GCSE Maths: Multiplying (Foundation)
Part of the GCSE Maths number basics chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Multiplying
Multiplication is repeated addition: 6 × 4 means six lots of 4, which is 4 + 4 + 4 + 4 + 4 + 4 = 24. The answer is called the product. The words times, multiplied by, lots of and product all mean multiply. Learn your times tables up to 10 × 10: every method below uses them.
Multiplying by multiples of 10. To work out 34 × 20, think of 20 as 2 × 10. Work out 34 × 2 = 68, then multiply by 10 by moving the digits one place to the left: 680.
Grid method. Split both numbers into hundreds, tens and ones. Draw a grid, multiply the parts in each box, then add up all the boxes. It works because 34 × 26 = (30 + 4) × (20 + 6), and every part of the first number is multiplied by every part of the second.
Long multiplication. This is the grid method written in a compact form. Write the bigger number on top. Multiply it by the units digit of the bottom number to make the first row. Then multiply it by the tens digit: the 2 in 26 is really 20, so first write a 0 place holder in the units column, then multiply by 2. Finally add the rows. Work from the right and carry the tens digit of every column.
Multiplying decimals. Ignore the decimal points and multiply the numbers as whole numbers. Then count the total number of decimal places in the two numbers in the question, and make the answer have that many. For example 0.3 × 0.04: 3 × 4 = 12; there are 1 + 2 = 3 decimal places altogether, so the answer is 0.012. This works because 0.3 is 3 ÷ 10 and 0.04 is 4 ÷ 100, so the answer is 12 ÷ (10 × 100) = 12 ÷ 1000.
Check by estimating. Round each number to one significant figure (one non-zero digit) and multiply the rounded numbers. 3.4 × 0.25 is about 3 × 0.3 = 0.9, so an answer of 8.5 or 0.085 would be wrong.
Key ideas
- Product means the answer to a multiplication. Use your times tables.
- × 20 means × 2 then × 10.
- Long multiplication: multiply by the units digit, then the tens digit (with a 0 place holder), then add the rows.
- Decimals: multiply as whole numbers, then give the answer as many decimal places as there are in the question altogether.
Key words. The product is the answer to a multiplication. A place holder is a 0 used to keep digits in the correct column.
Method. Long multiplication: multiply by each digit of the second number (units, tens, hundreds...) and add the rows.
- Set out the multiplication. Write 347 on top and 26 underneath, with the units under the units. We multiply the top number by each digit of the bottom number, then add the rows.347 × 26 -----
- Multiply 347 by the units digit, 6. Work from the right. Multiply each digit of the top number by 6. If the answer is 10 or more, write the units digit and carry the rest into the next column.7 × 6 = 42: write 2, carry 4
4 × 6 + 4 (carried) = 28: write 8, carry 2
3 × 6 + 2 (carried) = 20: write 0, carry 2
Write the last carry: 2
So 347 × 6 = 2082. - Multiply 347 by the tens digit, 2 (which means 20). The 2 is in the tens column, so it is really 20. First write 1 zero on the right as place holder, then multiply by 2.7 × 2 = 14: write 4, carry 1
4 × 2 + 1 (carried) = 9: write 9
3 × 2 = 6: write 6
347 × 2 = 694, then add 1 zero: 347 × 20 = 6940. - Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.2082 + 6940 = 9022
347 × 26 ----- 2082 6940 ----- 9022
Check. Estimate by rounding each number to 1 significant figure: 300 × 30 = 9000. This is close to 9022, so the size is sensible. ✓
Watch out. When you multiply by the tens digit, write the place-holder 0 first. Forgetting it makes the answer ten times too small.
Method. The grid method splits each number into hundreds, tens and ones, multiplies each pair of parts, then adds all the answers.
- Split each number into hundreds, tens and ones. Partitioning makes the numbers easy to multiply, because we can use times tables and add zeros.58 = 50 + 8
34 = 30 + 4 - Draw a grid and multiply the parts. Each box is one part of the first number times one part of the second number.50 × 30 = 1500
50 × 4 = 200
8 × 30 = 240
8 × 4 = 32 - Fill in the grid. Write each product in its box. The row totals will make adding easier.
× 30 4 Row total 50 1500 200 1700 8 240 32 272 - Add all the boxes. The whole answer is the sum of all the parts.1500 + 200 + 240 + 32 = 1972
Check. Estimate: 60 × 30 = 1800, which is close to 1972. ✓
Watch out. Every part of the first number must be multiplied by every part of the second. A grid helps you not to miss any.
