Year 6 Fractions: check-up and SATs-style questions

Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.

Method. Find the HCF of the top and the bottom, then divide both numbers by it.

1. Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.
   Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
   Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
   HCF = 12
2. Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.
   24 ÷ 12 = 2 and 36 ÷ 12 = 3
3. Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.
   Factors of 2: 1, 2. Factors of 3: 1, 3. Only 1 is shared, so the simplest form is 23.

Check. 2 × 12 = 24 and 3 × 12 = 36. ✓

Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.

Key words. A common denominator is a bottom number that two or more fractions share. The lowest common denominator is the lowest common multiple (LCM) of the bottom numbers.

Method. Make the bottom number 24 by multiplying the top and the bottom by 3.

1. Find what the bottom number 8 must be multiplied by to make 24. 24 ÷ 8 = 3.
   24 ÷ 8 = 3
2. Multiply the top and the bottom by the same number. Doing the same to both numbers keeps the fraction equal.
   58 × 3/3 = 1524

Check. 15 ÷ 3 = 5 and 24 ÷ 3 = 8. ✓

Watch out. Multiply the top as well as the bottom. Changing only the bottom changes the value.

Key words. A common denominator is a bottom number that two or more fractions share. The lowest common denominator is the lowest common multiple (LCM) of the bottom numbers.

Method. Write both fractions with the same bottom number, then compare the top numbers.

1. Find the lowest common denominator. Use the LCM of the two bottom numbers.
   LCM of 12 and 9 = 36
2. Write both with the same bottom number. Multiply the top and bottom of each fraction by the same number.
   712 = 2136
   59 = 2036
3. Compare the top numbers. When the bottoms are the same, the bigger top number is the bigger fraction.
   21 and 20: the bigger top number is 21
4. Say which fraction is bigger. Write the smaller one, then the bigger one.
   59 < 712

Check. 2136 is the bigger value of the two, so 712 is the bigger fraction. ✓

Watch out. Do not just compare the bottom numbers. A bigger bottom number means smaller pieces.

Key words. A common denominator is a bottom number that two or more fractions share. The lowest common denominator is the lowest common multiple (LCM) of the bottom numbers.

Method. Make the bottom numbers the same, then compare the top numbers.

1. Write them all with the same bottom number, 12. The lowest common denominator of the bottoms is 12. Change each fraction so that its bottom number is 12.
   34 = 912
   56 = 1012
   23 = 812
2. Compare the top numbers. The bigger top number is the bigger fraction. Put them smallest first.
   8, 9, 10 (out of 12)
3. Write the fractions in order, using the fractions from the question.
   23, 34, 56

Check. 23 is the smallest and 56 is the largest. ✓

Watch out. A bigger bottom number means smaller pieces. Compare the top numbers only after the bottoms are the same.

Key words. A common denominator is a bottom number that two or more fractions share. The lowest common denominator is the lowest common multiple (LCM) of the bottom numbers.

Method. Write both fractions with the LCM of their bottoms, then add the top numbers.

1. Find the lowest common denominator. Use the LCM of the two bottom numbers.
   LCM of 4 and 9 = 36
2. Write both with the same bottom number. Multiply the top and bottom of each fraction by the same number.
   34 = 2736
   29 = 836
3. Add the top numbers. The bottom number stays the same, because the pieces are the same size.
   2736 + 836 = 3536

Check. 3536 − 29 = 34. ✓

Watch out. Only the top numbers are added or subtracted. Do not add the bottom numbers.

Key words. A mixed number has a whole part and a fraction part, such as 2 1/3. An improper fraction has a top number that is bigger than its bottom number, such as 7/3. The two forms are the same amount.

Method. Write both fractions with the LCM of their bottoms, then subtract the top numbers.

