Year 5: Sharing, grouping, scaling and rates

Key words. Short division (the bus stop method): the number you divide goes inside the bus stop and the divisor goes outside. A remainder is what is left over. To exchange is to swap 1 of a place value for 10 of the next smaller one.

Method. Draw a bar model, choose the operation, then work it out.

1. Draw a bar model. There are 4 equal shares. Each share is the same size.
2. Choose the operation. 96 is shared equally between 4, so divide by 4.
   96 ÷ 4
3. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   4	9	6

4. Divide the tens. Always start on the left, with the biggest place value.
   9 ÷ 4 = 2, remainder 1
   Write 2 above the 9.
   Exchange the remainder: 1 ten becomes 10 ones. Carry 1 to the next digit.
5. Divide the ones. Move to the next digit on the right. Add the 1 that was carried.
   The carried 1 makes 10 ones. With the 6 ones there are 10 + 6 = 16 ones.
   16 ÷ 4: 4 × 4 = 16. Write 4 above the 6.
   Nothing is left over, so there is no remainder.
6. Read the answer. Read the digits above the line from left to right.

   	2	4
   4	9	16

   
   96 ÷ 4 = 24
7. Write the answer. Include the unit.
   24 sweets

Check. Multiply back: 24 × 4 = 20 × 4 + 4 × 4 = 80 + 16 = 96. ✓

Watch out. Start on the left, with the biggest place value. Carry any remainder to the next digit.

Key words. Short division (the bus stop method): the number you divide goes inside the bus stop and the divisor goes outside. A remainder is what is left over. To exchange is to swap 1 of a place value for 10 of the next smaller one.

Method. Draw a bar model, choose the operation, then work it out.

1. Draw a bar model. There are 5 equal shares. Each share is the same size.
2. Choose the operation. 415 is shared equally between 5, so divide by 5.
   415 ÷ 5
3. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   5	4	1	5

4. Divide the hundreds. Always start on the left, with the biggest place value.
   4 ÷ 5: 4 is smaller than 5, so we cannot give each share a whole hundred. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 4 hundreds for 40 tens. Carry 4 to the next digit.
5. Divide the tens. Move to the next digit on the right. Add the 4 that was carried.
   The carried 4 makes 40 tens. With the 1 tens there are 40 + 1 = 41 tens.
   41 ÷ 5: 8 × 5 = 40, and 41 − 40 = 1. Write 8 above the 1.
   Exchange: 1 ten becomes 10 ones. Carry 1 to the next digit.
6. Divide the ones. Move to the next digit on the right. Add the 1 that was carried.
   The carried 1 makes 10 ones. With the 5 ones there are 10 + 5 = 15 ones.
   15 ÷ 5: 3 × 5 = 15. Write 3 above the 5.
   Nothing is left over, so there is no remainder.
7. Read the answer. Ignore any 0 at the front of the answer.

   	0	8	3
   5	4	41	15

   
   415 ÷ 5 = 83
8. Write the answer. Include the unit.
   83 stamps

Check. Multiply back: 83 × 5 = 80 × 5 + 3 × 5 = 400 + 15 = 415. ✓

Watch out. If the first digit is smaller than 5, write 0 above it (not in the final answer) and carry the digit to the next place.

Key words. Short division (the bus stop method): the number you divide goes inside the bus stop and the divisor goes outside. A remainder is what is left over. To exchange is to swap 1 of a place value for 10 of the next smaller one.

Method. Draw a bar model, choose the operation, then work it out.

1. Draw a bar model. There are 6 equal shares. Each share is the same size.
2. Choose the operation. 1,368 is shared equally between 6, so divide by 6.
   1,368 ÷ 6
3. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   6	1	3	6	8

4. Divide the thousands. Always start on the left, with the biggest place value.
   1 ÷ 6: 1 is smaller than 6, so we cannot give each share a whole thousand. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 1 thousand for 10 hundreds. Carry 1 to the next digit.
5. Divide the hundreds. Move to the next digit on the right. Add the 1 that was carried.
   The carried 1 makes 10 hundreds. With the 3 hundreds there are 10 + 3 = 13 hundreds.
   13 ÷ 6: 2 × 6 = 12, and 13 − 12 = 1. Write 2 above the 3.
   Exchange: 1 hundred becomes 10 tens. Carry 1 to the next digit.
6. Divide the tens. Move to the next digit on the right. Add the 1 that was carried.
   The carried 1 makes 10 tens. With the 6 tens there are 10 + 6 = 16 tens.
   16 ÷ 6: 2 × 6 = 12, and 16 − 12 = 4. Write 2 above the 6.
   Exchange: 4 tens become 40 ones. Carry 4 to the next digit.
7. Divide the ones. Move to the next digit on the right. Add the 4 that was carried.
   The carried 4 makes 40 ones. With the 8 ones there are 40 + 8 = 48 ones.
   48 ÷ 6: 8 × 6 = 48. Write 8 above the 8.
   Nothing is left over, so there is no remainder.
8. Read the answer. Ignore any 0 at the front of the answer.

   	0	2	2	8
   6	1	13	16	48

   
   1,368 ÷ 6 = 228
9. Write the answer. Include the unit.
   228 seats

Check. Multiply back: 228 × 6 = 200 × 6 + 20 × 6 + 8 × 6 = 1,200 + 120 + 48 = 1,368. ✓

Watch out. If the first digit is smaller than 6, write 0 above it (not in the final answer) and carry the digit to the next place.

Key words. Short division (the bus stop method): the number you divide goes inside the bus stop and the divisor goes outside. A remainder is what is left over. To exchange is to swap 1 of a place value for 10 of the next smaller one.

Method. Draw a bar model, choose the operation, then work it out.

1. Draw a bar model. Each box is a group of 7. We count how many boxes make 84.
2. Choose the operation. We find how many groups of 7 fit into 84, so divide by 7.
   84 ÷ 7
3. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   7	8	4

4. Divide the tens. Always start on the left, with the biggest place value.
   8 ÷ 7 = 1, remainder 1
   Write 1 above the 8.
   Exchange the remainder: 1 ten becomes 10 ones. Carry 1 to the next digit.
5. Divide the ones. Move to the next digit on the right. Add the 1 that was carried.
   The carried 1 makes 10 ones. With the 4 ones there are 10 + 4 = 14 ones.
   14 ÷ 7: 2 × 7 = 14. Write 2 above the 4.
   Nothing is left over, so there is no remainder.
6. Read the answer. Read the digits above the line from left to right.