Key words. A decimal place is a digit after the decimal point. 3.45 has 2 decimal places.
Method. Multiply decimals as whole numbers first, then put the decimal point back using the total number of decimal places in the question.
- Ignore the decimal points and multiply as whole numbers. 4.6 is 46 divided by 10, and 3.2 is 32 divided by 10. So we can multiply 46 by 32, then fix the size at the end.4.6 × 3.2 → 46 × 32
- Multiply 46 by the units digit, 2. Work from the right. Multiply each digit of the top number by 2. If the answer is 10 or more, write the units digit and carry the rest into the next column.6 × 2 = 12: write 2, carry 1
4 × 2 + 1 (carried) = 9: write 9
So 46 × 2 = 92. - Multiply 46 by the tens digit, 3 (which means 30). The 3 is in the tens column, so it is really 30. First write 1 zero on the right as place holder, then multiply by 3.6 × 3 = 18: write 8, carry 1
4 × 3 + 1 (carried) = 13: write 3, carry 1
Write the last carry: 1
46 × 3 = 138, then add 1 zero: 46 × 30 = 1380. - Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.92 + 1380 = 1472
46 × 32 ---- 92 1380 ----- 1472 - Put the decimal point back. Count the decimal places in the question: 4.6 has 1 and 3.2 has 1, so the answer has 1 + 1 = 2 decimal places. Count that many digits in from the right.1472 with 2 decimal places: 14.72
Check. Estimate: 5 × 3 = 15. 14.72 is close to 15. ✓
Watch out. Count decimal places in both numbers and add them. The answer has as many as the two numbers together.
Watch out
When you multiply decimals, you do not line up the decimal points (that is only for adding and subtracting). Multiply as whole numbers, then count the decimal places in the question and put the point in the answer.Exercise 1C Multiplying whole numbers and decimals
Non-calculatorMethod. Multiply by the non-zero digit first, then by 10, 100 or 1000 by moving the digits.
- Split 20 into 2 and 10. 20 is 2 lots of 10. Multiplying by a number like 20 is the same as multiplying by 2 and then by 10.20 = 2 × 10
- Multiply by 2. This is an easy times-table style calculation.34 × 2 = 68
- Multiply by 10 by moving the digits 1 place to the left. Each place the digits move left makes the number ten times bigger.68 × 10 = 680
Check. Reverse: 680 ÷ 20 = 34. ✓
Watch out. Do not just stick zeros on the end of a decimal. 0.4 × 10 = 4, not 0.40.
Method. Multiply by the non-zero digit first, then by 10, 100 or 1000 by moving the digits.
- Split 30 into 3 and 10. 30 is 3 lots of 10. Multiplying by a number like 30 is the same as multiplying by 3 and then by 10.30 = 3 × 10
- Multiply by 3. This is an easy times-table style calculation.250 × 3 = 750
- Multiply by 10 by moving the digits 1 place to the left. Each place the digits move left makes the number ten times bigger.750 × 10 = 7500
Check. Reverse: 7500 ÷ 30 = 250. ✓
Watch out. Do not just stick zeros on the end of a decimal. 0.4 × 10 = 4, not 0.40.
Method. Multiply by the non-zero digit first, then by 10, 100 or 1000 by moving the digits.
- Split 200 into 2 and 100. 200 is 2 lots of 100. Multiplying by a number like 200 is the same as multiplying by 2 and then by 100.200 = 2 × 100
- Multiply by 2. This is an easy times-table style calculation.46 × 2 = 92
- Multiply by 100 by moving the digits 2 places to the left. Each place the digits move left makes the number ten times bigger.92 × 100 = 9200
Check. Reverse: 9200 ÷ 200 = 46. ✓
Watch out. Do not just stick zeros on the end of a decimal. 0.4 × 10 = 4, not 0.40.
Method. Multiply by the non-zero digit first, then by 10, 100 or 1000 by moving the digits.
- Split 40 into 4 and 10. 40 is 4 lots of 10. Multiplying by a number like 40 is the same as multiplying by 4 and then by 10.40 = 4 × 10
- Multiply by 4. This is an easy times-table style calculation.7.5 × 4 = 30
- Multiply by 10 by moving the digits 1 place to the left. Each place the digits move left makes the number ten times bigger.30 × 10 = 300
Check. Reverse: 300 ÷ 40 = 7.5. ✓
Watch out. Do not just stick zeros on the end of a decimal. 0.4 × 10 = 4, not 0.40.
