How to Multiply Decimals and Integers
To demonstrate how to multiply decimals and integers using long multiplication (without a calculator), we will use the following multiplication:
First I will list the steps to multiplying the numbers, then I will demonstrate each step.
- Write the number with the highest number of digits (multiplicand).
- Directly beneath the first number, write the second number (multiplier).
- Working from right to left, multiply by each digit of multiplicand by right-most digit of the multiplier.
- If a multiplication is less than 10, place the result in the first row beneath the multiplier, in the next available column cell.
- If a multiplication is greater than 9, place the one's portion of the result in the row beneath the multiplier, and place the ten's portion (carryover) at the top of the next column. This carryover is then added to the result of the next multiplication.
- For each subsequent multiplier digit, add a new row with the number of zeros indicating the power of 10 for that multiplier's digit.
- Repeat the multiplication for each multiplier digit.
- Once the multiplications are complete, you should have the same number of rows as the number of digits of the multiplier.
- Add up all result rows to get the product of the two numbers being multiplied.
- Count the number of decimal places in the multiplicand and multiplier and add them together to get the number of decimal places to create in the product.
- Determine if the result is positive or negative.
Here is how the above steps would play out for our example multiplication with decimals (132.5 x 4.6).
Step #1: Write and Position Numbers
Create a grid for the multiplication and enter the number with the greatest number of digits first, followed by the number with the least number of digits. Note that I have added vertical red lines to indicate where the decimal points exist.
| 1 | 3 | 2 | 5 | ||
| x | 4 | 6 |
Step #2: Multiply One's Digit In Multiplier
Multiply the right-most digit of the multiplier (6) by each digit of the multiplicand.
- 6 x 5 = 30, so write 0 in column #1 in the result row and write a 3 at the top of column #2.
- 6 x 2 = 12, + carryover of 3 = 15, so write 5 in column #2 in the result row and write a 1 at the top of column #3.
- 6 x 3 = 18, + carryover of 1 = 19, so write 9 in column #3 in the result row and write a 1 at the top of column #4.
- 6 x 1 = 6, + carryover of 1 = 7, so write 7 in column #5 in the result row.
| 1 | 1 | 3 | |||
| 1 | 3 | 2 | 5 | ||
| x | 4 | 6 | |||
| 7 | 9 | 5 | 0 | ||
| col | 5 | 4 | 3 | 2 | 1 |
Step #3: Multiply Ten's Digit In Multiplier
Multiply the next digit of the multiplier (4, which actually represents 40) by each digit of the multiplicand.
- Since we are now working with the tens column, write a 0 in column #1 on a new row.
- 4 x 5 = 20, so write 0 in column #2 of the second result row and write a 2 at the top of column #3.
- 4 x 2 = 8, + carryover of 2 = 10, so write 0 in column #3 of the second result row and write a 1 at the top of column #4.
- 4 x 3 = 12, + carryover of 1 = 13, so write 3 in column #4 of the second result row and write a 1 at the top of column #5.
- 4 x 1 = 4, + carryover of 1 = 5, so write 5 in column #5 of the second result row.
| 1 | 1 | 2 | |||
| 1 | 3 | 2 | 5 | ||
| x | 4 | 6 | |||
| 7 | 9 | 5 | 0 | ||
| + | 5 | 3 | 0 | 0 | 0 |
| col | 5 | 4 | 3 | 2 | 1 |
Step #4: Add Up The Result Rows
Add the two result rows together to get the product of the two numbers.
7950 + 53000 = 60950
| 1 | 1 | 3 | |||
| 1 | 3 | 2 | 5 | ||
| x | 4 | 6 | |||
| 7 | 9 | 5 | 0 | ||
| + | 5 | 3 | 0 | 0 | 0 |
| = | 6 | 0 | 9 | 5 | 0 |
Step #5: Insert Decimal Point If Applicable
Count the number of decimal places in the multiplicand (132.5 = 1 place) and in the multiplier (4.6 = 1 place) and add them together to get the number of decimal places to create in the product (1 + 1 = 2 places).
Product of 132.5 x 4.6 = 609.50 (or 609.5)
Note that if the long multiplication result does not have enough digits to create the required number of decimal places, simply add the needed number of zeros to the left side of the result.
For example, if the long multiplication result is 85091 (5 digits), but you need to add 6 decimal places, simply add a 0 to the left side of the result and insert the decimal point at the beginning, so it becomes .085901 (now has 6 decimal places).
Step #6: Determine If The Result Is Positive or Negative
- If you multiplied 2 positive numbers, the result is positive.
- If you multiplied a positive number by a negative number, the result is negative.
- If you multiplied 2 negative numbers, the result is positive.