   	1	2
   7	8	14

   
   84 ÷ 7 = 12
7. Write the answer. Include the unit.
   12 bags

Check. Multiply back: 12 × 7 = 10 × 7 + 2 × 7 = 70 + 14 = 84. ✓

Watch out. Start on the left, with the biggest place value. Carry any remainder to the next digit.

Key words. Short division (the bus stop method): the number you divide goes inside the bus stop and the divisor goes outside. A remainder is what is left over. To exchange is to swap 1 of a place value for 10 of the next smaller one.

Method. Draw a bar model, choose the operation, then work it out.

1. Draw a bar model. Each box is a group of 8. We count how many boxes make 456.
2. Choose the operation. We find how many groups of 8 fit into 456, so divide by 8.
   456 ÷ 8
3. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   8	4	5	6

4. Divide the hundreds. Always start on the left, with the biggest place value.
   4 ÷ 8: 4 is smaller than 8, so we cannot give each share a whole hundred. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 4 hundreds for 40 tens. Carry 4 to the next digit.
5. Divide the tens. Move to the next digit on the right. Add the 4 that was carried.
   The carried 4 makes 40 tens. With the 5 tens there are 40 + 5 = 45 tens.
   45 ÷ 8: 5 × 8 = 40, and 45 − 40 = 5. Write 5 above the 5.
   Exchange: 5 tens become 50 ones. Carry 5 to the next digit.
6. Divide the ones. Move to the next digit on the right. Add the 5 that was carried.
   The carried 5 makes 50 ones. With the 6 ones there are 50 + 6 = 56 ones.
   56 ÷ 8: 7 × 8 = 56. Write 7 above the 6.
   Nothing is left over, so there is no remainder.
7. Read the answer. Ignore any 0 at the front of the answer.

   	0	5	7
   8	4	45	56

   
   456 ÷ 8 = 57
8. Write the answer. Include the unit.
   57 rows

Check. Multiply back: 57 × 8 = 50 × 8 + 7 × 8 = 400 + 56 = 456. ✓

Watch out. If the first digit is smaller than 8, write 0 above it (not in the final answer) and carry the digit to the next place.

Key words. Short division (the bus stop method): the number you divide goes inside the bus stop and the divisor goes outside. A remainder is what is left over. To exchange is to swap 1 of a place value for 10 of the next smaller one.

Method. Draw a bar model, choose the operation, then work it out.

1. Draw a bar model. Each box is a group of 9. We count how many boxes make 2,250.
2. Choose the operation. We find how many groups of 9 fit into 2,250, so divide by 9.
   2,250 ÷ 9
3. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   9	2	2	5	0

4. Divide the thousands. Always start on the left, with the biggest place value.
   2 ÷ 9: 2 is smaller than 9, so we cannot give each share a whole thousand. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 2 thousands for 20 hundreds. Carry 2 to the next digit.
5. Divide the hundreds. Move to the next digit on the right. Add the 2 that was carried.
   The carried 2 makes 20 hundreds. With the 2 hundreds there are 20 + 2 = 22 hundreds.
   22 ÷ 9: 2 × 9 = 18, and 22 − 18 = 4. Write 2 above the 2.
   Exchange: 4 hundreds become 40 tens. Carry 4 to the next digit.
6. Divide the tens. Move to the next digit on the right. Add the 4 that was carried.
   The carried 4 makes 40 tens. With the 5 tens there are 40 + 5 = 45 tens.
   45 ÷ 9: 5 × 9 = 45. Write 5 above the 5.
7. Divide the ones. Move to the next digit on the right.
   Nothing was carried, so there are just 0 ones.
   0 ÷ 9 = 0. Write 0 above. This 0 is a place holder, and it must stay in the answer.
   Nothing is left over, so there is no remainder.
8. Read the answer. Ignore any 0 at the front of the answer.

   	0	2	5	0
   9	2	22	45	0

   
   2,250 ÷ 9 = 250
9. Write the answer. Include the unit.
   250 trips

Check. Multiply back: 250 × 9 = 200 × 9 + 50 × 9 = 1,800 + 450 = 2,250. ✓

Watch out. A 0 in the answer is a place holder. Leave it out and 250 would look like a much smaller number. Write a 0 every time a digit does not divide.

Key words. Sharing means splitting an amount into equal groups when you know how many groups there are. Grouping means finding how many groups of a given size you can make. Both are division.

Method. Sharing means you know the number of groups. Grouping means you know the size of each group. Either way, you divide.

1. Is it sharing or grouping? Read the story. Does it tell us how many groups, or how big each group is?
   The pencils are split into 8 equal pots. We know the number of pots and want to find how many pencils go in each. This is sharing.
2. Draw a bar model. The bar shows the whole amount split into equal parts.
3. Choose the operation. Both sharing and grouping are division.
   240 ÷ 8
4. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   8	2	4	0

5. Divide the hundreds. Always start on the left, with the biggest place value.
   2 ÷ 8: 2 is smaller than 8, so we cannot give each share a whole hundred. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 2 hundreds for 20 tens. Carry 2 to the next digit.
6. Divide the tens. Move to the next digit on the right. Add the 2 that was carried.
   The carried 2 makes 20 tens. With the 4 tens there are 20 + 4 = 24 tens.
   24 ÷ 8: 3 × 8 = 24. Write 3 above the 4.
7. Divide the ones. Move to the next digit on the right.
   Nothing was carried, so there are just 0 ones.
   0 ÷ 8 = 0. Write 0 above. This 0 is a place holder, and it must stay in the answer.
   Nothing is left over, so there is no remainder.
8. Read the answer. Ignore any 0 at the front of the answer.

   	0	3	0
   8	2	24	0

   
   240 ÷ 8 = 30
9. Write the answer. Say whether it was sharing or grouping.
   Sharing: 30 pencils in each pot

Check. 30 × 8 = 240. ✓

Watch out. The calculation is the same for sharing and grouping. The difference is in what the answer means.

Key words. Sharing means splitting an amount into equal groups when you know how many groups there are. Grouping means finding how many groups of a given size you can make. Both are division.

Method. Sharing means you know the number of groups. Grouping means you know the size of each group. Either way, you divide.