Method. Multiply by the non-zero digit first, then by 10, 100 or 1000 by moving the digits.
- Split 600 into 6 and 100. 600 is 6 lots of 100. Multiplying by a number like 600 is the same as multiplying by 6 and then by 100.600 = 6 × 100
- Multiply by 6. This is an easy times-table style calculation.0.8 × 6 = 4.8
- Multiply by 100 by moving the digits 2 places to the left. Each place the digits move left makes the number ten times bigger.4.8 × 100 = 480
Check. Reverse: 480 ÷ 600 = 0.8. ✓
Watch out. Do not just stick zeros on the end of a decimal. 0.4 × 10 = 4, not 0.40.
Method. Multiply by the non-zero digit first, then by 10, 100 or 1000 by moving the digits.
- Split 4000 into 4 and 1000. 4000 is 4 lots of 1000. Multiplying by a number like 4000 is the same as multiplying by 4 and then by 1000.4000 = 4 × 1000
- Multiply by 4. This is an easy times-table style calculation.125 × 4 = 500
- Multiply by 1000 by moving the digits 3 places to the left. Each place the digits move left makes the number ten times bigger.500 × 1000 = 500 000
Check. Reverse: 500 000 ÷ 4000 = 125. ✓
Watch out. Do not just stick zeros on the end of a decimal. 0.4 × 10 = 4, not 0.40.
Key words. The product is the answer to a multiplication. A place holder is a 0 used to keep digits in the correct column.
Method. Long multiplication: multiply by each digit of the second number (units, tens, hundreds...) and add the rows.
- Set out the multiplication. Write 245 on top and 6 underneath, with the units under the units. We multiply the top number by each digit of the bottom number, then add the rows.245 × 6 -----
- Multiply 245 by 6. Work from the right. Multiply each digit of the top number by 6. If the answer is 10 or more, write the units digit and carry the rest into the next column.5 × 6 = 30: write 0, carry 3
4 × 6 + 3 (carried) = 27: write 7, carry 2
2 × 6 + 2 (carried) = 14: write 4, carry 1
Write the last carry: 1
So 245 × 6 = 1470.
Check. Estimate by rounding each number to 1 significant figure: 200 × 6 = 1200. This is close to 1470, so the size is sensible. ✓
Watch out. When you multiply by the tens digit, write the place-holder 0 first. Forgetting it makes the answer ten times too small.
Key words. The product is the answer to a multiplication. A place holder is a 0 used to keep digits in the correct column.
Method. Long multiplication: multiply by each digit of the second number (units, tens, hundreds...) and add the rows.
- Set out the multiplication. Write 1308 on top and 7 underneath, with the units under the units. We multiply the top number by each digit of the bottom number, then add the rows.1308 × 7 ------
- Multiply 1308 by 7. Work from the right. Multiply each digit of the top number by 7. If the answer is 10 or more, write the units digit and carry the rest into the next column.8 × 7 = 56: write 6, carry 5
0 × 7 + 5 (carried) = 5: write 5
3 × 7 = 21: write 1, carry 2
1 × 7 + 2 (carried) = 9: write 9
So 1308 × 7 = 9156.
Check. Estimate by rounding each number to 1 significant figure: 1000 × 7 = 7000. This is close to 9156, so the size is sensible. ✓
Watch out. When you multiply by the tens digit, write the place-holder 0 first. Forgetting it makes the answer ten times too small.
Key words. The product is the answer to a multiplication. A place holder is a 0 used to keep digits in the correct column.
Method. Long multiplication: multiply by each digit of the second number (units, tens, hundreds...) and add the rows.
- Set out the multiplication. Write 86 on top and 9 underneath, with the units under the units. We multiply the top number by each digit of the bottom number, then add the rows.86 × 9 ----
- Multiply 86 by 9. Work from the right. Multiply each digit of the top number by 9. If the answer is 10 or more, write the units digit and carry the rest into the next column.6 × 9 = 54: write 4, carry 5
8 × 9 + 5 (carried) = 77: write 7, carry 7
Write the last carry: 7
So 86 × 9 = 774.
Check. Estimate by rounding each number to 1 significant figure: 90 × 9 = 810. This is close to 774, so the size is sensible. ✓
Watch out. When you multiply by the tens digit, write the place-holder 0 first. Forgetting it makes the answer ten times too small.
