer.
Check. A different route, as decimals: 0.6667 × 0.8 = 0.5333, and our answer 815 = 0.5333. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.38 × 49
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.38 × 49 = 12 × 13
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 1 = 1
2 × 3 = 6
16 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 6 have no common factor except 1, so 16 is the answer.
Check. A different route, as decimals: 0.375 × 0.4444 = 0.1667, and our answer 16 = 0.1667. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.212 × 115
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.212 = (2 × 2 + 1) over 2 = 52
115 = (1 × 5 + 1) over 5 = 65 - Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.52 × 65 = 11 × 31
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 3 = 3
1 × 1 = 1
31 - Write the answer. The bottom number is 1, so the answer is a whole number.31 = 3
Check. A different route, as decimals: 2.5 × 1.2 = 3, and our answer 3 = 3. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.5 × 29
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.5 = 51
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.Tops: 5, 2. Bottoms: 1, 9. Nothing cancels here.
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.5 × 2 = 10
1 × 9 = 9
109 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.10 ÷ 9 = 1 remainder 1
109 = 119
Check. A different route, as decimals: 5 × 0.2222 = 1.1111, and our answer 119 = 1.1111. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Decide what to calculate. The area of a rectangle is length × width, so we multiply the two lengths.Area = 56 × 310
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.56 × 310 = 12 × 12
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 1 = 1
2 × 2 = 4
14 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 4 have no common factor except 1, so 14 is the answer.
- Write the answer with its units. We multiplied two lengths in metres, so the area is in square metres.Area = 14 m²
Check. A different route, as decimals: 0.8333 × 0.3 = 0.25, and our answer 14 = 0.25. ✓
Watch out. Area is in square units (m²), not m. And multiplying two fractions less than 1 gives a smaller answer.
Watch out
Multiplying does not always make a number bigger. Multiplying by a fraction less than 1 makes it smaller: 12 × 12 = 14.Exercise 3F Multiplying fractions
Non-calculatorMethod. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.12 × 13
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.Tops: 1, 1. Bottoms: 2, 3. Nothing cancels here.
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 1 = 1
2 × 3 = 6
16 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 6 have no common factor except 1, so 16 is the answer.
Check. A different route, as decimals: 0.5 × 0.3333 = 0.1667, and our answer 16 = 0.1667. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.25 × 37
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.Tops: 2, 3. Bottoms: 5, 7. Nothing cancels here.
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.2 × 3 = 6
5 × 7 = 35
635 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.6 and 35 have no common factor except 1, so 635 is the answer.
Check. A different route, as decimals: 0.4 × 0.4286 = 0.1714, and our answer 635 = 0.1714. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.34 × 29
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.34 × 29 = 12 × 13
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 1 = 1
2 × 3 = 6
16 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 6 have no common factor except 1, so 16 is the answer.
Check. A different route, as decimals: 0.75 × 0.2222 = 0.1667, and our answer 16 = 0.1667. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.45 × 38
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.45 × 38 = 15 × 32
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 3 = 3
5 × 2 = 10
310 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.3 and 10 have no common factor except 1, so 310 is the answer.
Check. A different route, as decimals: 0.8 × 0.375 = 0.3, and our answer 310 = 0.3. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.56 × 310
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.56 × 310 = 12 × 12
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 1 = 1
2 × 2 = 4
14 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 4 have no common factor except 1, so 14 is the answer.
Check. A different route, as decimals: 0.8333 × 0.3 = 0.25, and our answer 14 = 0.25. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.79 × 314
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.79 × 314 = 13 × 12
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 1 = 1
3 × 2 = 6
16 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 6 have no common factor except 1, so 16 is the answer.
Check. A different route, as decimals: 0.7778 × 0.2143 = 0.1667, and our answer 16 = 0.1667. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.3 × 25
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.3 = 31
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.Tops: 3, 2. Bottoms: 1, 5. Nothing cancels here.