1. Is it sharing or grouping? Read the story. Does it tell us how many groups, or how big each group is?
   Each page has 5 stickers. We know the size of each page and want to find how many pages can be made. This is grouping.
2. Draw a bar model. The bar shows the whole amount split into equal parts.
3. Choose the operation. Both sharing and grouping are division.
   135 ÷ 5
4. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   5	1	3	5

5. Divide the hundreds. Always start on the left, with the biggest place value.
   1 ÷ 5: 1 is smaller than 5, so we cannot give each share a whole hundred. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 1 hundred for 10 tens. Carry 1 to the next digit.
6. Divide the tens. Move to the next digit on the right. Add the 1 that was carried.
   The carried 1 makes 10 tens. With the 3 tens there are 10 + 3 = 13 tens.
   13 ÷ 5: 2 × 5 = 10, and 13 − 10 = 3. Write 2 above the 3.
   Exchange: 3 tens become 30 ones. Carry 3 to the next digit.
7. Divide the ones. Move to the next digit on the right. Add the 3 that was carried.
   The carried 3 makes 30 ones. With the 5 ones there are 30 + 5 = 35 ones.
   35 ÷ 5: 7 × 5 = 35. Write 7 above the 5.
   Nothing is left over, so there is no remainder.
8. Read the answer. Ignore any 0 at the front of the answer.

   	0	2	7
   5	1	13	35

   
   135 ÷ 5 = 27
9. Write the answer. Say whether it was sharing or grouping.
   Grouping: 27 pages

Check. 27 × 5 = 135. ✓

Watch out. The calculation is the same for sharing and grouping. The difference is in what the answer means.

Key words. Sharing means splitting an amount into equal groups when you know how many groups there are. Grouping means finding how many groups of a given size you can make. Both are division.

Method. Sharing means you know the number of groups. Grouping means you know the size of each group. Either way, you divide.

1. Is it sharing or grouping? Read the story. Does it tell us how many groups, or how big each group is?
   Each tray has 6 eggs. We know the size of each tray and want to find how many trays can be made. This is grouping.
2. Draw a bar model. The bar shows the whole amount split into equal parts.
3. Choose the operation. Both sharing and grouping are division.
   672 ÷ 6
4. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   6	6	7	2

5. Divide the hundreds. Always start on the left, with the biggest place value.
   6 ÷ 6 = 1
   Write 1 above the 6.
6. Divide the tens. Move to the next digit on the right.
   Nothing was carried, so there are just 7 tens.
   7 ÷ 6: 1 × 6 = 6, and 7 − 6 = 1. Write 1 above the 7.
   Exchange: 1 ten becomes 10 ones. Carry 1 to the next digit.
7. Divide the ones. Move to the next digit on the right. Add the 1 that was carried.
   The carried 1 makes 10 ones. With the 2 ones there are 10 + 2 = 12 ones.
   12 ÷ 6: 2 × 6 = 12. Write 2 above the 2.
   Nothing is left over, so there is no remainder.
8. Read the answer. Read the digits above the line from left to right.

   	1	1	2
   6	6	7	12

   
   672 ÷ 6 = 112
9. Write the answer. Say whether it was sharing or grouping.
   Grouping: 112 trays

Check. 112 × 6 = 672. ✓

Watch out. The calculation is the same for sharing and grouping. The difference is in what the answer means.

Key words. Scaling means making an amount bigger or smaller. “A quarter as long” means divide by 4. “How many times as long” means divide the longer by the shorter.

Method. “A sixth as much” means divide by 6.

1. Draw a bar model. Tank is split into 6 equal parts. Jug is 1 of those parts.
2. Choose the operation. A sixth as much means splitting into 6 equal parts and taking one part, so divide by 6.
   480 litres ÷ 6
3. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   6	4	8	0

4. Divide the hundreds. Always start on the left, with the biggest place value.
   4 ÷ 6: 4 is smaller than 6, so we cannot give each share a whole hundred. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 4 hundreds for 40 tens. Carry 4 to the next digit.
5. Divide the tens. Move to the next digit on the right. Add the 4 that was carried.
   The carried 4 makes 40 tens. With the 8 tens there are 40 + 8 = 48 tens.
   48 ÷ 6: 8 × 6 = 48. Write 8 above the 8.
6. Divide the ones. Move to the next digit on the right.
   Nothing was carried, so there are just 0 ones.
   0 ÷ 6 = 0. Write 0 above. This 0 is a place holder, and it must stay in the answer.
   Nothing is left over, so there is no remainder.
7. Read the answer. Ignore any 0 at the front of the answer.

   	0	8	0
   6	4	48	0

   
   480 ÷ 6 = 80
8. Write the answer. Include the unit.
   80 litres

Check. 6 × 80 litres = 480 litres. ✓

Watch out. “A fifth as long” makes the amount smaller: divide. “5 times as long” makes it bigger: multiply.

Key words. Scaling means making an amount bigger or smaller. “A quarter as long” means divide by 4. “How many times as long” means divide the longer by the shorter.

Method. “An eighth as much” means divide by 8.

1. Draw a bar model. Large bag is split into 8 equal parts. Small bag is 1 of those parts.
2. Choose the operation. An eighth as much means splitting into 8 equal parts and taking one part, so divide by 8.
   640 g ÷ 8
3. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   8	6	4	0

4. Divide the hundreds. Always start on the left, with the biggest place value.
   6 ÷ 8: 6 is smaller than 8, so we cannot give each share a whole hundred. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 6 hundreds for 60 tens. Carry 6 to the next digit.
5. Divide the tens. Move to the next digit on the right. Add the 6 that was carried.
   The carried 6 makes 60 tens. With the 4 tens there are 60 + 4 = 64 tens.
   64 ÷ 8: 8 × 8 = 64. Write 8 above the 4.
6. Divide the ones. Move to the next digit on the right.
   Nothing was carried, so there are just 0 ones.
   0 ÷ 8 = 0. Write 0 above. This 0 is a place holder, and it must stay in the answer.
   Nothing is left over, so there is no remainder.
7. Read the answer. Ignore any 0 at the front of the answer.

   	0	8	0
   8	6	64	0

   
   640 ÷ 8 = 80
8. Write the answer. Include the unit.
   80 g

Check. 8 × 80 g = 640 g. ✓

Watch out. “A fifth as long” makes the amount smaller: divide. “5 times as long” makes it bigger: multiply.