1. Write each mixed number as an improper fraction. Multiply the whole number by the bottom, then add the top. The bottom stays the same.
   513: 5 × 3 + 1 = 16, so 163
   256: 2 × 6 + 5 = 17, so 176
2. Find the lowest common denominator. Use the LCM of the two bottom numbers.
   LCM of 3 and 6 = 6
3. Write both with the same bottom number. Multiply the top and bottom of each fraction by the same number.
   163 = 326
   176 = 176
4. Subtract the top numbers. The bottom number stays the same, because the pieces are the same size.
   326 − 176 = 156
5. Simplify. Divide the top and the bottom by their HCF, 3.
   156 = 52
6. Write the answer as a mixed number. The top is bigger than the bottom, so take out the whole parts.
   52 = 212

Check. 212 + 256 = 513. ✓

Watch out. Only the top numbers are added or subtracted. Do not add the bottom numbers.

Key words. To multiply two fractions, multiply the tops and multiply the bottoms. A fraction of an amount means the same as multiplying by that fraction.

Method. Multiply the tops, multiply the bottoms, then simplify.

1. Multiply the top numbers. The top of the answer is the product of the two tops.
   4 × 5 = 20
2. Multiply the bottom numbers. The bottom of the answer is the product of the two bottoms.
   5 × 8 = 40
3. Write the answer as one fraction. Put the two results together.
   45 × 58 = 2040
4. Simplify. Divide the top and the bottom by their HCF, 20.
   2040 = 12

Check. The answer is smaller than both 45 and 58, because we took a part of a part. ✓

Watch out. You may divide a top and a bottom by a common factor before you multiply, to keep the numbers small. The answer is the same.

Key words. Dividing by a whole number shares a fraction into equal parts. Each part is smaller, so the answer is a smaller fraction.

Method. Dividing by a whole number makes the bottom number bigger. Multiply the bottom, keep the top, then simplify.

1. Write the division. We are sharing the fraction into 5 equal parts.
   512 ÷ 5
2. Multiply the bottom number by the whole number. Each piece is cut into smaller parts, so the bottom gets bigger. The top stays the same.
   512 ÷ 5 = 560
3. Simplify. Divide the top and the bottom by their HCF, 5.
   560 = 112

Check. 112 × 5 = 512. ✓

Watch out. When the top divides exactly by the whole number, you can divide the top instead.

Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.

Method. Write the decimal as a fraction over a power of ten, then simplify.

1. Write the decimal over a power of ten. Each digit after the point is a tenth, hundredth or thousandth. There are 2 digits after the point, so the bottom number is 100.
   0.45 = 45100
2. Find the HCF of the top and the bottom. Divide both by the biggest number that goes into both.
   HCF of 45 and 100 = 5
3. Simplify. Divide the top and the bottom by the HCF.
   45 ÷ 5 = 9 and 100 ÷ 5 = 20, so 920

Check. 9 ÷ 20 = 0.45, and 0.45 is the same. ✓

Watch out. Simplify the fraction fully. A decimal like 0.6 is 6/10, which simplifies to 3/5.

Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.

Method. Use long division. Keep adding zeros until the remainder is 0.

1. Add a decimal point and zeros to the top number. The top number is smaller than the bottom, so the answer starts with 0 and a decimal point. Adding zeros after the point does not change the value.
   7 becomes 7.000
2. Divide one step at a time. Multiply the remainder by 10, then divide by the bottom number. Each answer is the next digit.
   70 ÷ 8 = 8, remainder 6
   60 ÷ 8 = 7, remainder 4
   40 ÷ 8 = 5, remainder 0
3. Write the digits after the point. Stop when the remainder is 0. The decimal stops.
   78 = 0.875

Check. 0.875 × 8 = 7. ✓

Watch out. If the remainder never reaches 0, the decimal goes on forever. Check the bottom number has only factors 2 and 5.

Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.

Method. Find the HCF of the top and the bottom, then divide both numbers by it.

1. Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.
   Factors of 18: 1, 2, 3, 6, 9, 18
   Factors of 27: 1, 3, 9, 27
   HCF = 9
2. Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.
   18 ÷ 9 = 2 and 27 ÷ 9 = 3
3. Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.
   Factors of 2: 1, 2. Factors of 3: 1, 3. Only 1 is shared, so the simplest form is 23.

