GCSE Maths: Fraction arithmetic with confidence (Higher)
Part of the GCSE Maths fractions chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Fraction arithmetic with confidence
At Higher level you are expected to handle fractions quickly and accurately inside longer calculations, with brackets, mixed numbers and negative values. Everything here builds on the four operations from Foundation.
Key ideas
- Follow BIDMAS: brackets, indices, then division and multiplication, then addition and subtraction.
- Work out the top and bottom of a “big” fraction separately, then divide: ab ÷ cd.
- The sign rules for integers apply to fractions: a negative × a negative is positive; a negative ÷ a positive is negative.
- Always give the answer in its simplest form, and as a mixed number when the question works with mixed numbers.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).315 − 134
- 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.315 = (3 × 5 + 1) over 5 = 165
134 = (1 × 4 + 3) over 4 = 74 - Find a common denominator: the LCM of 5 and 4. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 5: 5, 10, 15, 20
Multiples of 4: 4, 8, 12, 16, 20
LCM = 20 - Rewrite both fractions with denominator 20. Multiply the top and the bottom by the same number so the value does not change.165 = 6420 (× 4)
74 = 3520 (× 5) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.6420 − 3520 = 2920
- 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.29 ÷ 20 = 1 remainder 9
2920 = 1920
Check. A different route, as decimals: 3.2 − 1.75 = 1.45, and our answer 1920 = 1.45. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
Method. Follow BIDMAS: work out each bracket completely first, then divide by flipping the second fraction.
- Decide the order of working. BIDMAS says brackets come first. There are two brackets, so each must become a single fraction before we divide.(23 + 14) ÷ (56 − 12)
- Work out the first bracket. To add we need a common denominator. The LCM of 3 and 4 is 12.23 = 812 and 14 = 312
812 + 312 = 1112 - Work out the second bracket. The LCM of 6 and 2 is 6, so write a half as three sixths. Then subtract and simplify.12 = 36
56 − 36 = 26 = 13 - Keep, change, flip. Dividing by a fraction is multiplying by its reciprocal. Flip only the second fraction.1112 ÷ 13 = 1112 × 31
- Cancel and multiply. The 3 on top and the 12 on the bottom share a factor of 3.1112 × 31 = 114
- Write as a mixed number. The top is bigger than the bottom, so divide: 11 ÷ 4 = 2 remainder 3.114 = 234
Check. A different route, as decimals: 0.667 + 0.25 = 0.917 and 0.833 − 0.5 = 0.333. Then 0.917 ÷ 0.333 = 2.75, and 2 3/4 = 2.75. ✓
Watch out. Do not flip until both brackets are single fractions. Flip only the fraction you are dividing by.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).25 − 34
- Find a common denominator: the LCM of 5 and 4. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 5: 5, 10, 15, 20
Multiples of 4: 4, 8, 12, 16, 20
LCM = 20 - Rewrite both fractions with denominator 20. Multiply the top and the bottom by the same number so the value does not change.25 = 820 (× 4)
34 = 1520 (× 5) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.820 − 1520 = −720
- Write the answer. The result is negative because the second fraction was bigger.−720
Check. Add back: −720 + 1520 = 820 = 25, the number we started with. ✓
Watch out. The answer is negative because the second fraction is bigger than the first. Keep the order: first minus second.
Method. Work out the bracket first. Then dividing 1 by a fraction gives its reciprocal.
- Do the bracket first. BIDMAS says brackets come before division.12 + 13
- Find a common denominator. The LCM of 2 and 3 is 6.12 = 36 (× 3) and 13 = 26 (× 2)
- Add the numerators. The pieces are now the same size, so keep the denominator.36 + 26 = 56
- Divide 1 by the result. Write 1 as a fraction over 1, then keep, change, flip.1 ÷ 56 = 11 × 65 = 65
- Write as a mixed number. 6 ÷ 5 = 1 remainder 1.65 = 115
Check. As decimals: 0.5 + 0.333 = 0.833 and 1 ÷ 0.833 = 1.2, which is 115. ✓
Watch out. 11 ÷ (12 + 13) is not 11 ÷ 12 + 11 ÷ 13. You cannot split a division over a sum.
Method. Decide the sign first with the rules for negatives, then divide the fractions as normal.