Key words. Scaling means making an amount bigger or smaller. “A quarter as long” means divide by 4. “How many times as long” means divide the longer by the shorter.

Method. “Half as much” means divide by 2.

1. Draw a bar model. Film is split into 2 equal parts. TV show is 1 of those parts.
2. Choose the operation. Half as much means splitting into 2 equal parts and taking one part, so divide by 2.
   156 minutes ÷ 2
3. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   2	1	5	6

4. Divide the hundreds. Always start on the left, with the biggest place value.
   1 ÷ 2: 1 is smaller than 2, so we cannot give each share a whole hundred. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 1 hundred for 10 tens. Carry 1 to the next digit.
5. Divide the tens. Move to the next digit on the right. Add the 1 that was carried.
   The carried 1 makes 10 tens. With the 5 tens there are 10 + 5 = 15 tens.
   15 ÷ 2: 7 × 2 = 14, and 15 − 14 = 1. Write 7 above the 5.
   Exchange: 1 ten becomes 10 ones. Carry 1 to the next digit.
6. Divide the ones. Move to the next digit on the right. Add the 1 that was carried.
   The carried 1 makes 10 ones. With the 6 ones there are 10 + 6 = 16 ones.
   16 ÷ 2: 8 × 2 = 16. Write 8 above the 6.
   Nothing is left over, so there is no remainder.
7. Read the answer. Ignore any 0 at the front of the answer.

   	0	7	8
   2	1	15	16

   
   156 ÷ 2 = 78
8. Write the answer. Include the unit.
   78 minutes

Check. 2 × 78 minutes = 156 minutes. ✓

Watch out. “A fifth as long” makes the amount smaller: divide. “5 times as long” makes it bigger: multiply.

Key words. Scaling means making an amount bigger or smaller. “A quarter as long” means divide by 4. “How many times as long” means divide the longer by the shorter.

Method. “How many times as big?” asks how many groups of the smaller amount fit into the bigger amount, so divide.

1. Draw a bar model. The wall’s amount is 1 box. We count how many boxes of that size make The tower’s amount.
2. Choose the operation. To find how many times as big, divide the bigger amount by the smaller amount. (This is grouping: how many 9s are in 72?)
   72 ÷ 9
3. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   9	7	2

4. Divide the tens. Always start on the left, with the biggest place value.
   7 ÷ 9: 7 is smaller than 9, so we cannot give each share a whole ten. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 7 tens for 70 ones. Carry 7 to the next digit.
5. Divide the ones. Move to the next digit on the right. Add the 7 that was carried.
   The carried 7 makes 70 ones. With the 2 ones there are 70 + 2 = 72 ones.
   72 ÷ 9: 8 × 9 = 72. Write 8 above the 2.
   Nothing is left over, so there is no remainder.
6. Read the answer. Ignore any 0 at the front of the answer.

   	0	8
   9	7	72

   
   72 ÷ 9 = 8
7. Write the answer.
   8 times as big

Check. 8 × 9 = 72. ✓

Watch out. Divide the bigger amount by the smaller one, not the other way round.

Key words. Scaling means making an amount bigger or smaller. “A quarter as long” means divide by 4. “How many times as long” means divide the longer by the shorter.

Method. “How many times as big?” asks how many groups of the smaller amount fit into the bigger amount, so divide.

1. Choose the operation. To find how many times as big, divide the bigger amount by the smaller amount. (This is grouping: how many 9s are in 360?)
   360 ÷ 9
2. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   9	3	6	0

3. Divide the hundreds. Always start on the left, with the biggest place value.
   3 ÷ 9: 3 is smaller than 9, so we cannot give each share a whole hundred. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 3 hundreds for 30 tens. Carry 3 to the next digit.
4. Divide the tens. Move to the next digit on the right. Add the 3 that was carried.
   The carried 3 makes 30 tens. With the 6 tens there are 30 + 6 = 36 tens.
   36 ÷ 9: 4 × 9 = 36. Write 4 above the 6.
5. Divide the ones. Move to the next digit on the right.
   Nothing was carried, so there are just 0 ones.
   0 ÷ 9 = 0. Write 0 above. This 0 is a place holder, and it must stay in the answer.
   Nothing is left over, so there is no remainder.
6. Read the answer. Ignore any 0 at the front of the answer.

   	0	4	0
   9	3	36	0

   
   360 ÷ 9 = 40
7. Write the answer.
   40 times as big

Check. 40 × 9 = 360. ✓

Watch out. Divide the bigger amount by the smaller one, not the other way round.

Key words. Scaling means making an amount bigger or smaller. “A quarter as long” means divide by 4. “How many times as long” means divide the longer by the shorter.

Method. “How many times as big?” asks how many groups of the smaller amount fit into the bigger amount, so divide.

1. Choose the operation. To find how many times as big, divide the bigger amount by the smaller amount. (This is grouping: how many 7s are in 84?)
   84 ÷ 7
2. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   7	8	4

3. Divide the tens. Always start on the left, with the biggest place value.
   8 ÷ 7 = 1, remainder 1
   Write 1 above the 8.
   Exchange the remainder: 1 ten becomes 10 ones. Carry 1 to the next digit.
4. Divide the ones. Move to the next digit on the right. Add the 1 that was carried.
   The carried 1 makes 10 ones. With the 4 ones there are 10 + 4 = 14 ones.
   14 ÷ 7: 2 × 7 = 14. Write 2 above the 4.
   Nothing is left over, so there is no remainder.
5. Read the answer. Read the digits above the line from left to right.

   	1	2
   7	8	14

   
   84 ÷ 7 = 12
6. Write the answer.
   12 times as big

Check. 12 × 7 = 84. ✓

Watch out. Divide the bigger amount by the smaller one, not the other way round.

Key words. A rate compares two amounts, such as £12 for 4 pens or 60 km each hour. A unit rate is the amount for just one (one pen, one hour, one litre). The word “per” means “for each” and tells you to divide.

Method. Change the total to pence, divide by the number of items, then change back to pounds if you need to.

