Year 5 Long Multiplication: check-up and SATs-style questions

Key words. Place value tells you what each digit is worth. A place holder is a zero that keeps the other digits in the right columns. Multiplying by 10, 100 or 1,000 makes a number 10, 100 or 1,000 times bigger.

Method. To multiply by 100, move every digit 2 places to the left, and fill any empty places on the right with zeros. The decimal point does not move.

1. Say what multiplying by 100 does. Each digit becomes 100 times bigger, so it moves 2 places to the left. The digits move. The decimal point does not.
   3 ones become 3 hundreds
   4 tenths become 4 tens
   2 hundredths become 2 ones
2. Move every digit 2 places to the left. Use a place-value chart and follow the arrows.
3. Look at what is left after the decimal point. Nothing is left, so we do not write the point or any zeros after it.
   3.42 × 100 = 342, not 342.0
4. Write the answer.
   3.42 × 100 = 342

Check. Go back: 342 ÷ 100 = 3.42 (the digits move 2 places to the right). ✓

Watch out. Do not just add zeros to a decimal. That does not change its value (3.42 and 3.420 are the same number). Move the digits to the left instead.

Key words. Place value tells you what each digit is worth. A place holder is a zero that keeps the other digits in the right columns. Multiplying by 10, 100 or 1,000 makes a number 10, 100 or 1,000 times bigger.

Method. Use the fact 4 × 7 = 28. Multiply the numbers without their zeros, then put all the zeros back.

1. Find the multiplication fact you know. Take the zeros off both numbers. What is left is a times table fact.
   40 = 4 × 10  and  70 = 7 × 10
   The fact is 4 × 7 = 28
2. Count the zeros you took off. Each zero means one × 10. Together they make 100.
   1 zero + 1 zero = 2 zeros, so × 100
3. Multiply the fact by that power of ten. Move the digits of the fact to the left, filling the gaps with zeros.
   28 × 100 = 2,800
4. Write the answer.
   40 × 70 = 2,800

Check. Check by dividing: 2,800 ÷ 70 = 40. ✓

Watch out. Count ALL the zeros from BOTH numbers, then put them after 28.

Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To double is to multiply by 2. To halve is to divide by 2.

Method. Multiplying by 25 is the same as multiplying by 100 and then dividing by 4 (halving twice), because 25 is a quarter of 100.

1. Multiply by 100. 25 is a quarter of 100, so first multiply by 100: move the digits two places to the left.
   36 × 100 = 3,600
2. Halve the answer. To divide by 4, halve twice. First halve.
   3,600 ÷ 2 = 1,800
3. Halve again. A quarter is half of a half.
   1,800 ÷ 2 = 900
4. Write the answer.
   36 × 25 = 900

Check. Estimate: 36 is about 40; 25 is about 30. 40 × 30 = 1,200. Our answer, 900, is a bit less than 1,200. ✓

Watch out. Halve TWICE. Dividing by 4 means halving, then halving again.

Key words. Short multiplication is a written method for multiplying by a one-digit number. Write the numbers in columns and multiply one column at a time, starting with the ones. If a column makes 10 or more, write the ones digit and carry the tens digit to the next column.

Method. Write the numbers in columns. Multiply each digit by the one-digit number, starting with the ones. If a column makes 10 or more, write the ones digit and carry the tens digit.

1. Write the numbers in columns. Ones go under ones, tens under tens and so on. The one-digit number goes under the ones.

   	TTh	Th	H	T	O
   		4	7	2	5
   ×					6

2. Multiply the ones. Always start with the ones, on the right.
   5 × 6 = 30
   30 is 3 tens and 0 ones. Write 0 in the ones column and carry 3.
3. Multiply the tens. Multiply, then add the 3 we carried.
   2 × 6 = 12, and 12 + 3 = 15
   15 is 1 hundred and 5 tens. Write 5 in the tens column and carry 1.
4. Multiply the hundreds. Multiply, then add the 1 we carried.
   7 × 6 = 42, and 42 + 1 = 43
   43 is 4 thousands and 3 hundreds. Write 3 in the hundreds column and carry 4.
5. Multiply the thousands. Multiply, then add the 4 we carried.
   4 × 6 = 24, and 24 + 4 = 28
   28 is 2 ten thousands and 8 thousands. Write 8 in the thousands column and carry 2.
6. Write the carried digit. There is nothing left to multiply in the ten thousands column, but we carried 2 into it.
   Write 2 in the ten thousands column.
7. Read the answer from the final layout. Check that it has about the right number of digits.