- Decide the sign of the answer. A negative divided by a negative is positive. So we can work with the positive values and the answer is positive.− ÷ − = +, so we work out 58 ÷ 1516
- Keep, change, flip. Turn the second fraction upside down and change the division to a multiplication.58 ÷ 1516 = 58 × 1615
- Cancel common factors. 5 on the top and 15 on the bottom share a factor of 5. 16 on the top and 8 on the bottom share a factor of 8.58 × 1615 = 11 × 23
- Multiply across. Top times top and bottom times bottom.Tops: 1 × 2 = 2
Bottoms: 1 × 3 = 3
23 - Write the answer with its sign. The sign was decided in step 1: positive.23
Check. Multiply back: 23 × −1516 = −3048 = −58, which is the number we started with. ✓
Watch out. Two negatives in a division give a positive. Do not put a minus sign in the answer.
Watch out
In (23 + 14) ÷ (56 − 12), do not flip until both brackets are single fractions. Flip only the fraction you are dividing by.Exercise 3A Fraction arithmetic
Non-calculatorMethod. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).315 − 134
- 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.315 = (3 × 5 + 1) over 5 = 165
134 = (1 × 4 + 3) over 4 = 74 - Find a common denominator: the LCM of 5 and 4. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 5: 5, 10, 15, 20
Multiples of 4: 4, 8, 12, 16, 20
LCM = 20 - Rewrite both fractions with denominator 20. Multiply the top and the bottom by the same number so the value does not change.165 = 6420 (× 4)
74 = 3520 (× 5) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.6420 − 3520 = 2920
- 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.29 ÷ 20 = 1 remainder 9
2920 = 1920
Check. A different route, as decimals: 3.2 − 1.75 = 1.45, and our answer 1920 = 1.45. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.
- Read the question and decide the plan. We can only add pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).223 + 456
- 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.223 = (2 × 3 + 2) over 3 = 83
456 = (4 × 6 + 5) over 6 = 296 - Find a common denominator: the LCM of 3 and 6. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 3: 3, 6
Multiples of 6: 6
LCM = 6 - Rewrite both fractions with denominator 6. Multiply the top and the bottom by the same number so the value does not change.83 = 166 (× 2)
296 = 296 (× 1) - Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.166 + 296 = 456
- Simplify the answer. 45 and 6 have a common factor of 3 (their HCF), so divide the top and the bottom by 3.45 ÷ 3 = 15 and 6 ÷ 3 = 2
456 = 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: 2.6667 + 4.8333 = 7.5, and our answer 712 = 7.5. ✓
Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).518 − 234
- 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.518 = (5 × 8 + 1) over 8 = 418
234 = (2 × 4 + 3) over 4 = 114 - Find a common denominator: the LCM of 8 and 4. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 8: 8
Multiples of 4: 4, 8
LCM = 8 - Rewrite both fractions with denominator 8. Multiply the top and the bottom by the same number so the value does not change.418 = 418 (× 1)
114 = 228 (× 2) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.418 − 228 = 198
- 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.19 ÷ 8 = 2 remainder 3
198 = 238
Check. A different route, as decimals: 5.125 − 2.75 = 2.375, and our answer 238 = 2.375. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).629 − 356
- 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.629 = (6 × 9 + 2) over 9 = 569
356 = (3 × 6 + 5) over 6 = 236 - Find a common denominator: the LCM of 9 and 6. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 9: 9, 18
Multiples of 6: 6, 12, 18
LCM = 18 - Rewrite both fractions with denominator 18. Multiply the top and the bottom by the same number so the value does not change.569 = 11218 (× 2)
236 = 6918 (× 3) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.11218 − 6918 = 4318
- 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.43 ÷ 18 = 2 remainder 7
4318 = 2718
Check. A different route, as decimals: 6.2222 − 3.8333 = 2.3889, and our answer 2718 = 2.3889. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
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 × 159
- 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
159 = (1 × 9 + 5) over 9 = 149 - 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 × 149 = 12 × 71
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 7 = 7
2 × 1 = 2
72 - 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.7 ÷ 2 = 3 remainder 1
72 = 312
Check. A different route, as decimals: 2.25 × 1.5556 = 3.5, and our answer 312 = 3.5. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.335 ÷ 225
- 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.335 = (3 × 5 + 3) over 5 = 185
225 = (2 × 5 + 2) over 5 = 125 - Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.185 ÷ 125 = 185 × 512
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.185 × 512 = 31 × 12
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.3 × 1 = 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. Multiply back: 112 × 225 should give 335. 1.5 × 2.4 = 3.6 and 335 = 3.6. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the 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.412 × 223
- 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.412 = (4 × 2 + 1) over 2 = 92
223 = (2 × 3 + 2) over 3 = 83 - 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.92 × 83 = 31 × 41
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.3 × 4 = 12
1 × 1 = 1
121 - Write the answer. The bottom number is 1, so the answer is a whole number.121 = 12
Check. A different route, as decimals: 4.5 × 2.6667 = 12, and our answer 12 = 12. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.178 ÷ 334
- 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.178 = (1 × 8 + 7) over 8 = 158
334 = (3 × 4 + 3) over 4 = 154 - Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.158 ÷ 154 = 158 × 415
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.158 × 415 = 12 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: 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.