1. Change the price to pence. Working in pence means no decimal point. £1 = 100p, so multiply by 100.
   £6.00 = 600p
2. Choose the operation. 8 pens cost 600p altogether. The cost of one is the total shared equally between 8.
   600p ÷ 8
3. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   8	6	0	0

4. Divide the hundreds. Always start on the left, with the biggest place value.
   6 ÷ 8: 6 is smaller than 8, so we cannot give each share a whole hundred. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 6 hundreds for 60 tens. Carry 6 to the next digit.
5. Divide the tens. Move to the next digit on the right. Add the 6 that was carried.
   The carried 6 makes 60 tens. With the 0 tens there are 60 + 0 = 60 tens.
   60 ÷ 8: 7 × 8 = 56, and 60 − 56 = 4. Write 7 above the 0.
   Exchange: 4 tens become 40 ones. Carry 4 to the next digit.
6. Divide the ones. Move to the next digit on the right. Add the 4 that was carried.
   The carried 4 makes 40 ones. With the 0 ones there are 40 + 0 = 40 ones.
   40 ÷ 8: 5 × 8 = 40. Write 5 above the 0.
   Nothing is left over, so there is no remainder.
7. Read the answer. Ignore any 0 at the front of the answer.

   	0	7	5
   8	6	60	40

   
   600 ÷ 8 = 75
8. Write the answer. Include the unit.
   75p for 1 pen

Check. 75p × 8 = 600p. ✓

Watch out. Make sure your answer is in the right unit: pence or pounds.

Key words. A rate compares two amounts, such as £12 for 4 pens or 60 km each hour. A unit rate is the amount for just one (one pen, one hour, one litre). The word “per” means “for each” and tells you to divide.

Method. Divide the total cost by the number of items to find the cost of one.

1. Choose the operation. 5 notebooks cost £20.00 altogether. The cost of one is the total shared equally between 5.
   £20.00 ÷ 5
2. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   5	2	0

3. Divide the tens. Always start on the left, with the biggest place value.
   2 ÷ 5: 2 is smaller than 5, so we cannot give each share a whole ten. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 2 tens for 20 ones. Carry 2 to the next digit.
4. Divide the ones. Move to the next digit on the right. Add the 2 that was carried.
   The carried 2 makes 20 ones. With the 0 ones there are 20 + 0 = 20 ones.
   20 ÷ 5: 4 × 5 = 20. Write 4 above the 0.
   Nothing is left over, so there is no remainder.
5. Read the answer. Ignore any 0 at the front of the answer.

   	0	4
   5	2	20

   
   20 ÷ 5 = 4
6. Write the answer. Include the pound sign.
   £4 for 1 notebook

Check. £4 × 5 = £20. ✓

Watch out. Cost of one is always less than the total. Divide the total by the number of items.

Key words. A rate compares two amounts, such as £12 for 4 pens or 60 km each hour. A unit rate is the amount for just one (one pen, one hour, one litre). The word “per” means “for each” and tells you to divide.

Method. Change the total to pence, divide by the number of items, then change back to pounds if you need to.

1. Change the price to pence. Working in pence means no decimal point. £1 = 100p, so multiply by 100.
   £2.40 = 240p
2. Choose the operation. 6 apples cost 240p altogether. The cost of one is the total shared equally between 6.
   240p ÷ 6
3. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   6	2	4	0

4. Divide the hundreds. Always start on the left, with the biggest place value.
   2 ÷ 6: 2 is smaller than 6, so we cannot give each share a whole hundred. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 2 hundreds for 20 tens. Carry 2 to the next digit.
5. Divide the tens. Move to the next digit on the right. Add the 2 that was carried.
   The carried 2 makes 20 tens. With the 4 tens there are 20 + 4 = 24 tens.
   24 ÷ 6: 4 × 6 = 24. Write 4 above the 4.
6. Divide the ones. Move to the next digit on the right.
   Nothing was carried, so there are just 0 ones.
   0 ÷ 6 = 0. Write 0 above. This 0 is a place holder, and it must stay in the answer.
   Nothing is left over, so there is no remainder.
7. Read the answer. Ignore any 0 at the front of the answer.

   	0	4	0
   6	2	24	0

   
   240 ÷ 6 = 40
8. Write the answer. Include the unit.
   40p for 1 apple

Check. 40p × 6 = 240p. ✓

Watch out. Make sure your answer is in the right unit: pence or pounds.

Key words. A rate compares two amounts, such as £12 for 4 pens or 60 km each hour. A unit rate is the amount for just one (one pen, one hour, one litre). The word “per” means “for each” and tells you to divide.

Method. Use a table. Divide both amounts by the same number and the rate stays the same.

1. Put the facts in a table. 24 sandwiches go with £60. We want to know about 6 sandwiches.

   		÷ 4 →
   Sandwiches	24		6
   Cost	£60		?
   		÷ 4 →

2. Find the scale factor. Look at the top row. What do we divide 24 by to get 6?
   24 ÷ 4 = 6, so divide by 4
3. Divide the bottom row by 4 as well. The rate stays the same only if both rows are divided by the same number.
   £60 ÷ 4
4. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   4	6	0

5. Divide the tens. Always start on the left, with the biggest place value.
   6 ÷ 4 = 1, remainder 2
   Write 1 above the 6.
   Exchange the remainder: 2 tens become 20 ones. Carry 2 to the next digit.
6. Divide the ones. Move to the next digit on the right. Add the 2 that was carried.
   The carried 2 makes 20 ones. With the 0 ones there are 20 + 0 = 20 ones.
   20 ÷ 4: 5 × 4 = 20. Write 5 above the 0.
   Nothing is left over, so there is no remainder.
7. Read the answer. Read the digits above the line from left to right.

   	1	5
   4	6	20

   
   60 ÷ 4 = 15
8. Write the answer.
   £15

Check. £15 × 4 = £60. ✓

Watch out. Divide the top and the bottom by the SAME number.

Key words. A rate compares two amounts, such as £12 for 4 pens or 60 km each hour. A unit rate is the amount for just one (one pen, one hour, one litre). The word “per” means “for each” and tells you to divide.

Method. Use a table. Divide both amounts by the same number and the rate stays the same.

1. Put the facts in a table. 15 oranges go with £6. We want to know about 5 oranges.

   		÷ 3 →
   Oranges	15		5
   Cost	£6		?
   		÷ 3 →

2. Find the scale factor. Look at the top row. What do we divide 15 by to get 5?
   15 ÷ 3 = 5, so divide by 3
3. Divide the bottom row by 3 as well. The rate stays the same only if both rows are divided by the same number.
   £6 ÷ 3
4. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   3	6

5. Divide the ones. Always start on the left, with the biggest place value.
   6 ÷ 3 = 2
   Write 2 above the 6.
   Nothing is left over, so there is no remainder.
6. Read the answer. Read the digits above the line from left to right.

