GCSE Maths: Adding and subtracting fractions (Foundation)

Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.

1. 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).
   27 + 37
2. Look at the denominators. They are both 7, so the pieces are the same size and we can add straight away.
   Denominators: 7 and 7
3. Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   27 + 37 = 57
4. 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.
   5 and 7 have no common factor except 1, so 57 is the answer.

Check. A different route, as decimals: 0.2857 + 0.4286 = 0.7143, and our answer 57 = 0.7143. ✓

Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.

Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.

1. 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).
   59 + 29
2. Look at the denominators. They are both 9, so the pieces are the same size and we can add straight away.
   Denominators: 9 and 9
3. Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   59 + 29 = 79
4. 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.
   7 and 9 have no common factor except 1, so 79 is the answer.

Check. A different route, as decimals: 0.5556 + 0.2222 = 0.7778, and our answer 79 = 0.7778. ✓

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.

1. 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).
   710 − 310
2. Look at the denominators. They are both 10, so the pieces are the same size and we can subtract straight away.
   Denominators: 10 and 10
3. Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   710 − 310 = 410
4. Simplify the answer. 4 and 10 have a common factor of 2 (their HCF), so divide the top and the bottom by 2.
   4 ÷ 2 = 2 and 10 ÷ 2 = 5
   410 = 25

Check. A different route, as decimals: 0.7 − 0.3 = 0.4, and our answer 25 = 0.4. ✓

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.

1. 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).
   1112 − 512
2. Look at the denominators. They are both 12, so the pieces are the same size and we can subtract straight away.
   Denominators: 12 and 12
3. Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   1112 − 512 = 612
4. Simplify the answer. 6 and 12 have a common factor of 6 (their HCF), so divide the top and the bottom by 6.
   6 ÷ 6 = 1 and 12 ÷ 6 = 2
   612 = 12

Check. A different route, as decimals: 0.9167 − 0.4167 = 0.5, and our answer 12 = 0.5. ✓

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.

1. 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).
   12 + 14
2. Find a common denominator: the LCM of 2 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 2: 2, 4
   Multiples of 4: 4
   LCM = 4
3. Rewrite both fractions with denominator 4. Multiply the top and the bottom by the same number so the value does not change.
   12 = 24 (× 2)
   14 = 14 (× 1)
4. Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   24 + 14 = 34
5. 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 4 have no common factor except 1, so 34 is the answer.

Check. A different route, as decimals: 0.5 + 0.25 = 0.75, and our answer 34 = 0.75. ✓

Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.

Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.

1. 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).
   13 + 16
2. 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
3. Rewrite both fractions with denominator 6. Multiply the top and the bottom by the same number so the value does not change.
   13 = 26 (× 2)
   16 = 16 (× 1)
4. Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   26 + 16 = 36
5. Simplify the answer. 3 and 6 have a common factor of 3 (their HCF), so divide the top and the bottom by 3.
   3 ÷ 3 = 1 and 6 ÷ 3 = 2
   36 = 12

Check. A different route, as decimals: 0.3333 + 0.1667 = 0.5, and our answer 12 = 0.5. ✓

Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.

Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.

1. 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).
   25 + 110
2. Find a common denominator: the LCM of 5 and 10. 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
   Multiples of 10: 10
   LCM = 10
3. Rewrite both fractions with denominator 10. Multiply the top and the bottom by the same number so the value does not change.
   25 = 410 (× 2)
   110 = 110 (× 1)
4. Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   410 + 110 = 510
5. Simplify the answer. 5 and 10 have a common factor of 5 (their HCF), so divide the top and the bottom by 5.
   5 ÷ 5 = 1 and 10 ÷ 5 = 2
   510 = 12

Check. A different route, as decimals: 0.4 + 0.1 = 0.5, and our answer 12 = 0.5. ✓

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.

1. 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).
   34 − 18
2. Find a common denominator: the LCM of 4 and 8. 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 4: 4, 8
   Multiples of 8: 8
   LCM = 8
3. Rewrite both fractions with denominator 8. Multiply the top and the bottom by the same number so the value does not change.
   34 = 68 (× 2)
   18 = 18 (× 1)
4. Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   68 − 18 = 58
5. 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.
   5 and 8 have no common factor except 1, so 58 is the answer.

Check. A different route, as decimals: 0.75 − 0.125 = 0.625, and our answer 58 = 0.625. ✓

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.