Key words. The product is the answer to a multiplication. A place holder is a 0 used to keep digits in the correct column.
Method. Long multiplication: multiply by each digit of the second number (units, tens, hundreds...) and add the rows.
- Set out the multiplication. Write 4125 on top and 8 underneath, with the units under the units. We multiply the top number by each digit of the bottom number, then add the rows.4125 × 8 ------
- Multiply 4125 by 8. Work from the right. Multiply each digit of the top number by 8. If the answer is 10 or more, write the units digit and carry the rest into the next column.5 × 8 = 40: write 0, carry 4
2 × 8 + 4 (carried) = 20: write 0, carry 2
1 × 8 + 2 (carried) = 10: write 0, carry 1
4 × 8 + 1 (carried) = 33: write 3, carry 3
Write the last carry: 3
So 4125 × 8 = 33 000.
Check. Estimate by rounding each number to 1 significant figure: 4000 × 8 = 32 000. This is close to 33 000, so the size is sensible. ✓
Watch out. When you multiply by the tens digit, write the place-holder 0 first. Forgetting it makes the answer ten times too small.
Method. The grid method splits each number into hundreds, tens and ones, multiplies each pair of parts, then adds all the answers.
- Split each number into hundreds, tens and ones. Partitioning makes the numbers easy to multiply, because we can use times tables and add zeros.23 = 20 + 3
14 = 10 + 4 - Draw a grid and multiply the parts. Each box is one part of the first number times one part of the second number.20 × 10 = 200
20 × 4 = 80
3 × 10 = 30
3 × 4 = 12 - Fill in the grid. Write each product in its box. The row totals will make adding easier.
× 10 4 Row total 20 200 80 280 3 30 12 42 - Add all the boxes. The whole answer is the sum of all the parts.200 + 80 + 30 + 12 = 322
Check. Estimate: 20 × 10 = 200, which is close to 322. ✓
Watch out. Every part of the first number must be multiplied by every part of the second. A grid helps you not to miss any.
Method. The grid method splits each number into hundreds, tens and ones, multiplies each pair of parts, then adds all the answers.
- Split each number into hundreds, tens and ones. Partitioning makes the numbers easy to multiply, because we can use times tables and add zeros.46 = 40 + 6
35 = 30 + 5 - Draw a grid and multiply the parts. Each box is one part of the first number times one part of the second number.40 × 30 = 1200
40 × 5 = 200
6 × 30 = 180
6 × 5 = 30 - Fill in the grid. Write each product in its box. The row totals will make adding easier.
× 30 5 Row total 40 1200 200 1400 6 180 30 210 - Add all the boxes. The whole answer is the sum of all the parts.1200 + 200 + 180 + 30 = 1610
Check. Estimate: 50 × 40 = 2000, which is close to 1610. ✓
Watch out. Every part of the first number must be multiplied by every part of the second. A grid helps you not to miss any.
Method. The grid method splits each number into hundreds, tens and ones, multiplies each pair of parts, then adds all the answers.
- Split each number into hundreds, tens and ones. Partitioning makes the numbers easy to multiply, because we can use times tables and add zeros.124 = 100 + 20 + 4
23 = 20 + 3 - Draw a grid and multiply the parts. Each box is one part of the first number times one part of the second number.100 × 20 = 2000
100 × 3 = 300
20 × 20 = 400
20 × 3 = 60
4 × 20 = 80
4 × 3 = 12 - Fill in the grid. Write each product in its box. The row totals will make adding easier.
× 20 3 Row total 100 2000 300 2300 20 400 60 460 4 80 12 92 - Add all the boxes. The whole answer is the sum of all the parts.2000 + 300 + 400 + 60 + 80 + 12 = 2852
Check. Estimate: 100 × 20 = 2000, which is close to 2852. ✓
Watch out. Every part of the first number must be multiplied by every part of the second. A grid helps you not to miss any.
Key words. The product is the answer to a multiplication. A place holder is a 0 used to keep digits in the correct column.
Method. Long multiplication: multiply by each digit of the second number (units, tens, hundreds...) and add the rows.