   	2
   3	6

   
   6 ÷ 3 = 2
7. Write the answer.
   £2

Check. £2 × 3 = £6. ✓

Watch out. Divide the top and the bottom by the SAME number.

Key words. A rate compares two amounts, such as £12 for 4 pens or 60 km each hour. A unit rate is the amount for just one (one pen, one hour, one litre). The word “per” means “for each” and tells you to divide.

Method. Use a table. Divide both amounts by the same number and the rate stays the same.

1. Put the facts in a table. 36 stickers go with £9. We want to know how many stickers go with £3.

   		÷ 3 →
   Stickers	36		?
   Cost	£9		£3
   		÷ 3 →

2. Find the scale factor. Look at the bottom row. What do we divide £9 by to get £3?
   9 ÷ 3 = 3, so divide by 3
3. Divide the top row by 3 as well. Both rows must be divided by the same number.
   36 ÷ 3
4. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   3	3	6

5. Divide the tens. Always start on the left, with the biggest place value.
   3 ÷ 3 = 1
   Write 1 above the 3.
6. Divide the ones. Move to the next digit on the right.
   Nothing was carried, so there are just 6 ones.
   6 ÷ 3: 2 × 3 = 6. Write 2 above the 6.
   Nothing is left over, so there is no remainder.
7. Read the answer. Read the digits above the line from left to right.

   	1	2
   3	3	6

   
   36 ÷ 3 = 12
8. Write the answer.
   12

Check. 12 × 3 = 36. ✓

Watch out. Divide the top and the bottom by the SAME number.

Key words. A rate compares two amounts, such as £12 for 4 pens or 60 km each hour. A unit rate is the amount for just one (one pen, one hour, one litre). The word “per” means “for each” and tells you to divide.

Method. Find the cost of one first (divide), then multiply to find the cost of the number you want.

1. Put the facts in a table. Find the cost of 1 first. Then scale up.

   		÷ 5 →		× 8 →
   Buns	5		1		8
   Cost	£2		?		?
   		÷ 5 →		× 8 →

2. Find the cost of 1. Divide the cost of 5 buns by 5.
   200p ÷ 5
3. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   5	2	0	0

4. Divide the hundreds. Always start on the left, with the biggest place value.
   2 ÷ 5: 2 is smaller than 5, so we cannot give each share a whole hundred. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 2 hundreds for 20 tens. Carry 2 to the next digit.
5. Divide the tens. Move to the next digit on the right. Add the 2 that was carried.
   The carried 2 makes 20 tens. With the 0 tens there are 20 + 0 = 20 tens.
   20 ÷ 5: 4 × 5 = 20. Write 4 above the 0.
6. Divide the ones. Move to the next digit on the right.
   Nothing was carried, so there are just 0 ones.
   0 ÷ 5 = 0. Write 0 above. This 0 is a place holder, and it must stay in the answer.
   Nothing is left over, so there is no remainder.
7. Read the answer. Ignore any 0 at the front of the answer.

   	0	4	0
   5	2	20	0

   
   200 ÷ 5 = 40
8. Multiply by 8. Now find the cost of 8: 8 lots of one.
   40p × 8 = 320p
9. Write the answer. Change to pounds if the number of pence is 100 or more.
   £3.20

Check. 320p ÷ 8 = 40p, and 40p × 5 = 200p. ✓

Watch out. Pence or pounds? Keep the same unit all the way through, then change at the end.

Key words. A rate compares two amounts, such as £12 for 4 pens or 60 km each hour. A unit rate is the amount for just one (one pen, one hour, one litre). The word “per” means “for each” and tells you to divide.

Method. Find the cost of one first (divide), then multiply to find the cost of the number you want.

1. Put the facts in a table. Find the cost of 1 first. Then scale up.

   		÷ 6 →		× 10 →
   Pencils	6		1		10
   Cost	90p		?		?
   		÷ 6 →		× 10 →

2. Find the cost of 1. Divide the cost of 6 pencils by 6.
   90p ÷ 6
3. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   6	9	0

4. Divide the tens. Always start on the left, with the biggest place value.
   9 ÷ 6 = 1, remainder 3
   Write 1 above the 9.
   Exchange the remainder: 3 tens become 30 ones. Carry 3 to the next digit.
5. Divide the ones. Move to the next digit on the right. Add the 3 that was carried.
   The carried 3 makes 30 ones. With the 0 ones there are 30 + 0 = 30 ones.
   30 ÷ 6: 5 × 6 = 30. Write 5 above the 0.
   Nothing is left over, so there is no remainder.
6. Read the answer. Read the digits above the line from left to right.

   	1	5
   6	9	30

   
   90 ÷ 6 = 15
7. Multiply by 10. Now find the cost of 10: 10 lots of one.
   15p × 10 = 150p
8. Write the answer. Change to pounds if the number of pence is 100 or more.
   £1.50

Check. 150p ÷ 10 = 15p, and 15p × 6 = 90p. ✓

Watch out. Pence or pounds? Keep the same unit all the way through, then change at the end.

Key words. A rate compares two amounts, such as £12 for 4 pens or 60 km each hour. A unit rate is the amount for just one (one pen, one hour, one litre). The word “per” means “for each” and tells you to divide.

Method. To find a rate, divide the total amount by the number of hours.

1. Choose the operation. “A train travels 540 km in 6 hours” asks for the amount for one hour. The word “per” means “for each”, so divide the total by the number of hours.
   540 km ÷ 6 hours
2. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   6	5	4	0

3. Divide the hundreds. Always start on the left, with the biggest place value.
   5 ÷ 6: 5 is smaller than 6, so we cannot give each share a whole hundred. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 5 hundreds for 50 tens. Carry 5 to the next digit.
4. Divide the tens. Move to the next digit on the right. Add the 5 that was carried.
   The carried 5 makes 50 tens. With the 4 tens there are 50 + 4 = 54 tens.
   54 ÷ 6: 9 × 6 = 54. Write 9 above the 4.
5. Divide the ones. Move to the next digit on the right.
   Nothing was carried, so there are just 0 ones.
   0 ÷ 6 = 0. Write 0 above. This 0 is a place holder, and it must stay in the answer.
   Nothing is left over, so there is no remainder.
6. Read the answer. Ignore any 0 at the front of the answer.