1. 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).
   56 − 13
2. Find a common denominator: the LCM of 6 and 3. 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 6: 6
   Multiples of 3: 3, 6
   LCM = 6
3. Rewrite both fractions with denominator 6. Multiply the top and the bottom by the same number so the value does not change.
   56 = 56 (× 1)
   13 = 26 (× 2)
4. Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   56 − 26 = 36
5. Simplify the answer. 3 and 6 have a common factor of 3 (their HCF), so divide the top and the bottom by 3.
   3 ÷ 3 = 1 and 6 ÷ 3 = 2
   36 = 12

Check. A different route, as decimals: 0.8333 − 0.3333 = 0.5, and our answer 12 = 0.5. ✓

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.

1. 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).
   78 − 34
2. 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
3. Rewrite both fractions with denominator 8. Multiply the top and the bottom by the same number so the value does not change.
   78 = 78 (× 1)
   34 = 68 (× 2)
4. Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   78 − 68 = 18
5. 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 8 have no common factor except 1, so 18 is the answer.

Check. A different route, as decimals: 0.875 − 0.75 = 0.125, and our answer 18 = 0.125. ✓

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.

1. 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).
   13 + 14
2. Find a common denominator: the LCM of 3 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 3: 3, 6, 9, 12
   Multiples of 4: 4, 8, 12
   LCM = 12
3. Rewrite both fractions with denominator 12. Multiply the top and the bottom by the same number so the value does not change.
   13 = 412 (× 4)
   14 = 312 (× 3)
4. Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   412 + 312 = 712
5. 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.
   7 and 12 have no common factor except 1, so 712 is the answer.

Check. A different route, as decimals: 0.3333 + 0.25 = 0.5833, and our answer 712 = 0.5833. ✓

Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.

Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.

1. 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).
   25 + 13
2. Find a common denominator: the LCM of 5 and 3. 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
   Multiples of 3: 3, 6, 9, 12, 15
   LCM = 15
3. Rewrite both fractions with denominator 15. Multiply the top and the bottom by the same number so the value does not change.
   25 = 615 (× 3)
   13 = 515 (× 5)
4. Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   615 + 515 = 1115
5. 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.
   11 and 15 have no common factor except 1, so 1115 is the answer.

Check. A different route, as decimals: 0.4 + 0.3333 = 0.7333, and our answer 1115 = 0.7333. ✓

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.

1. 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).
   34 − 25
2. Find a common denominator: the LCM of 4 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 4: 4, 8, 12, 16, 20
   Multiples of 5: 5, 10, 15, 20
   LCM = 20
3. Rewrite both fractions with denominator 20. Multiply the top and the bottom by the same number so the value does not change.
   34 = 1520 (× 5)
   25 = 820 (× 4)
4. Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   1520 − 820 = 720
5. 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.
   7 and 20 have no common factor except 1, so 720 is the answer.

Check. A different route, as decimals: 0.75 − 0.4 = 0.35, and our answer 720 = 0.35. ✓

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.

1. 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).
   78 − 13
2. Find a common denominator: the LCM of 8 and 3. 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, 16, 24
   Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24
   LCM = 24
3. Rewrite both fractions with denominator 24. Multiply the top and the bottom by the same number so the value does not change.
   78 = 2124 (× 3)
   13 = 824 (× 8)
4. Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   2124 − 824 = 1324
5. 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.
   13 and 24 have no common factor except 1, so 1324 is the answer.

Check. A different route, as decimals: 0.875 − 0.3333 = 0.5417, and our answer 1324 = 0.5417. ✓

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.

1. 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).
   56 + 34
2. Find a common denominator: the LCM of 6 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 6: 6, 12
   Multiples of 4: 4, 8, 12
   LCM = 12
3. Rewrite both fractions with denominator 12. Multiply the top and the bottom by the same number so the value does not change.
   56 = 1012 (× 2)
   34 = 912 (× 3)
4. Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   1012 + 912 = 1912
5. 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 ÷ 12 = 1 remainder 7
   1912 = 1712

Check. A different route, as decimals: 0.8333 + 0.75 = 1.5833, and our answer 1712 = 1.5833. ✓

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.

1. 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).
   45 − 310
2. Find a common denominator: the LCM of 5 and 10. 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
   Multiples of 10: 10
   LCM = 10
3. Rewrite both fractions with denominator 10. Multiply the top and the bottom by the same number so the value does not change.
   45 = 810 (× 2)
   310 = 310 (× 1)
4. Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   810 − 310 = 510
5. Simplify the answer. 5 and 10 have a common factor of 5 (their HCF), so divide the top and the bottom by 5.
   5 ÷ 5 = 1 and 10 ÷ 5 = 2
   510 = 12

Check. A different route, as decimals: 0.8 − 0.3 = 0.5, and our answer 12 = 0.5. ✓

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.