Check. 2 × 9 = 18 and 3 × 9 = 27. ✓

Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.

Key words. A common denominator is a bottom number that two or more fractions share. The lowest common denominator is the lowest common multiple (LCM) of the bottom numbers.

Method. Write both fractions with the LCM of their bottoms, then add the top numbers.

1. Find the lowest common denominator. Use the LCM of the two bottom numbers.
   LCM of 4 and 8 = 8
2. Write both with the same bottom number. Multiply the top and bottom of each fraction by the same number.
   34 = 68
   18 = 18
3. Add the top numbers. The bottom number stays the same, because the pieces are the same size.
   68 + 18 = 78

Check. 78 − 18 = 34. ✓

Watch out. Only the top numbers are added or subtracted. Do not add the bottom numbers.

Key words. A common denominator is a bottom number that two or more fractions share. The lowest common denominator is the lowest common multiple (LCM) of the bottom numbers.

Method. Write both fractions with the same bottom number, then compare the top numbers.

1. Find the lowest common denominator. Use the LCM of the two bottom numbers.
   LCM of 7 and 9 = 63
2. Write both with the same bottom number. Multiply the top and bottom of each fraction by the same number.
   47 = 3663
   59 = 3563
3. Compare the top numbers. When the bottoms are the same, the bigger top number is the bigger fraction.
   36 and 35: the bigger top number is 36
4. Say which fraction is bigger. Write the smaller one, then the bigger one.
   59 < 47

Check. 3663 is the bigger value of the two, so 47 is the bigger fraction. ✓

Watch out. Do not just compare the bottom numbers. A bigger bottom number means smaller pieces.

Key words. A common denominator is a bottom number that two or more fractions share. The lowest common denominator is the lowest common multiple (LCM) of the bottom numbers.

Method. Make the bottom numbers the same, then compare the top numbers.

1. Write them all with the same bottom number, 30. The lowest common denominator of the bottoms is 30. Change each fraction so that its bottom number is 30.
   35 = 1830
   23 = 2030
   710 = 2130
2. Compare the top numbers. The bigger top number is the bigger fraction. Put them smallest first.
   18, 20, 21 (out of 30)
3. Write the fractions in order, using the fractions from the question.
   35, 23, 710

Check. 35 is the smallest and 710 is the largest. ✓

Watch out. A bigger bottom number means smaller pieces. Compare the top numbers only after the bottoms are the same.

Key words. A mixed number has a whole part and a fraction part, such as 2 1/3. An improper fraction has a top number that is bigger than its bottom number, such as 7/3. The two forms are the same amount.

Method. Write both fractions with the LCM of their bottoms, then add the top numbers.

1. Write each mixed number as an improper fraction. Multiply the whole number by the bottom, then add the top. The bottom stays the same.
   214: 2 × 4 + 1 = 9, so 94
   156: 1 × 6 + 5 = 11, so 116
2. Find the lowest common denominator. Use the LCM of the two bottom numbers.
   LCM of 4 and 6 = 12
3. Write both with the same bottom number. Multiply the top and bottom of each fraction by the same number.
   94 = 2712
   116 = 2212
4. Add the top numbers. The bottom number stays the same, because the pieces are the same size.
   2712 + 2212 = 4912
5. Write the answer as a mixed number. The top is bigger than the bottom, so take out the whole parts.
   4912 = 4112

Check. 4112 − 156 = 214. ✓

Watch out. Only the top numbers are added or subtracted. Do not add the bottom numbers.

Key words. A mixed number has a whole part and a fraction part, such as 2 1/3. An improper fraction has a top number that is bigger than its bottom number, such as 7/3. The two forms are the same amount.

Method. Write both fractions with the LCM of their bottoms, then subtract the top numbers.