Check. Multiply back: 12 × 334 should give 178. 0.5 × 3.75 = 1.875 and 178 = 1.875. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. Follow BIDMAS: Brackets, Indices (powers), Division and Multiplication, then Addition and Subtraction.
- Decide the order of working. There is a multiplication and an addition. Multiplication comes before addition, so do the multiplication first.23 + 14 × 25: multiply first
- Multiply first. Cancel the common factor 2 (between the 2 on top and the 4 on the bottom), then multiply across.14 × 25 = 12 × 15 = 110
- Find a common denominator for the addition. The LCM of 3 and 10 is 30.23 = 2030 (× 10)
110 = 330 (× 3) - Add the numerators. The pieces are now the same size, so keep the denominator.2030 + 330 = 2330
Check. As decimals: 0.667 + 0.25 × 0.4 = 0.667 + 0.1 = 0.767, and 2330 = 0.767. ✓
Watch out. Do not add first. If you add before multiplying you get a different (wrong) answer.
Method. Follow BIDMAS: Brackets, Indices (powers), Division and Multiplication, then Addition and Subtraction.
- Decide the order of working. Brackets first. There are two brackets, so each must become a single fraction before we divide.(23 + 14) ÷ (56 − 12)
- Work out the first bracket. The LCM of 3 and 4 is 12.23 + 14 = 812 + 312 = 1112
- Work out the second bracket. The LCM of 6 and 2 is 6. Then simplify the answer.56 − 12 = 56 − 36 = 26 = 13
- Keep, change, flip. Dividing by a fraction is multiplying by its reciprocal. Flip only the second fraction.1112 ÷ 13 = 1112 × 31 = 3312 = 114
- Write as a mixed number. 11 ÷ 4 = 2 remainder 3.234
Check. As decimals: 0.917 ÷ 0.333 = 2.75, which is 2 3/4. ✓
Watch out. Work out both brackets completely before dividing.
Method. Follow BIDMAS: Brackets, Indices (powers), Division and Multiplication, then Addition and Subtraction.
- Decide the order of working. Multiplication comes before subtraction, so do the multiplication first.34 − 25 × 56: multiply first
- Multiply first. Cancel the 5s and then the common factor 2 (2 on top, 6 on the bottom).25 × 56 = 26 = 13
- Find a common denominator for the subtraction. The LCM of 4 and 3 is 12.34 = 912 (× 3)
13 = 412 (× 4) - Subtract the numerators. Take the second fraction away from the first.912 − 412 = 512
Check. As decimals: 0.75 − 0.4 × 0.833 = 0.75 − 0.333 = 0.417, and 512 = 0.417. ✓
Watch out. Do not subtract first. Multiply before you subtract.
Method. Follow BIDMAS: Brackets, Indices (powers), Division and Multiplication, then Addition and Subtraction.
- Decide the order of working. The bracket comes first, then the power (squaring).(112 − 23)²
- Change the mixed number to an improper fraction. Whole × denominator + numerator, over the same denominator.112 = 32
- Work out the bracket. The LCM of 2 and 3 is 6.32 − 23 = 96 − 46 = 56
- Square the result. To square a fraction, square the top and square the bottom.(56)² = 5²6² = 2536
Check. As decimals: (1.5 − 0.667)² = 0.833² = 0.694, and 2536 = 0.694. ✓
Watch out. Square the top and the bottom. The square of 5/6 is not 10/12.
Method. Follow BIDMAS: Brackets, Indices (powers), Division and Multiplication, then Addition and Subtraction.
- Decide the order of working. The bracket must be worked out first. Then divide 1 by the result.1 ÷ (12 + 13)
- Work out the bracket. The LCM of 2 and 3 is 6.12 + 13 = 36 + 26 = 56
- Divide 1 by the result. Dividing 1 by a fraction gives its reciprocal. Keep, change, flip.1 ÷ 56 = 11 × 65 = 65
- Write as a mixed number. 6 ÷ 5 = 1 remainder 1.115
Check. As decimals: 1 ÷ 0.833 = 1.2, which is 1 1/5. ✓
Watch out. You cannot split the division over the sum: 1 ÷ (12 + 13) is not 1 ÷ 12 + 1 ÷ 13.