- Set out the multiplication. Write 47 on top and 36 underneath, with the units under the units. We multiply the top number by each digit of the bottom number, then add the rows.47 × 36 ----
- Multiply 47 by the units digit, 6. Work from the right. Multiply each digit of the top number by 6. If the answer is 10 or more, write the units digit and carry the rest into the next column.7 × 6 = 42: write 2, carry 4
4 × 6 + 4 (carried) = 28: write 8, carry 2
Write the last carry: 2
So 47 × 6 = 282. - Multiply 47 by the tens digit, 3 (which means 30). The 3 is in the tens column, so it is really 30. First write 1 zero on the right as place holder, then multiply by 3.7 × 3 = 21: write 1, carry 2
4 × 3 + 2 (carried) = 14: write 4, carry 1
Write the last carry: 1
47 × 3 = 141, then add 1 zero: 47 × 30 = 1410. - Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.282 + 1410 = 1692
47 × 36 ---- 282 1410 ----- 1692
Check. Estimate by rounding each number to 1 significant figure: 50 × 40 = 2000. This is close to 1692, so the size is sensible. ✓
Watch out. When you multiply by the tens digit, write the place-holder 0 first. Forgetting it makes the answer ten times too small.
Key words. The product is the answer to a multiplication. A place holder is a 0 used to keep digits in the correct column.
Method. Long multiplication: multiply by each digit of the second number (units, tens, hundreds...) and add the rows.
- Set out the multiplication. Write 308 on top and 25 underneath, with the units under the units. We multiply the top number by each digit of the bottom number, then add the rows.308 × 25 -----
- Multiply 308 by the units digit, 5. Work from the right. Multiply each digit of the top number by 5. If the answer is 10 or more, write the units digit and carry the rest into the next column.8 × 5 = 40: write 0, carry 4
0 × 5 + 4 (carried) = 4: write 4
3 × 5 = 15: write 5, carry 1
Write the last carry: 1
So 308 × 5 = 1540. - Multiply 308 by the tens digit, 2 (which means 20). The 2 is in the tens column, so it is really 20. First write 1 zero on the right as place holder, then multiply by 2.8 × 2 = 16: write 6, carry 1
0 × 2 + 1 (carried) = 1: write 1
3 × 2 = 6: write 6
308 × 2 = 616, then add 1 zero: 308 × 20 = 6160. - Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.1540 + 6160 = 7700
308 × 25 ----- 1540 6160 ----- 7700
Check. Estimate by rounding each number to 1 significant figure: 300 × 30 = 9000. This is close to 7700, so the size is sensible. ✓
Watch out. When you multiply by the tens digit, write the place-holder 0 first. Forgetting it makes the answer ten times too small.
Key words. The product is the answer to a multiplication. A place holder is a 0 used to keep digits in the correct column.
Method. Long multiplication: multiply by each digit of the second number (units, tens, hundreds...) and add the rows.
- Set out the multiplication. Write 726 on top and 48 underneath, with the units under the units. We multiply the top number by each digit of the bottom number, then add the rows.726 × 48 -----
- Multiply 726 by the units digit, 8. Work from the right. Multiply each digit of the top number by 8. If the answer is 10 or more, write the units digit and carry the rest into the next column.6 × 8 = 48: write 8, carry 4
2 × 8 + 4 (carried) = 20: write 0, carry 2
7 × 8 + 2 (carried) = 58: write 8, carry 5
Write the last carry: 5
So 726 × 8 = 5808. - Multiply 726 by the tens digit, 4 (which means 40). The 4 is in the tens column, so it is really 40. First write 1 zero on the right as place holder, then multiply by 4.6 × 4 = 24: write 4, carry 2
2 × 4 + 2 (carried) = 10: write 0, carry 1
7 × 4 + 1 (carried) = 29: write 9, carry 2
Write the last carry: 2
726 × 4 = 2904, then add 1 zero: 726 × 40 = 29 040. - Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.5808 + 29 040 = 34 848
726 × 48 ----- 5808 29040 ------ 34848
Check. Estimate by rounding each number to 1 significant figure: 700 × 50 = 35 000. This is close to 34 848, so the size is sensible. ✓
Watch out. When you multiply by the tens digit, write the place-holder 0 first. Forgetting it makes the answer ten times too small.
Key words. The product is the answer to a multiplication. A place holder is a 0 used to keep digits in the correct column.
Method. Long multiplication: multiply by each digit of the second number (units, tens, hundreds...) and add the rows.