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.3 × 2 = 6
1 × 5 = 5
65 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.6 ÷ 5 = 1 remainder 1
65 = 115
Check. A different route, as decimals: 3 × 0.4 = 1.2, and our answer 115 = 1.2. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.4 × 38
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.4 = 41
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.41 × 38 = 11 × 32
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 3 = 3
1 × 2 = 2
32 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.3 ÷ 2 = 1 remainder 1
32 = 112
Check. A different route, as decimals: 4 × 0.375 = 1.5, and our answer 112 = 1.5. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.6 × 59
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.6 = 61
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.61 × 59 = 21 × 53
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.2 × 5 = 10
1 × 3 = 3
103 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.10 ÷ 3 = 3 remainder 1
103 = 313
Check. A different route, as decimals: 6 × 0.5556 = 3.3333, and our answer 313 = 3.3333. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.23 × 12
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.12 = 121
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.23 × 121 = 21 × 41
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.2 × 4 = 8
1 × 1 = 1
81 - Write the answer. The bottom number is 1, so the answer is a whole number.81 = 8
Check. A different route, as decimals: 0.6667 × 12 = 8, and our answer 8 = 8. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.57 × 14
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.14 = 141
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.57 × 141 = 51 × 21
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.5 × 2 = 10
1 × 1 = 1
101 - Write the answer. The bottom number is 1, so the answer is a whole number.101 = 10
Check. A different route, as decimals: 0.7143 × 14 = 10, and our answer 10 = 10. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.34 × 10
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.10 = 101
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.34 × 101 = 32 × 51
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.3 × 5 = 15
2 × 1 = 2
152 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.15 ÷ 2 = 7 remainder 1
152 = 712
Check. A different route, as decimals: 0.75 × 10 = 7.5, and our answer 712 = 7.5. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.112 × 23
- Change the mixed number to improper fraction. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.112 = (1 × 2 + 1) over 2 = 32
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.32 × 23 = 11 × 11
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 1 = 1
1 × 1 = 1
11 - Write the answer. The bottom number is 1, so the answer is a whole number.11 = 1
Check. A different route, as decimals: 1.5 × 0.6667 = 1, and our answer 1 = 1. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.214 × 113
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.214 = (2 × 4 + 1) over 4 = 94
113 = (1 × 3 + 1) over 3 = 43 - Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.94 × 43 = 31 × 11
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.3 × 1 = 3
1 × 1 = 1
31 - Write the answer. The bottom number is 1, so the answer is a whole number.31 = 3
Check. A different route, as decimals: 2.25 × 1.3333 = 3, and our answer 3 = 3. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.313 × 145
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.313 = (3 × 3 + 1) over 3 = 103
145 = (1 × 5 + 4) over 5 = 95 - Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.103 × 95 = 21 × 31
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.2 × 3 = 6
1 × 1 = 1
61 - Write the answer. The bottom number is 1, so the answer is a whole number.61 = 6
Check. A different route, as decimals: 3.3333 × 1.8 = 6, and our answer 6 = 6. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.225 × 114
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.225 = (2 × 5 + 2) over 5 = 125
114 = (1 × 4 + 1) over 4 = 54 - Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.125 × 54 = 31 × 11
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.3 × 1 = 3
1 × 1 = 1
31 - Write the answer. The bottom number is 1, so the answer is a whole number.31 = 3
Check. A different route, as decimals: 2.4 × 1.25 = 3, and our answer 3 = 3. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.134 × 227
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.134 = (1 × 4 + 3) over 4 = 74
227 = (2 × 7 + 2) over 7 = 167 - Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.74 × 167 = 11 × 41
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 4 = 4
1 × 1 = 1
41 - Write the answer. The bottom number is 1, so the answer is a whole number.41 = 4
Check. A different route, as decimals: 1.75 × 2.2857 = 4, and our answer 4 = 4. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. In an area model, the fraction of a fraction is the overlap of the two shadings.