Method. Follow BIDMAS: Brackets, Indices (powers), Division and Multiplication, then Addition and Subtraction.
- Decide the order of working. Multiplication and division come before subtraction. Do each division separately, then subtract.34 ÷ 12 − 23 ÷ 49
- Work out the first division. Keep, change, flip.34 ÷ 12 = 34 × 21 = 64 = 32
- Work out the second division. Keep, change, flip. Then cancel the 3 and the 9, and the 2 and the 4.23 ÷ 49 = 23 × 94 = 1812 = 32
- Subtract. Both answers are the same, so the difference is zero.32 − 32 = 0
Check. As decimals: 0.75 ÷ 0.5 = 1.5 and 0.667 ÷ 0.444 = 1.5. 1.5 − 1.5 = 0. ✓
Watch out. Check each division separately before you subtract.
Method. Rewrite with a common denominator, then add the numerators, paying attention to the signs.
- Find a common denominator. The LCM of 4 and 2 is 4. Write 1/2 as quarters.12 = 24
- Rewrite the calculation. The negative sign stays with the first fraction.−34 + 24
- Add the numerators. A negative plus a positive: take the smaller size away from the bigger size and keep the sign of the bigger one. 3 is bigger than 2 and is negative.−3 + 2 = −1
- Write the answer. Keep the denominator.−14
Check. On a number line: start at −0.75 and move 0.5 to the right. You land on −0.25. ✓
Watch out. Be careful with the sign of the larger number. The answer is negative here because the negative part is bigger.
Method. Rewrite with a common denominator, then subtract the numerators.
- Find a common denominator. The LCM of 3 and 6 is 6.13 = 26
- Rewrite the calculation. Now the pieces are the same size.26 − 56
- Subtract the numerators. 2 − 5 is below zero, so the answer is negative.2 − 5 = −3
- Simplify. Divide the top and the bottom by 3.−36 = −12
Check. As decimals: 0.333 − 0.833 = −0.5. ✓
Watch out. Subtract in the order given: first fraction minus second fraction. Do not swap them.
Method. Negative × positive = negative. Then multiply the fractions as normal.
- Decide the sign. The signs are different, so the answer is negative.negative × positive = negative
- Cancel common factors. 2 (on top) and 10 (on the bottom) share a factor of 2. 9 (on top) and 3 (on the bottom) share a factor of 3.23 × 910 = 11 × 35
- Multiply across. Top × top and bottom × bottom.Tops: 1 × 3 = 3
Bottoms: 1 × 5 = 5
35 - Write the answer with its sign. The sign was decided in step 1.−35
Check. As decimals: −0.667 × 0.9 = −0.6, and −3/5 = −0.6. ✓
Watch out. Decide the sign first. Do not forget the minus sign in the answer.
Method. Negative ÷ negative = positive. Then divide the fractions as normal.
- Decide the sign. The signs are the same, so the answer is positive.negative ÷ negative = positive
- Keep, change, flip. Turn the second fraction upside down and change the division to a multiplication.58 ÷ 1516 = 58 × 1615
- Cancel common factors. 5 and 15 share a factor of 5. 16 and 8 share a factor of 8.58 × 1615 = 11 × 23
- Multiply across. Top × top and bottom × bottom.23
Check. Multiply back: 23 × −1516 = −3048 = −58. ✓
Watch out. A negative divided by a negative is positive.
Method. The perimeter of a shape is the total distance around it: add all the sides.
- Write the sum. There are three sides, so add all three.213 + 134 + 356
- Change the mixed numbers to improper fractions. Whole × denominator + numerator, over the same denominator.213 = 73, 134 = 74, 356 = 236
- Find a common denominator. The LCM of 3, 4 and 6 is 12.73 = 2812, 74 = 2112, 236 = 4612
- Add the numerators. Keep the denominator.28 + 21 + 46 = 95, so 9512
- Write as a mixed number with units. 95 ÷ 12 = 7 remainder 11.71112 cm
Check. As decimals: 2.333 + 1.75 + 3.833 = 7.917 and 7 11/12 = 7.917. ✓
Watch out. Check the units: perimeter is a length, so the answer is in cm.