   	TTh	Th	H	T	O
   		4	7	2	5
   ×					6
   	2	8	3	5	0
   	2	4	1	3

8. Write the answer.
   4,725 × 6 = 28,350

Check. Estimate: 4,725 is about 5,000. 5,000 × 6 = 30,000. Our answer, 28,350, is close to 30,000. ✓

Watch out. Write each carry small, then add it in the next column AFTER you have multiplied that column.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Multiply by the ones digit on one row. Write a 0 place holder, then multiply by the tens digit on the next row. Add the two rows.

1. Split the multiplier. Multiplying by 46 means multiplying by 6 and by 40, then adding. Write the numbers in columns.
   46 = 40 + 6, so 38 × 46 = (38 × 6) + (38 × 40)
   

   			3	8
   ×			4	6

2. Row 1: multiply by the 6 (the ones digit). This is short multiplication, starting with the ones. Work out 38 × 6.
   Ones: 8 × 6 = 48. Write 8 and carry 4.
   Tens: 3 × 6 = 18, and 18 + 4 (carried) = 22. Write 2 and carry 2.
   Hundreds: nothing left to multiply. Write the carried 2.
   Row 1 = 228
3. Row 2: write a 0 first, then multiply by the 4. The 4 is really 40, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 38 by 4.
   Write 0 in the ones column.
   Ones: 8 × 4 = 32. Write 2 and carry 3.
   Tens: 3 × 4 = 12, and 12 + 3 (carried) = 15. Write 5 and carry 1.
   Hundreds: nothing left to multiply. Write the carried 1.
   Row 2 = 152 with a 0 on the end = 1,520
4. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   			3	8
   ×			4	6
   		2	2	8	38 × 6
   +	1	5	2	0	38 × 40
   	1	7	4	8

   
   228 + 1,520 = 1,748
5. Write the answer.
   38 × 46 = 1,748

Check. Estimate: 38 is about 40; 46 is about 50. 40 × 50 = 2,000. Our answer, 1,748, is close to 2,000. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 4, not by 40, and the answer is ten times too small.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Multiply by the ones digit on one row. Write a 0 place holder, then multiply by the tens digit on the next row. Add the two rows.

1. Split the multiplier. Multiplying by 37 means multiplying by 7 and by 30, then adding. Write the numbers in columns.
   37 = 30 + 7, so 524 × 37 = (524 × 7) + (524 × 30)
   

   			5	2	4
   ×				3	7

2. Row 1: multiply by the 7 (the ones digit). This is short multiplication, starting with the ones. Work out 524 × 7.
   Ones: 4 × 7 = 28. Write 8 and carry 2.
   Tens: 2 × 7 = 14, and 14 + 2 (carried) = 16. Write 6 and carry 1.
   Hundreds: 5 × 7 = 35, and 35 + 1 (carried) = 36. Write 6 and carry 3.
   Thousands: nothing left to multiply. Write the carried 3.
   Row 1 = 3,668
3. Row 2: write a 0 first, then multiply by the 3. The 3 is really 30, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 524 by 3.
   Write 0 in the ones column.
   Ones: 4 × 3 = 12. Write 2 and carry 1.
   Tens: 2 × 3 = 6, and 6 + 1 (carried) = 7. Write 7.
   Hundreds: 5 × 3 = 15. Write 5 and carry 1.
   Thousands: nothing left to multiply. Write the carried 1.
   Row 2 = 1,572 with a 0 on the end = 15,720
4. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   			5	2	4
   ×				3	7
   		3	6	6	8	524 × 7
   +	1	5	7	2	0	524 × 30
   	1	9	3	8	8

   
   3,668 + 15,720 = 19,388
5. Write the answer.
   524 × 37 = 19,388

Check. Estimate: 524 is about 500; 37 is about 40. 500 × 40 = 20,000. Our answer, 19,388, is close to 20,000. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 3, not by 30, and the answer is ten times too small.

Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.

Method. Round each number to one digit followed by zeros, then multiply the rounded numbers.

1. Round each number. Keep the first digit and make the others zeros. Rounded numbers are easy to multiply.
   487 to the nearest 100 is 500
   62 to the nearest 10 is 60
2. Multiply the rounded numbers. Use a times table fact and put the zeros back.
   500 × 60: 5 × 6 = 30, then 3 zeros.  30,000
3. Write the estimate. Use the “about equal to” sign.
   487 × 62 ≈ 30,000
4. Think: is it too big or too small? It helps to know which way the rounding went.
   We rounded one number up and one number down, so we cannot tell for sure whether the real answer is bigger or smaller than the estimate.

Check. The real answer is 30,194, which is close to 30,000. ✓

Watch out. An estimate is not exact, so use ≈ (about equal to) and not =.