- Set out the multiplication. Write 1204 on top and 63 underneath, with the units under the units. We multiply the top number by each digit of the bottom number, then add the rows.1204 × 63 ------
- Multiply 1204 by the units digit, 3. Work from the right. Multiply each digit of the top number by 3. If the answer is 10 or more, write the units digit and carry the rest into the next column.4 × 3 = 12: write 2, carry 1
0 × 3 + 1 (carried) = 1: write 1
2 × 3 = 6: write 6
1 × 3 = 3: write 3
So 1204 × 3 = 3612. - Multiply 1204 by the tens digit, 6 (which means 60). The 6 is in the tens column, so it is really 60. First write 1 zero on the right as place holder, then multiply by 6.4 × 6 = 24: write 4, carry 2
0 × 6 + 2 (carried) = 2: write 2
2 × 6 = 12: write 2, carry 1
1 × 6 + 1 (carried) = 7: write 7
1204 × 6 = 7224, then add 1 zero: 1204 × 60 = 72 240. - Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.3612 + 72 240 = 75 852
1204 × 63 ------ 3612 72240 ------ 75852
Check. Estimate by rounding each number to 1 significant figure: 1000 × 60 = 60 000. This is close to 75 852, so the size is sensible. ✓
Watch out. When you multiply by the tens digit, write the place-holder 0 first. Forgetting it makes the answer ten times too small.
Key words. A decimal place is a digit after the decimal point. 3.45 has 2 decimal places.
Method. Multiply decimals as whole numbers first, then put the decimal point back using the total number of decimal places in the question.
- Ignore the decimal points and multiply as whole numbers. 0.6 is 6 divided by 10, and 0.8 is 8 divided by 10. So we can multiply 6 by 8, then fix the size at the end.0.6 × 0.8 → 6 × 8
- Multiply 8 by 6. Work from the right. Multiply each digit of the top number by 6. If the answer is 10 or more, write the units digit and carry the rest into the next column.8 × 6 = 48: write 8, carry 4
Write the last carry: 4
So 8 × 6 = 48. - Put the decimal point back. Count the decimal places in the question: 0.6 has 1 and 0.8 has 1, so the answer has 1 + 1 = 2 decimal places. Count that many digits in from the right.48 with 2 decimal places: 0.48
Check. Estimate: 0.6 × 0.8 = 0.48. 0.48 is close to 0.48. ✓
Watch out. Count decimal places in both numbers and add them. The answer has as many as the two numbers together.
Key words. A decimal place is a digit after the decimal point. 3.45 has 2 decimal places.
Method. Multiply decimals as whole numbers first, then put the decimal point back using the total number of decimal places in the question.
- Ignore the decimal points and multiply as whole numbers. 2.5 is 25 divided by 10, and 4.2 is 42 divided by 10. So we can multiply 25 by 42, then fix the size at the end.2.5 × 4.2 → 25 × 42
- Multiply 42 by the units digit, 5. Work from the right. Multiply each digit of the top number by 5. If the answer is 10 or more, write the units digit and carry the rest into the next column.2 × 5 = 10: write 0, carry 1
4 × 5 + 1 (carried) = 21: write 1, carry 2
Write the last carry: 2
So 42 × 5 = 210. - Multiply 42 by the tens digit, 2 (which means 20). The 2 is in the tens column, so it is really 20. First write 1 zero on the right as place holder, then multiply by 2.2 × 2 = 4: write 4
4 × 2 = 8: write 8
42 × 2 = 84, then add 1 zero: 42 × 20 = 840. - Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.210 + 840 = 1050
42 × 25 ---- 210 840 ----- 1050 - Put the decimal point back. Count the decimal places in the question: 2.5 has 1 and 4.2 has 1, so the answer has 1 + 1 = 2 decimal places. Count that many digits in from the right.1050 with 2 decimal places: 10.5
Check. Estimate: 3 × 4 = 12. 10.5 is close to 12. ✓
Watch out. Count decimal places in both numbers and add them. The answer has as many as the two numbers together.
Key words. A decimal place is a digit after the decimal point. 3.45 has 2 decimal places.
Method. Multiply decimals as whole numbers first, then put the decimal point back using the total number of decimal places in the question.