1. Write each mixed number as an improper fraction. Multiply the whole number by the bottom, then add the top. The bottom stays the same.
   416: 4 × 6 + 1 = 25, so 256
   234: 2 × 4 + 3 = 11, so 114
2. Find the lowest common denominator. Use the LCM of the two bottom numbers.
   LCM of 6 and 4 = 12
3. Write both with the same bottom number. Multiply the top and bottom of each fraction by the same number.
   256 = 5012
   114 = 3312
4. Subtract the top numbers. The bottom number stays the same, because the pieces are the same size.
   5012 − 3312 = 1712
5. Write the answer as a mixed number. The top is bigger than the bottom, so take out the whole parts.
   1712 = 1512

Check. 1512 + 234 = 416. ✓

Watch out. Only the top numbers are added or subtracted. Do not add the bottom numbers.

Key words. To multiply two fractions, multiply the tops and multiply the bottoms. A fraction of an amount means the same as multiplying by that fraction.

Method. Multiply the tops, multiply the bottoms, then simplify.

1. Multiply the top numbers. The top of the answer is the product of the two tops.
   2 × 9 = 18
2. Multiply the bottom numbers. The bottom of the answer is the product of the two bottoms.
   3 × 10 = 30
3. Write the answer as one fraction. Put the two results together.
   23 × 910 = 1830
4. Simplify. Divide the top and the bottom by their HCF, 6.
   1830 = 35

Check. The answer is smaller than both 23 and 910, because we took a part of a part. ✓

Watch out. You may divide a top and a bottom by a common factor before you multiply, to keep the numbers small. The answer is the same.

Key words. A common denominator is a bottom number that two or more fractions share. The lowest common denominator is the lowest common multiple (LCM) of the bottom numbers.

Method. Write both fractions with the LCM of their bottoms, then subtract the top numbers.

1. Find the lowest common denominator. Use the LCM of the two bottom numbers.
   LCM of 6 and 8 = 24
2. Write both with the same bottom number. Multiply the top and bottom of each fraction by the same number.
   56 = 2024
   38 = 924
3. Subtract the top numbers. The bottom number stays the same, because the pieces are the same size.
   2024 − 924 = 1124

Check. 1124 + 38 = 56. ✓

Watch out. Only the top numbers are added or subtracted. Do not add the bottom numbers.

Key words. To multiply two fractions, multiply the tops and multiply the bottoms. A fraction of an amount means the same as multiplying by that fraction.

Method. Multiply the tops, multiply the bottoms, then simplify.

1. Multiply the top numbers. The top of the answer is the product of the two tops.
   1 × 2 = 2
2. Multiply the bottom numbers. The bottom of the answer is the product of the two bottoms.
   2 × 3 = 6
3. Write the answer as one fraction. Put the two results together.
   12 × 23 = 26
4. Simplify. Divide the top and the bottom by their HCF, 2.
   26 = 13

Check. The answer is smaller than both 12 and 23, because we took a part of a part. ✓

Watch out. You may divide a top and a bottom by a common factor before you multiply, to keep the numbers small. The answer is the same.

Key words. A mixed number has a whole part and a fraction part, such as 2 1/3. An improper fraction has a top number that is bigger than its bottom number, such as 7/3. The two forms are the same amount.

Method. Add the two amounts used. Then take the total away from the 3 litres. Use a common bottom number of 12.

1. Write the amounts with a bottom number of 12. The bottoms 4 and 6 both divide into 12.
   34 = 912
   1 56 = 116 = 2212
2. Add the amounts used. Add the top numbers, because the bottom is the same.
   912 + 2212 = 3112 = 2712
3. Take the total used away from 3 litres. Write 3 as twelfths: 3 = 36/12.
   3612 − 3112 = 512
4. Write the answer. This is the amount of paint left.
   512 litre

Check. 3112 + 512 = 3612, which is 3. ✓

Watch out. Check that the amounts used are less than 3 litres. If they are not, the answer cannot be positive.