Key words. A rate compares two amounts, such as £3 for 5 apples or 12 km each hour. A unit rate is the amount for just one (one apple, one hour).

Method. Write the amounts in a table. Find the number you multiply by to get from one amount to the one you want. Multiply the other amount by the same number.

1. Write what you know in a table. A ratio table keeps matching amounts side by side.

   Buns	6	18
   Cost	£4	?

2. Find the multiplier. How many times bigger is 18 than 6? Count in 6s or use a times table.
   6 × 3 = 18, so the multiplier is 3
3. Multiply the other amount by the same number. If you buy 3 times as much, you pay 3 times as much.
   £4 × 3
4. Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.
   4 = 4
5. Multiply each part by 3. Use a times table fact, then put the zeros back.
   4 × 3 = 12
6. Add the answers. Add them one at a time, starting with the biggest.
   12
7. Write the answer. Include the unit.
   £12

Check. Work back: £12 is 3 times the first amount, and £4 × 3 = £12. ✓

Watch out. Multiply BOTH amounts by the same number so that the rate stays the same.

Key words. Area is the amount of flat space inside a shape. It is measured in square units, such as cm². The area of a rectangle is its length multiplied by its width.

Method. Area of a rectangle = length × width. Multiply the two sides.

1. Draw the rectangle and label the sides. The area is the space inside. Rows of squares: 12 rows with 17 in each row.
2. Write the rule for area. Area of a rectangle = length × width.
   17 × 12
3. Split the multiplier. Multiplying by 12 means multiplying by 2 and by 10, then adding. Write the numbers in columns.
   12 = 10 + 2, so 17 × 12 = (17 × 2) + (17 × 10)
   

   		1	7
   ×		1	2

4. Row 1: multiply by the 2 (the ones digit). This is short multiplication, starting with the ones. Work out 17 × 2.
   Ones: 7 × 2 = 14. Write 4 and carry 1.
   Tens: 1 × 2 = 2, and 2 + 1 (carried) = 3. Write 3.
   Row 1 = 34
5. Row 2: write a 0 first, then multiply by the 1. The 1 is really 10, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 17 by 1.
   Write 0 in the ones column.
   Ones: 7 × 1 = 7. Write 7.
   Tens: 1 × 1 = 1. Write 1.
   Row 2 = 17 with a 0 on the end = 170
6. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   		1	7
   ×		1	2
   		3	4	17 × 2
   +	1	7	0	17 × 10
   	2	0	4

   
   34 + 170 = 204
7. Write the answer with square units. Area is measured in square cm, written cm².
   204 cm²

Check. Estimate: 17 is about 20; 12 is about 10. 20 × 10 = 200. Our answer, 204, is close to 200. ✓

Watch out. Area is in square units (cm²). Perimeter is a length (cm), so do not mix them up.

Key words. The equals sign (=) means “is the same value as”. The two sides of an equals sign balance, like the two sides of a see-saw. It does not mean “the answer comes next”.

Method. Work out the value of the side with no missing number. The other side must have the same value.

1. Work out the left side. The equals sign means both sides are worth the same. Find the value of the side with no missing number.
   6 × 8 = 48
2. Say what the right side must be. It has to balance the left side.
   12 × ? = 48
3. Ask: how many 12s make 48? Count in steps, or use the inverse operation.
   48 ÷ 12 = 4
4. Spot the pattern. If one number gets bigger, the other must get smaller by the same factor to keep the answer.
   12 is 2 times 6, so the missing number must be 2 times smaller than 8: 8 ÷ 2 = 4.
5. Read the sentence back. Both sides must be equal.
   6 × 8 = 12 × 4 (both are 48)

Check. 12 × 4 = 48 and 6 × 8 = 48. ✓

Watch out. If you double one number you must halve the other to keep the same answer. The two sides need the same VALUE, not the same numbers.

Key words. Place value tells you what each digit is worth. A place holder is a zero that keeps the other digits in the right columns. Multiplying by 10, 100 or 1,000 makes a number 10, 100 or 1,000 times bigger.

Method. First work out 4.5 × 100. Then find the option with the same value.

1. Work out the calculation in the question. We need its value to compare with the options.
   4.5 × 100 = 450
2. Work out every option. Do not stop at the first one that looks right. Use place value or a known fact.

   Option	Calculation	Value	Equal to 450?
   A	45 × 10	450	yes
   B	4.5 × 1,000	4,500	no
   C	0.45 × 10	4.5	no
   D	450 × 100	45,000	no

3. Choose the option that matches. Only one option has the right value.
   A (45 × 10)

Check. Check the chosen option again: it has the value 450. ✓

Watch out. Work out all four options. Two of them might look similar.