- Count the squares in the whole grid. The grid is 4 columns by 2 rows. The total is the denominator of the answer.4 × 2 = 8 squares
- Count the squares shaded by the first fraction. 34 means 3 of the 4 columns are shaded.3 columns
- Count the squares shaded by the second fraction. 12 means 1 of the 2 rows is shaded.1 row
- Count the overlap (the dark squares). Where a shaded column and the shaded row cross, squares are shaded twice. Multiply columns by rows.3 × 1 = 3 dark squares
- Write the fraction. Dark squares over total squares.38
Check. Multiply the fractions: 12 × 34 = 38 (tops 1 × 3 = 3, bottoms 2 × 4 = 8). ✓
Watch out. The overlap is the answer, not the total shaded area.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Decide the operation. The area of a rectangle is length × width, so multiply the two sides.Area = 34 × 23
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.34 × 23 = 12 × 11
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 1 = 1
2 × 1 = 2
12 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 2 have no common factor except 1, so 12 is the answer.
- Write the answer with its units. Area is measured in square units, here square metres.Area = 12 m²
Check. A different route, as decimals: 0.75 × 0.6667 = 0.5, and our answer 12 = 0.5. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Decide the operation. “A fraction of” means multiply. The water used is one quarter of the water that is in the tank, which is three fifths of the whole tank.14 of 35 = 35 × 14
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.Tops: 3, 1. Bottoms: 5, 4. Nothing cancels here.
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.3 × 1 = 3
5 × 4 = 20
320 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.3 and 20 have no common factor except 1, so 320 is the answer.
- Interpret the answer. The product is a fraction of the whole tank, because the three fifths were already a fraction of the whole tank.320 of the whole tank is used.
Check. A different route, as decimals: 0.6 × 0.25 = 0.15, and our answer 320 = 0.15. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To show a statement is false you need just one counter-example: a case where it does not work.
- Say what Sam claims. Sam says the product of two numbers is always bigger than both of them.Claim: a × b > a and a × b > b
- Choose two small fractions. Fractions less than 1 will make the product smaller. Pick the simplest one.12 and 12
- Multiply them. Multiply the tops and the bottoms.12 × 12 = 1 × 12 × 2 = 14
- Compare the answer with the numbers. 14 is smaller than 12, so the answer is smaller than both numbers.14 < 12
- Conclude. One counter-example is enough to prove the claim is false.Sam is wrong.
Check. Think of it as “half of a half”. Half of something is less than it, so the answer 14 must be smaller than 12. ✓
Watch out. The claim is true for whole numbers bigger than 1, but not for fractions. One counter-example is enough to disprove it.
Method. Area of a triangle = ½ × base × height. Use improper fractions and multiply two at a time.
- Change the mixed numbers to improper fractions. Whole × denominator + numerator, over the same denominator.112 = 32 and 223 = 83
- Write the calculation. Multiply the three numbers.Area = 12 × 32 × 83
- Multiply the first two. Multiply the tops and the bottoms.12 × 32 = 34
- Multiply by the third and cancel. Cancel the 3s, and cancel 4 with 8 (both divide by 4).34 × 83 = 11 × 21 = 2
- Write the answer with its units. Area is in square centimetres.Area = 2 cm²
Check. As decimals: 0.5 × 1.5 × 2.667 = 2. ✓
Watch out. Do not forget the ½ in the triangle formula.
Method. To find a missing fraction in a multiplication, divide the answer by the known fraction.
- Write the problem as an equation. The missing fraction times 2/3 gives 1/6.23 × ? = 16
- Use the inverse operation. The inverse of multiplying is dividing. So the missing fraction is the answer divided by the known fraction.? = 16 ÷ 23
- Keep, change, flip. Dividing by a fraction is multiplying by its reciprocal.16 × 32 = 312
- Simplify. Divide the top and the bottom by 3.312 = 14
Check. Multiply back: 23 × 14 = 212 = 16. ✓
Watch out. Flip only the fraction you are dividing by (here 2/3), not the 1/6.