Introduction to Multiplying and Dividing Decimals
Can you do division? Divide a loaf by a knife—what's the answer to that?
—From Through the Looking Glass, by LEWIS CARROLL, English author and mathematician (1832–1898)
You may not have to multiply and divide decimals as often as you have to add and subtract them—though the word problems in this lesson show some practical examples of multiplication and division of decimals. However, questions on multiplying and dividing decimals often show up on tests, so it's important to know how to handle them.
Multiplying Decimals
To multiply decimals:
 Ignore the decimal points and multiply as you would whole numbers.
 Count the number of decimal digits (the digits to the right of the decimal point) in both of the numbers you multiplied.
 Beginning at the right side of the product (the answer), count left that number of digits, and put the decimal point to the left of the last digit you counted.
Example: 1.57 × 2.4
1. 
Multiply 157 times 24: 

2. 
Because there are a total of three decimal digits in 1.57 and 2.4, count off 3 places from the right in 3768 and place the decimal point to the left of the third digit you counted (7): 

To check the reasonableness of your work, estimate the product by using the rounding technique you learned in Lesson 6. Round each number you multiplied to the nearest whole number, and then multiply the results. If the product is close to your answer, your answer is in the ballpark. Otherwise, you may have made a mistake in placing the decimal point or in multiplying. Rounding 1.57 and 2.4 to the nearest whole numbers gives you 2 and 2. Their product is 4, which is close to your answer. Thus, your actual answer of 3.768 seems reasonable.
Tip
Always think about your answers! Remember that multiplying by a decimal that has a value less than one will give you a smaller number. 
In multiplying decimals, you may get a product that doesn't have enough digits for you to put in a decimal point. In that case, tack zeros onto the left of the product to give your answer enough digits; then add the decimal point.
Example: 0.03 × 0.006
1. 
Multiply 3 times 6: 
3 × 6 =18 
2. 
The answer requires 5 decimal digits because there are a total of five decimal digits in 0.03 and 0.006. 

Because there are only 2 digits in the answer (18), tack three zeros onto the left: 
00018 
3. 
Put the decimal point at the front of the number (which is 5 digits in from the right): 

Multiplication Shortcut
To quickly multiply a number by 10, just move the decimal point one digit to the right. To multiply a number by 100, move the decimal point two digits to the right. To multiply a number by 1,000, move the decimal point three digits to the right. In general, just count the number of zeros, and move the decimal point that number of digits to the right. If you don't have enough digits, first tack zeros onto the right.
Example: 1,000 × 3.82
1. 
Since there are three zeros in 1,000, move the decimal point in 3.82 three digits to the right. 
2. 
Since 3.82 has only two decimal digits to the right of the decimal point, add one zero on the right before moving the decimal point: 
3.820 
Thus, 1,000 × 3.82 × 3,820
Tip
To multiply by any multiple of 10, you can ignore the last zero digits, and add them back on to your answer in the end. For example, with 220 × 3,000, think of it to start as 22 × 3 = 66. Then, add back the four zeros that were temporarily removed: 220 × 3,000 = 660,000.

Dividing Decimals
Dividing Decimals by Whole Numbers
To divide a decimal by a whole number, bring the decimal point straight up into the answer (the quotient), and then divide as you would normally divide whole numbers.
Example:
1. 
Move the decimal point straight up into the quotient area: 

2. 
Divide: 










3. 
To check your division, multiply the quotient (0.128) by the divisor (4). If you get back the dividend (0.512), you know you divided correctly. 
0.128 


Dividing by Decimals
To divide any number by a decimal, first change the problem into one in which you're dividing by a whole number.
1. 
Move the decimal point to the right of the number you're dividing by (the divisor). 
2. 
Move the decimal point the same number of places to the right in the number you're dividing into (the dividend). 
3. 
Bring the decimal point straight up into the answer (the quotient) and divide. 
Tip
Does your answer make sense? Remember that dividing by a decimal that has a value less than one will give you a bigger number!

Example:
1. 
Because there are two decimal digits in .03, move the decimal point two places to the right in both numbers: 

2. 
Move the decimal point straight up into the quotient: 

3. 
Divide using the new numbers: 










Under the following conditions, you'll have to tack zeros onto the right of the last decimal digit in the dividend, the number you're dividing into:
Case 1. 
There aren't enough digits to move the decimal point to the right. 
Case 2. 
The answer doesn't come out evenly when you divide. 
Case 3. 
You're dividing a whole number by a decimal. In this case, you'll have to tack on the decimal point as well as some zeroes. 
Case 1
There aren't enough digits to move the decimal point to the right.
Example:
1. 
Because there are two decimal digits in 0.03, the decimal point must be moved two places to the right in both numbers. Since there aren't enough decimal digits in 1.2, tack a zero onto the end of 1.2 before moving the decimal point: 

2. 
Move the decimal point straight up into the quotient: 

3. 
Divide using the new numbers: 

Case 2
The answer doesn't come out evenly when you divide.
Example:
1. 
Because there is one decimal digit in 0.5, the decimal point must be moved one place to the right in both numbers: 

2. 
Move the decimal point straight up into the quotient: 

3. 
Divide, but notice that the division doesn't come out evenly 

4. 
Add a zero to the end of the dividend (12.) and continue dividing: 

Example:
1. 
Because there is one decimal digit in 0.3, the decimal point must be moved one place to the right in both numbers: 

2. 
Move the decimal point straight up into the quotient: 

3. 
Divide, but notice that the division doesn't come out evenly: 

4. 
Add a zero to the end of the dividend (1.0) and continue dividing: 

5. 
Since the division still did not come out evenly, add another zero to the end of the dividend (1.00) and continue dividing: 

6. 
By this point, you have probably noticed that the quotient is a repeating decimal. Thus, you can stop dividing and write the quotient like this: 

Case 3
When you're dividing a whole number by a decimal, you have to tack on the decimal point as well as some zeros.
Example:
1. 
There are two decimals in 0.02, so we have to move the decimal point to the right two places in both numbers. Because 19 is a whole number, put its decimal point at the end (19.), add two zeros to the end (19.00), and then move the decimal point to the right twice (1900.): 

2. 
Move the decimal point straight up into the quotient: 

3. 
Divide using the new numbers: 

Division Shortcut
To divide a number by 10, just move the decimal point in the number one digit to the left. To divide a number by 100, move the decimal point two digits to the left. Just count the number of zeros and move the decimal point that number of digits to the left. If you don't have enough digits, tack zeros onto the left before moving the decimal point.
Example: Divide 12.345 by 1,000.
1. 
Since there are three zeroes in 1,000, move the decimal point in 12.345 three digits to the left. 
2. 
Since 12.345 only has two digits to the left of its decimal point, add one zero at the left, and then move the decimal point: 

Thus, 12.345 ÷ 1,000 = 0.012345
Tip
Write down how much money you earn per hour (include both dollars and cents). If you earn a monthly or weekly salary, divide your salary by the number of hours in a month or week to get your hourly wage. If you don't have a job right now, invent a wage for yourself—and make it generous. Divide your hourly wage by 60 to see how much money you earn each minute. Then multiply your hourly wage by the number of hours you work per week to find your weekly wage; round your answer to the nearest dollar.