Key words. A square number is a number multiplied by itself: 6² = 6 × 6 = 36. A cube number is a number multiplied by itself twice: 4³ = 4 × 4 × 4 = 64.

Method. A cube number is a number multiplied by itself twice. 4³ = 4 × 4 × 4. Do it in two steps.

1. Write the cube as a multiplication. 4³ means three 4s multiplied together.
   4³ = 4 × 4 × 4
2. Do the first two numbers. Work in two steps. First multiply the first two numbers.
   4 × 4 = 16
3. Multiply the answer by 4 again. There is one more 4 to multiply by.
   16 × 4
4. Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.
   16 = 10 + 6
5. Multiply each part by 4. Use a times table fact, then put the zeros back.
   10 × 4 = 40
   6 × 4 = 24
6. Add the answers. Add them one at a time, starting with the biggest.
   40 + 24 = 64
7. Write the answer.
   4³ = 64

Check. 64 ÷ 4 = 16 and 16 ÷ 4 = 4. ✓

Watch out. The small 3 does not mean “times 3”. 4³ is 4 × 4 × 4, not 4 × 3.

Key words. Place value tells you what each digit is worth. A place holder is a zero that keeps the other digits in the right columns. Multiplying by 10, 100 or 1,000 makes a number 10, 100 or 1,000 times bigger.

Method. First work out 36 × 25. Then find the option with the same value.

1. Work out the calculation in the question. We need its value to compare with the options.
   36 × 25 = 900
2. Work out every option. Do not stop at the first one that looks right. Use place value or a known fact.

   Option	Calculation	Value	Equal to 900?
   A	36 × 100 ÷ 2	1,800	no
   B	36 × 100 ÷ 4	900	yes
   C	36 × 10 × 5	1,800	no
   D	36 × 20 + 5	725	no

3. Choose the option that matches. Only one option has the right value.
   B (36 × 100 ÷ 4)

Check. Check the chosen option again: it has the value 900. ✓

Watch out. Work out all four options. Two of them might look similar.

Key words. Short multiplication is a written method for multiplying by a one-digit number. Write the numbers in columns and multiply one column at a time, starting with the ones. If a column makes 10 or more, write the ones digit and carry the tens digit to the next column.

Method. Write the numbers in columns. Multiply each digit by the one-digit number, starting with the ones. If a column makes 10 or more, write the ones digit and carry the tens digit.

1. Write the numbers in columns. Ones go under ones, tens under tens and so on. The one-digit number goes under the ones.

   	TTh	Th	H	T	O
   		3	6	8	2
   ×					4

2. Multiply the ones. Always start with the ones, on the right.
   2 × 4 = 8
   8 is less than 10, so write 8. Nothing to carry.
3. Multiply the tens. Now the next column to the left.
   8 × 4 = 32
   32 is 3 hundreds and 2 tens. Write 2 in the tens column and carry 3.
4. Multiply the hundreds. Multiply, then add the 3 we carried.
   6 × 4 = 24, and 24 + 3 = 27
   27 is 2 thousands and 7 hundreds. Write 7 in the hundreds column and carry 2.
5. Multiply the thousands. Multiply, then add the 2 we carried.
   3 × 4 = 12, and 12 + 2 = 14
   14 is 1 ten thousand and 4 thousands. Write 4 in the thousands column and carry 1.
6. Write the carried digit. There is nothing left to multiply in the ten thousands column, but we carried 1 into it.
   Write 1 in the ten thousands column.
7. Read the answer from the final layout. Check that it has about the right number of digits.

   	TTh	Th	H	T	O
   		3	6	8	2
   ×					4
   	1	4	7	2	8
   	1	2	3

8. Write the answer.
   3,682 × 4 = 14,728

Check. Estimate: 3,682 is about 4,000. 4,000 × 4 = 16,000. Our answer, 14,728, is close to 16,000. ✓

Watch out. Write each carry small, then add it in the next column AFTER you have multiplied that column.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Multiply by the ones digit on one row. Write a 0 place holder, then multiply by the tens digit on the next row. Add the two rows.