   	0	9	0
   6	5	54	0

   
   540 ÷ 6 = 90
7. Write the answer. Include the unit.
   90 km per hour

Check. 90 × 6 = 540. ✓

Watch out. Check that you divided the total by the number of units, not the other way round.

Key words. A rate compares two amounts, such as £12 for 4 pens or 60 km each hour. A unit rate is the amount for just one (one pen, one hour, one litre). The word “per” means “for each” and tells you to divide.

Method. To find a rate, divide the total amount by the number of litres.

1. Choose the operation. “A car travels 392 km on 8 litres” asks for the amount for one litre. The word “per” means “for each”, so divide the total by the number of litres.
   392 km ÷ 8 litres
2. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   8	3	9	2

3. Divide the hundreds. Always start on the left, with the biggest place value.
   3 ÷ 8: 3 is smaller than 8, so we cannot give each share a whole hundred. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 3 hundreds for 30 tens. Carry 3 to the next digit.
4. Divide the tens. Move to the next digit on the right. Add the 3 that was carried.
   The carried 3 makes 30 tens. With the 9 tens there are 30 + 9 = 39 tens.
   39 ÷ 8: 4 × 8 = 32, and 39 − 32 = 7. Write 4 above the 9.
   Exchange: 7 tens become 70 ones. Carry 7 to the next digit.
5. Divide the ones. Move to the next digit on the right. Add the 7 that was carried.
   The carried 7 makes 70 ones. With the 2 ones there are 70 + 2 = 72 ones.
   72 ÷ 8: 9 × 8 = 72. Write 9 above the 2.
   Nothing is left over, so there is no remainder.
6. Read the answer. Ignore any 0 at the front of the answer.

   	0	4	9
   8	3	39	72

   
   392 ÷ 8 = 49
7. Write the answer. Include the unit.
   49 km per litre

Check. 49 × 8 = 392. ✓

Watch out. Check that you divided the total by the number of units, not the other way round.

Key words. A rate compares two amounts, such as £12 for 4 pens or 60 km each hour. A unit rate is the amount for just one (one pen, one hour, one litre). The word “per” means “for each” and tells you to divide.

Method. To find a rate, divide the total amount by the number of minutes.

1. Choose the operation. “A machine makes 2,160 bolts in 9 minutes” asks for the amount for one minute. The word “per” means “for each”, so divide the total by the number of minutes.
   2,160 bolts ÷ 9 minutes
2. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   9	2	1	6	0

3. Divide the thousands. Always start on the left, with the biggest place value.
   2 ÷ 9: 2 is smaller than 9, so we cannot give each share a whole thousand. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 2 thousands for 20 hundreds. Carry 2 to the next digit.
4. Divide the hundreds. Move to the next digit on the right. Add the 2 that was carried.
   The carried 2 makes 20 hundreds. With the 1 hundreds there are 20 + 1 = 21 hundreds.
   21 ÷ 9: 2 × 9 = 18, and 21 − 18 = 3. Write 2 above the 1.
   Exchange: 3 hundreds become 30 tens. Carry 3 to the next digit.
5. Divide the tens. Move to the next digit on the right. Add the 3 that was carried.
   The carried 3 makes 30 tens. With the 6 tens there are 30 + 6 = 36 tens.
   36 ÷ 9: 4 × 9 = 36. Write 4 above the 6.
6. Divide the ones. Move to the next digit on the right.
   Nothing was carried, so there are just 0 ones.
   0 ÷ 9 = 0. Write 0 above. This 0 is a place holder, and it must stay in the answer.
   Nothing is left over, so there is no remainder.
7. Read the answer. Ignore any 0 at the front of the answer.

   	0	2	4	0
   9	2	21	36	0

   
   2,160 ÷ 9 = 240
8. Write the answer. Include the unit.
   240 bolts per minute

Check. 240 × 9 = 2,160. ✓

Watch out. Check that you divided the total by the number of units, not the other way round.

Key words. A rate compares two amounts, such as £12 for 4 pens or 60 km each hour. A unit rate is the amount for just one (one pen, one hour, one litre). The word “per” means “for each” and tells you to divide.

Method. To find a rate, divide the total amount by the number of seconds.

1. Choose the operation. “A tap fills 4,500 ml in 5 seconds” asks for the amount for one second. The word “per” means “for each”, so divide the total by the number of seconds.
   4,500 ml ÷ 5 seconds
2. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   5	4	5	0	0

3. Divide the thousands. Always start on the left, with the biggest place value.
   4 ÷ 5: 4 is smaller than 5, so we cannot give each share a whole thousand. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 4 thousands for 40 hundreds. Carry 4 to the next digit.
4. Divide the hundreds. Move to the next digit on the right. Add the 4 that was carried.
   The carried 4 makes 40 hundreds. With the 5 hundreds there are 40 + 5 = 45 hundreds.
   45 ÷ 5: 9 × 5 = 45. Write 9 above the 5.
5. Divide the tens. Move to the next digit on the right.
   Nothing was carried, so there are just 0 tens.
   0 ÷ 5 = 0. Write 0 above. This 0 is a place holder, and it must stay in the answer.
6. Divide the ones. Move to the next digit on the right.
   Nothing was carried, so there are just 0 ones.
   0 ÷ 5 = 0. Write 0 above. This 0 is a place holder, and it must stay in the answer.
   Nothing is left over, so there is no remainder.
7. Read the answer. Ignore any 0 at the front of the answer.

   	0	9	0	0
   5	4	45	0	0

   
   4,500 ÷ 5 = 900
8. Write the answer. Include the unit.
   900 ml per second

Check. 900 × 5 = 4,500. ✓

Watch out. Check that you divided the total by the number of units, not the other way round.

Key words. A rate compares two amounts, such as £12 for 4 pens or 60 km each hour. A unit rate is the amount for just one (one pen, one hour, one litre). The word “per” means “for each” and tells you to divide.

Method. Divide the total cost by the number of items to find the cost of one.