- Ignore the decimal points and multiply as whole numbers. 12.4 is 124 divided by 10, and 0.35 is 35 divided by 100. So we can multiply 124 by 35, then fix the size at the end.12.4 × 0.35 → 124 × 35
- Multiply 124 by the units digit, 5. Work from the right. Multiply each digit of the top number by 5. If the answer is 10 or more, write the units digit and carry the rest into the next column.4 × 5 = 20: write 0, carry 2
2 × 5 + 2 (carried) = 12: write 2, carry 1
1 × 5 + 1 (carried) = 6: write 6
So 124 × 5 = 620. - Multiply 124 by the tens digit, 3 (which means 30). The 3 is in the tens column, so it is really 30. First write 1 zero on the right as place holder, then multiply by 3.4 × 3 = 12: write 2, carry 1
2 × 3 + 1 (carried) = 7: write 7
1 × 3 = 3: write 3
124 × 3 = 372, then add 1 zero: 124 × 30 = 3720. - Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.620 + 3720 = 4340
124 × 35 ----- 620 3720 ----- 4340 - Put the decimal point back. Count the decimal places in the question: 12.4 has 1 and 0.35 has 2, so the answer has 1 + 2 = 3 decimal places. Count that many digits in from the right.4340 with 3 decimal places: 4.34
Check. Estimate: 10 × 0.4 = 4. 4.34 is close to 4. ✓
Watch out. Count decimal places in both numbers and add them. The answer has as many as the two numbers together.
Key words. A decimal place is a digit after the decimal point. 3.45 has 2 decimal places.
Method. Multiply decimals as whole numbers first, then put the decimal point back using the total number of decimal places in the question.
- Ignore the decimal points and multiply as whole numbers. 0.09 is 9 divided by 100, and 0.7 is 7 divided by 10. So we can multiply 9 by 7, then fix the size at the end.0.09 × 0.7 → 9 × 7
- Multiply 9 by 7. Work from the right. Multiply each digit of the top number by 7. If the answer is 10 or more, write the units digit and carry the rest into the next column.9 × 7 = 63: write 3, carry 6
Write the last carry: 6
So 9 × 7 = 63. - Put the decimal point back. Count the decimal places in the question: 0.09 has 2 and 0.7 has 1, so the answer has 2 + 1 = 3 decimal places. Count that many digits in from the right.63 with 3 decimal places: 0.063
Check. Estimate: 0.09 × 0.7 = 0.063. 0.063 is close to 0.063. ✓
Watch out. Count decimal places in both numbers and add them. The answer has as many as the two numbers together.
Key words. A decimal place is a digit after the decimal point. 3.45 has 2 decimal places.
Method. Multiply decimals as whole numbers first, then put the decimal point back using the total number of decimal places in the question.
- Ignore the decimal points and multiply as whole numbers. 6.8 is 68 divided by 10, and 5 is 5 divided by 1. So we can multiply 68 by 5, then fix the size at the end.6.8 × 5 → 68 × 5
- Multiply 68 by 5. Work from the right. Multiply each digit of the top number by 5. If the answer is 10 or more, write the units digit and carry the rest into the next column.8 × 5 = 40: write 0, carry 4
6 × 5 + 4 (carried) = 34: write 4, carry 3
Write the last carry: 3
So 68 × 5 = 340. - Put the decimal point back. Count the decimal places in the question: 6.8 has 1 and 5 has 0, so the answer has 1 + 0 = 1 decimal places. Count that many digits in from the right.340 with 1 decimal places: 34
Check. Estimate: 7 × 5 = 35. 34 is close to 35. ✓
Watch out. Count decimal places in both numbers and add them. The answer has as many as the two numbers together.
Key words. A decimal place is a digit after the decimal point. 3.45 has 2 decimal places.
Method. Multiply decimals as whole numbers first, then put the decimal point back using the total number of decimal places in the question.
- Ignore the decimal points and multiply as whole numbers. 0.25 is 25 divided by 100, and 0.12 is 12 divided by 100. So we can multiply 25 by 12, then fix the size at the end.0.25 × 0.12 → 25 × 12
- Multiply 25 by the units digit, 2. Work from the right. Multiply each digit of the top number by 2. If the answer is 10 or more, write the units digit and carry the rest into the next column.5 × 2 = 10: write 0, carry 1
2 × 2 + 1 (carried) = 5: write 5
So 25 × 2 = 50. - Multiply 25 by the tens digit, 1 (which means 10). The 1 is in the tens column, so it is really 10. First write 1 zero on the right as place holder, then multiply by 1.5 × 1 = 5: write 5
2 × 1 = 2: write 2
25 × 1 = 25, then add 1 zero: 25 × 10 = 250. - Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.50 + 250 = 300
25 × 12 ---- 50 250 ---- 300 - Put the decimal point back. Count the decimal places in the question: 0.25 has 2 and 0.12 has 2, so the answer has 2 + 2 = 4 decimal places. Count that many digits in from the right.300 with 4 decimal places: 0.03
Check. Estimate: 0.3 × 0.1 = 0.03. 0.03 is close to 0.03. ✓
Watch out. Count decimal places in both numbers and add them. The answer has as many as the two numbers together.