1. 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).
   112 + 214
2. 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.
   112 = (1 × 2 + 1) over 2 = 32
   214 = (2 × 4 + 1) over 4 = 94
3. Find a common denominator: the LCM of 2 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 2: 2, 4
   Multiples of 4: 4
   LCM = 4
4. Rewrite both fractions with denominator 4. Multiply the top and the bottom by the same number so the value does not change.
   32 = 64 (× 2)
   94 = 94 (× 1)
5. Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   64 + 94 = 154
6. 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 ÷ 4 = 3 remainder 3
   154 = 334

Check. A different route, as decimals: 1.5 + 2.25 = 3.75, and our answer 334 = 3.75. ✓

Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.

Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.

1. 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).
   325 + 112
2. 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.
   325 = (3 × 5 + 2) over 5 = 175
   112 = (1 × 2 + 1) over 2 = 32
3. Find a common denominator: the LCM of 5 and 2. 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
   Multiples of 2: 2, 4, 6, 8, 10
   LCM = 10
4. Rewrite both fractions with denominator 10. Multiply the top and the bottom by the same number so the value does not change.
   175 = 3410 (× 2)
   32 = 1510 (× 5)
5. Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   3410 + 1510 = 4910
6. 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.
   49 ÷ 10 = 4 remainder 9
   4910 = 4910

Check. A different route, as decimals: 3.4 + 1.5 = 4.9, and our answer 4910 = 4.9. ✓

Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.

Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.

1. 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 + 134
2. 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
   134 = (1 × 4 + 3) over 4 = 74
3. Find a common denominator: the LCM of 3 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 3: 3, 6, 9, 12
   Multiples of 4: 4, 8, 12
   LCM = 12
4. Rewrite both fractions with denominator 12. Multiply the top and the bottom by the same number so the value does not change.
   83 = 3212 (× 4)
   74 = 2112 (× 3)
5. Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   3212 + 2112 = 5312
6. 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.
   53 ÷ 12 = 4 remainder 5
   5312 = 4512

Check. A different route, as decimals: 2.6667 + 1.75 = 4.4167, and our answer 4512 = 4.4167. ✓

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.

1. 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).
   412 − 113
2. 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
   113 = (1 × 3 + 1) over 3 = 43
3. Find a common denominator: the LCM of 2 and 3. 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 2: 2, 4, 6
   Multiples of 3: 3, 6
   LCM = 6
4. Rewrite both fractions with denominator 6. Multiply the top and the bottom by the same number so the value does not change.
   92 = 276 (× 3)
   43 = 86 (× 2)
5. Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   276 − 86 = 196
6. 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 ÷ 6 = 3 remainder 1
   196 = 316

Check. A different route, as decimals: 4.5 − 1.3333 = 3.1667, and our answer 316 = 3.1667. ✓

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.

1. 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).
   314 − 112
2. 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.
   314 = (3 × 4 + 1) over 4 = 134
   112 = (1 × 2 + 1) over 2 = 32
3. Find a common denominator: the LCM of 4 and 2. 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 4: 4
   Multiples of 2: 2, 4
   LCM = 4
4. Rewrite both fractions with denominator 4. Multiply the top and the bottom by the same number so the value does not change.
   134 = 134 (× 1)
   32 = 64 (× 2)
5. Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   134 − 64 = 74
6. 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 ÷ 4 = 1 remainder 3
   74 = 134

Check. A different route, as decimals: 3.25 − 1.5 = 1.75, and our answer 134 = 1.75. ✓

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.

1. 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).
   516 − 223
2. 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.
   516 = (5 × 6 + 1) over 6 = 316
   223 = (2 × 3 + 2) over 3 = 83
3. Find a common denominator: the LCM of 6 and 3. 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 6: 6
   Multiples of 3: 3, 6
   LCM = 6
4. Rewrite both fractions with denominator 6. Multiply the top and the bottom by the same number so the value does not change.
   316 = 316 (× 1)
   83 = 166 (× 2)
5. Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.
   316 − 166 = 156
6. Simplify the answer. 15 and 6 have a common factor of 3 (their HCF), so divide the top and the bottom by 3.
   15 ÷ 3 = 5 and 6 ÷ 3 = 2
   156 = 52
7. 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.
   5 ÷ 2 = 2 remainder 1
   52 = 212

Check. A different route, as decimals: 5.1667 − 2.6667 = 2.5, and our answer 212 = 2.5. ✓

Watch out. Subtract in the right order: the first fraction minus the second.