1. Choose the operation. 4 pens cost £12.00 altogether. The cost of one is the total shared equally between 4.
   £12.00 ÷ 4
2. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   4	1	2

3. Divide the tens. Always start on the left, with the biggest place value.
   1 ÷ 4: 1 is smaller than 4, so we cannot give each share a whole ten. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 1 ten for 10 ones. Carry 1 to the next digit.
4. Divide the ones. Move to the next digit on the right. Add the 1 that was carried.
   The carried 1 makes 10 ones. With the 2 ones there are 10 + 2 = 12 ones.
   12 ÷ 4: 3 × 4 = 12. Write 3 above the 2.
   Nothing is left over, so there is no remainder.
5. Read the answer. Ignore any 0 at the front of the answer.

   	0	3
   4	1	12

   
   12 ÷ 4 = 3
6. Write the answer. Include the pound sign.
   £3 for 1 pen

Check. £3 × 4 = £12. ✓

Watch out. Cost of one is always less than the total. Divide the total by the number of items.

Key words. A rate compares two amounts, such as £12 for 4 pens or 60 km each hour. A unit rate is the amount for just one (one pen, one hour, one litre). The word “per” means “for each” and tells you to divide.

Method. Find the cost of one first (divide), then multiply to find the cost of the number you want.

1. Put the facts in a table. Find the cost of 1 first. Then scale up.

   		÷ 4 →		× 7 →
   Pens	4		1		7
   Cost	£12		?		?
   		÷ 4 →		× 7 →

2. Find the cost of 1. Divide the cost of 4 pens by 4.
   1,200p ÷ 4
3. Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

   4	1	2	0	0

4. Divide the thousands. Always start on the left, with the biggest place value.
   1 ÷ 4: 1 is smaller than 4, so we cannot give each share a whole thousand. Write 0 (a 0 at the front is not part of the answer).
   Exchange the 1 thousand for 10 hundreds. Carry 1 to the next digit.
5. Divide the hundreds. Move to the next digit on the right. Add the 1 that was carried.
   The carried 1 makes 10 hundreds. With the 2 hundreds there are 10 + 2 = 12 hundreds.
   12 ÷ 4: 3 × 4 = 12. Write 3 above the 2.
6. Divide the tens. Move to the next digit on the right.
   Nothing was carried, so there are just 0 tens.
   0 ÷ 4 = 0. Write 0 above. This 0 is a place holder, and it must stay in the answer.
7. Divide the ones. Move to the next digit on the right.
   Nothing was carried, so there are just 0 ones.
   0 ÷ 4 = 0. Write 0 above. This 0 is a place holder, and it must stay in the answer.
   Nothing is left over, so there is no remainder.
8. Read the answer. Ignore any 0 at the front of the answer.

   	0	3	0	0
   4	1	12	0	0

   
   1,200 ÷ 4 = 300
9. Multiply by 7. Now find the cost of 7: 7 lots of one.
   300p × 7 = 2,100p
10. Write the answer. Change to pounds if the number of pence is 100 or more.
    £21

Check. 2,100p ÷ 7 = 300p, and 300p × 4 = 1,200p. ✓

Watch out. Pence or pounds? Keep the same unit all the way through, then change at the end.

Key words. A rate compares two amounts, such as £12 for 4 pens or 60 km each hour. A unit rate is the amount for just one (one pen, one hour, one litre). The word “per” means “for each” and tells you to divide.

Method. Work out the price of one item in each pack. The pack with the lower price for one item is the best buy.

1. Find the price of one in each pack. To compare fairly, we need the same amount of each. Work out the price of ONE pencil. Work in pence.
   Pack A: £1.50 = 150p for 6
   Pack B: £1.84 = 184p for 8
2. Pack A: divide 150p by 6. Price of one = price of the pack ÷ number in the pack.

   	0	2	5
   6	1	15	30

   
   150p ÷ 6 = 25p
3. Pack B: divide 184p by 8.

   	0	2	3
   8	1	18	24

   
   184p ÷ 8 = 23p
4. Compare. The smaller price for one is the better value.
   25p > 23p, so Pack B is better value.

Check. Pack A: 25p × 6 = 150p. Pack B: 23p × 8 = 184p. ✓

Watch out. The bigger pack is not always cheaper per item. Work it out.

Key words. A rate compares two amounts, such as £12 for 4 pens or 60 km each hour. A unit rate is the amount for just one (one pen, one hour, one litre). The word “per” means “for each” and tells you to divide.

Method. Work out the price of one item in each pack. The pack with the lower price for one item is the best buy.

1. Find the price of one in each pack. To compare fairly, we need the same amount of each. Work out the price of ONE yoghurt. Work in pence.
   Box of 4: £1.80 = 180p for 4
   Box of 6: £2.52 = 252p for 6
2. Box of 4: divide 180p by 4. Price of one = price of the pack ÷ number in the pack.

   	0	4	5
   4	1	18	20

   
   180p ÷ 4 = 45p
3. Box of 6: divide 252p by 6.

   	0	4	2
   6	2	25	12

   
   252p ÷ 6 = 42p
4. Compare. The smaller price for one is the better value.
   45p > 42p, so Box of 6 is better value.

Check. Box of 4: 45p × 4 = 180p. Box of 6: 42p × 6 = 252p. ✓

Watch out. The bigger pack is not always cheaper per item. Work it out.

Key words. A counter-example is one example that shows a statement is false.

Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.

1. Try some numbers. Test the statement with some examples.
   12 ÷ 3 = 4 (smaller than 12)
   100 ÷ 10 = 10 (smaller than 100)
   5 ÷ 2 = 2 r 1 (2 is smaller than 5)
2. Explain why. A reason shows what is happening for every number, not just the ones we tried.
   When you share an amount into 2 or more equal parts, each part is smaller than the whole. So the whole-number answer is always smaller than the number you started with.
3. Decide.
   Always

Check. Every answer was smaller. ✓

Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.

Key words. A counter-example is one example that shows a statement is false.

Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.

1. Try some numbers. Test the statement with some examples.
   £12 for 4 pens: £3 for 1 pen (a whole number of pounds)
   £10 for 4 pens: £2.50 for 1 pen (not a whole number of pounds)
2. Look for one that does not work. One example that breaks the rule is called a counter-example.
   £10 ÷ 4 = £2.50 is not a whole number of pounds.
3. Explain why. A reason shows what is happening for every number, not just the ones we tried.
   It depends on whether the total divides exactly by the number of items.
4. Decide.
   Sometimes

Check. We found one example each way. ✓

Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.