1. Split the multiplier. Multiplying by 36 means multiplying by 6 and by 30, then adding. Write the numbers in columns.
   36 = 30 + 6, so 47 × 36 = (47 × 6) + (47 × 30)
   

   			4	7
   ×			3	6

2. Row 1: multiply by the 6 (the ones digit). This is short multiplication, starting with the ones. Work out 47 × 6.
   Ones: 7 × 6 = 42. Write 2 and carry 4.
   Tens: 4 × 6 = 24, and 24 + 4 (carried) = 28. Write 8 and carry 2.
   Hundreds: nothing left to multiply. Write the carried 2.
   Row 1 = 282
3. Row 2: write a 0 first, then multiply by the 3. The 3 is really 30, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 47 by 3.
   Write 0 in the ones column.
   Ones: 7 × 3 = 21. Write 1 and carry 2.
   Tens: 4 × 3 = 12, and 12 + 2 (carried) = 14. Write 4 and carry 1.
   Hundreds: nothing left to multiply. Write the carried 1.
   Row 2 = 141 with a 0 on the end = 1,410
4. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   			4	7
   ×			3	6
   		2	8	2	47 × 6
   +	1	4	1	0	47 × 30
   	1	6	9	2

   
   282 + 1,410 = 1,692
5. Write the answer.
   47 × 36 = 1,692

Check. Estimate: 47 is about 50; 36 is about 40. 50 × 40 = 2,000. Our answer, 1,692, is close to 2,000. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 3, not by 30, and the answer is ten times too small.

Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.

Method. Estimate first. If the given answer is far from the estimate, it must be wrong.

1. Estimate the answer. Round each number and multiply.
   48 ≈ 50 and 52 ≈ 50
   50 × 50 = 2,500
2. Compare with the answer in the question. A sensible answer is close to the estimate.
   Estimate: 2,500.   Given answer: 2,496.
3. Decide.
   2,496 is close to 2,500, so the answer could be right.

Check. The real answer is 2,496. ✓

Watch out. An estimate can show that an answer is WRONG. It cannot prove that an answer is exactly right.

Key words. A rate compares two amounts, such as £3 for 5 apples or 12 km each hour. A unit rate is the amount for just one (one apple, one hour).

Method. Write the amounts in a table. Find the number you multiply by to get from one amount to the one you want. Multiply the other amount by the same number.

1. Write what you know in a table. A ratio table keeps matching amounts side by side.

   Apples	5	35
   Cost	£3	?

2. Find the multiplier. How many times bigger is 35 than 5? Count in 5s or use a times table.
   5 × 7 = 35, so the multiplier is 7
3. Multiply the other amount by the same number. If you buy 7 times as much, you pay 7 times as much.
   £3 × 7
4. Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.
   3 = 3
5. Multiply each part by 7. Use a times table fact, then put the zeros back.
   3 × 7 = 21
6. Add the answers. Add them one at a time, starting with the biggest.
   21
7. Write the answer. Include the unit.
   £21

Check. Work back: £21 is 7 times the first amount, and £3 × 7 = £21. ✓

Watch out. Multiply BOTH amounts by the same number so that the rate stays the same.

Key words. A bar model is a drawing that shows the amounts in a problem as bars, so you can see which calculation to do.

Method. Break the problem into steps. Work out one thing at a time and write down what each answer means.

1. Draw a bar model. The empty chairs are all the chairs less the 145 that are taken.
2. Find the number of chairs. 24 rows of 28 chairs. Use long multiplication.
   28 × 24: 28 × 4 = 112 and 28 × 20 = 560
   112 + 560 = 672
3. Subtract the taken seats. 145 seats are taken, so subtract.
   672 − 145 = 527
4. Write the answer. Include the unit.
   527 chairs

Check. 527 + 145 = 672. ✓

Watch out. Marks: 1 for 24 × 28 = 672, 1 for subtracting 145, 1 for the correct answer.

Key words. A factor pair is two whole numbers that multiply to make a number. The product is the answer to a multiplication. The sum is the answer to an addition.

Method. List every pair of whole numbers that multiply to make the product (the factor pairs). Then find the pair that also has the right sum.

1. List the factor pairs. These are the pairs of whole numbers with a product of 96. Start with 1 and work up.
   96 = 1 × 96, 2 × 48, 3 × 32, 4 × 24, 6 × 16, 8 × 12
2. Add each pair. We need the pair with a sum of 20.

   Pair	Sum	Sum is 20?
   1 × 96 = 96	1 + 96 = 97	no
   2 × 48 = 96	2 + 48 = 50	no
   3 × 32 = 96	3 + 32 = 35	no
   4 × 24 = 96	4 + 24 = 28	no
   6 × 16 = 96	6 + 16 = 22	no
   8 × 12 = 96	8 + 12 = 20	yes

3. Write the answer. Only one pair works.
   The numbers are 8 and 12.

Check. 8 × 12 = 96 and 8 + 12 = 20. ✓

Watch out. Check BOTH clues, not just one.