Key words. A decimal place is a digit after the decimal point. 3.45 has 2 decimal places.
Method. Multiply decimals as whole numbers first, then put the decimal point back using the total number of decimal places in the question.
- Ignore the decimal points and multiply as whole numbers. 14.5 is 145 divided by 10, and 36 is 36 divided by 1. So we can multiply 145 by 36, then fix the size at the end.14.5 × 36 → 145 × 36
- Multiply 145 by the units digit, 6. Work from the right. Multiply each digit of the top number by 6. If the answer is 10 or more, write the units digit and carry the rest into the next column.5 × 6 = 30: write 0, carry 3
4 × 6 + 3 (carried) = 27: write 7, carry 2
1 × 6 + 2 (carried) = 8: write 8
So 145 × 6 = 870. - Multiply 145 by the tens digit, 3 (which means 30). The 3 is in the tens column, so it is really 30. First write 1 zero on the right as place holder, then multiply by 3.5 × 3 = 15: write 5, carry 1
4 × 3 + 1 (carried) = 13: write 3, carry 1
1 × 3 + 1 (carried) = 4: write 4
145 × 3 = 435, then add 1 zero: 145 × 30 = 4350. - Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.870 + 4350 = 5220
145 × 36 ----- 870 4350 ----- 5220 - Put the decimal point back. Count the decimal places in the question: 14.5 has 1 and 36 has 0, so the answer has 1 + 0 = 1 decimal places. Count that many digits in from the right.5220 with 1 decimal places: 522
Check. Estimate: 10 × 40 = 400. 522 is close to 400. ✓
Watch out. Count decimal places in both numbers and add them. The answer has as many as the two numbers together.
Method. Each box has 24 eggs, so multiply the number of eggs in a box by the number of boxes.
- Decide the calculation. Equal groups (boxes of the same size) means multiply.24 × 125
- Multiply by the units digit, then tens, then hundreds of 125, and add. Long multiplication, with the larger number first: 125 × 24 gives the same answer.125 × 4 = 500, 125 × 20 = 2500, 500 + 2500 = 3000
- Write the answer with its unit. 3000 eggs
Check. Estimate: 20 × 100 = 2000, which is close to 3000. ✓
Method. Think about what multiplying by a number less than 1 does.
- Compare 0.9 with 1. 0.9 is less than 1.0.9 < 1
- Think about what that means. Multiplying by 1 leaves a number the same. Multiplying by something smaller than 1 makes the number smaller.7.3 × 0.9 < 7.3 × 1 = 7.3
- Write the conclusion. 7.3 is bigger than 7.3 × 0.9
Check. 7.3 × 0.9 = 6.57, which is less than 7.3. ✓
Watch out. Multiplying does not always make a number bigger. Multiplying by a number between 0 and 1 makes it smaller.
Method. Round each number to one significant figure, multiply, and compare with Ravi's answer.
- Round each number to one significant figure. This gives numbers that are easy to multiply.38.6 is about 40 and 4.7 is about 5
- Multiply the rounded numbers. The product of the rounded numbers gives an estimate of the answer.40 × 5 = 200
- Compare. Ravi's answer is about nine times too big.1814.2 is far bigger than 200, so it is wrong.
- Work out the correct answer. Multiply 386 × 47, then give the answer 1 + 1 = 2 decimal places.386 × 47 = 18142, so 38.6 × 4.7 = 181.42
Check. The correct answer, 181.42, is close to the estimate of 200. ✓
Watch out. Ravi put the decimal point in the wrong place. An estimate catches this kind of mistake.
Method. The product has 2 decimal places, so the whole numbers behind the decimals multiply to 24.
- Count decimal places. Each number has 1 decimal place, so the product has 1 + 1 = 2 decimal places. 0.24 = 24 hundredths.24 hundredths
- Find two whole numbers that multiply to 24. List pairs of factors of 24, with each number 9 or less so that it gives a number below 1 (tenths).3 × 8 = 24 and 4 × 6 = 24
- Make them into decimals with one decimal place. Divide each whole number by 10.0.4 × 0.6 = 0.24 and 0.3 × 0.8 = 0.24
Check. 0.4 × 0.6 = 0.24. ✓