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.Area = 212 × 135
- 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
135 = (1 × 5 + 3) over 5 = 85 - 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 × 85 = 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
- Write the answer with units. Area is in square metres.Area = 4 m²
Check. A different route, as decimals: 2.5 × 1.6 = 4, and our answer 4 = 4. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. Perimeter = 2 × (length + width), because there are two lengths and two widths.
- Change to improper fractions. Whole × denominator + numerator, over the same denominator.212 = 52 and 135 = 85
- Add the length and the width. The LCM of 2 and 5 is 10.2510 + 1610 = 4110
- Double it. Two lengths and two widths.2 × 4110 = 8210 = 415
- Write as a mixed number with units. 41 ÷ 5 = 8 remainder 1.815 m
Check. As decimals: 2.5 + 1.6 = 4.1 and 2 × 4.1 = 8.2, which is 8 1/5. ✓
Watch out. Perimeter is in m, not m².
Method. Work out both products and compare them.
- Say what “of” means. In maths, “of” between fractions means multiply.34 of 23 = 34 × 23
- Work out 34 of 23. Cancel the 3s, then multiply the tops and the bottoms.34 × 23 = 12 × 11 = 12
- Work out 23 of 34. Same numbers in the other order.23 × 34 = 11 × 12 = 12
- Compare. Both products are equal.12 = 12
- Explain why. Multiplication can be done in either order (it is commutative): a × b = b × a. Multiplying the tops gives 6 and multiplying the bottoms gives 12 either way.They are equal.
Check. Draw it: split a bar into 3 equal parts and then each into 4, giving 12 parts. 34 of 23 and 23 of 34 both cover 6 of the 12 parts. ✓
Watch out. Multiplication is commutative but division is not.
Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.
- Plan the proof. In a “show that” question the answer is given, so every step must be shown. Start from the left side and work out its value.Left side = 37 + 25
- Find a common denominator: the LCM of 7 and 5. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 7: 7, 14, 21, 28, 35
Multiples of 5: 5, 10, 15, 20, 25, 30, 35
LCM = 35 - Rewrite both fractions with denominator 35. Multiply the top and the bottom by the same number so the value does not change.37 = 1535 (× 5)
25 = 1435 (× 7) - Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.1535 + 1435 = 2935
- 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.29 and 35 have no common factor except 1, so 2935 is the answer.
- State the result. The last line must match the result we were asked to show.37 + 25 = 2935 ✓
Check. A different route, as decimals: 0.4286 + 0.4 = 0.8286, and our answer 2935 = 0.8286. ✓
Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.
Method. Divide the amount of water by the size of one jug to find how many jugs, and read the remainder as a fraction of a jug.
- Decide the operation. “How many jugs can be filled” asks how many times 3/4 goes into 12 1/2, so divide.1212 ÷ 34
- Change the mixed number to an improper fraction. Whole × denominator + numerator, over the same denominator.1212 = 252
- Keep, change, flip. Dividing by a fraction is multiplying by its reciprocal.252 ÷ 34 = 252 × 43 = 1006 = 503
- Write as a mixed number. 50 ÷ 3 = 16 remainder 2. The whole number is the number of full jugs.503 = 1623, so 16 full jugs
- Find the water left over. The fraction 23 is part of a jug. One jug is 34 litre, so the water left is 23 of 34 litre.23 × 34 = 12 litre
Check. 16 jugs hold 16 × 0.75 = 12 litres, and 12.5 − 12 = 0.5 litre, which is 12 litre. ✓
Watch out. The whole-number part is the number of full jugs. The fraction left over is a fraction of a jug, not of a litre.
Method. Look for unit fractions 1/a and 1/b that add to 5/12, by finding the possible values of a systematically.
- Find the smallest possible a. A unit fraction must be smaller than 512. 12 = 612 is too big, and 13 = 412 is smaller than 512, so a is at least 3.a ≥ 3
- Find the largest possible a. If a < b then 1a is the bigger of the two, so it must be at least half of 512, which is 524. 15 = 0.2 is smaller than 524 = 0.208, so a cannot be 5 or more.a ≤ 4
- Try a = 3. Subtract 1/3 from 5/12 to see what is left for 1/b.512 − 13 = 512 − 412 = 112, so b = 12
- Try a = 4. Subtract 1/4 from 5/12.512 − 14 = 512 − 312 = 212 = 16, so b = 6
- State the answers. Both work, so there are two answers.13 + 112 and 14 + 16
Check. 13 + 112 = 412 + 112 = 512 and 14 + 16 = 312 + 212 = 512. ✓
Watch out. If you take a bigger than b you just get the same answer swapped round.