Multiplying and Dividing Decimals

Multiplying Decimals:

Multiplying decimals is much like multiplying whole numbers, except a decimal must be inserted into the final answer. We must decide where to place the decimal in the answer. To decide where to place the decimal in the answer, we must count the number of total places behind the decimals in the multiplicands (the numbers being multiplied). The answer will equal the total number of places behind the answer in the product (answer).

Examples:

1.) Multiply: 12.4 × 1.5

12.4 × 1.5

Multiply as if the numbers were whole numbers

124 × 15 620 124 0 1860

12.4 x 1.5 620 124 0 1860

There are two total numbers behind the decimal points (two in the first number + one in the second number). So, there will be three total decimal places in the final answer. 12.4 x 1.5 620 124 0 18.60 Count from right to left. or 18.6 2.) Find the product of: 9.73 × 6.4

9.73 Multiply as if the numbers are whole numbers. × 6.4 3892 5838 0 62272

9.73 × 6.4 3892 5838 0 62.272 (There are three total places behind the decimal).

Using Order of Operations with Decimals

Simplify: 429 – 32.6 × 3.2

= 429 – 104.32 The multiplication must be done before the subtraction.

Now subtract.

8 9 10 4 2 9 . 0 0 Fill in zeros, line up the decimal points, and regroup. -1 0 4. 3 2 3 2 4. 6 8 Final answer Dividing Decimals:

I. Changing a fraction to a decimal.

To change a fraction to a decimal, we must use long division. The denominator is divided into the numerator. You must add a decimal and zeros to the dividend. The decimal point is placed in the quotient (answer) directly above the decimal point in the dividend. If the remainder is not zero, keep adding zeros until the division comes out with a remainder of zero OR round as indicated in the instructions.

Examples:

11 1.) Change to a decimal. 20

Divisor Dividend Quotient

20 11 Put the decimal in the answer above the decimal in the divisor. 20 11.0 Ask: How many times does 20 go into 110? The answer is 5. Place the 8 behind the decimal, above the zero.

0.5 20 11.0

Now multiply back out. 5 × 20 = 100. Then subtract.

0.5 20 11.0 -100 10 Add another zero. 0.55 20 11.00 -100 The remainder is zero, so the problem is completed. 100 -100 0

Final Answer: .55 25 2.) Change to a decimal. 4

6 4 25 Since the division does not come out evenly, add a decimal point and zero after the 25 and continue dividing.

6. 4 25.0 Since the remainder is not zero, add another zero in the dividend. - 24 10

6.2 4 25.0 - 24 10 - 8 2

6.25 4 25.00 - 24 10 The remainder is now zero, so the problem is completed. - 8 20 20 0

The final answer is: 6.25 1 3.) Change 7 to a decimal. 4

1 29 7 = Now divide as in example 3 above. 4 4

7 4 29 The remainder is not zero, so add another zero to the dividend. - 28 1

7.2 4 29.0 - 28 10 - 8

7.2 4 29.0 - 28 The remainder is not zero, so add another zero to the dividend. 10 - 8 2

7.25 4 29.00 - 28 10 The remainder is now zero so the problem is completed. - 8 20 - 20 0 Answer: 7.25 Sometimes the remainder will never come out to be zero. In these cases, you will be told to round to a particular place value. When this occurs, you should carry the division out one more place than what you will be rounding to. For example, if you are to round to the hundredths place, divide until you have a digit in the thousandths place.

Examples:

5 1.) Convert to a decimal. Round to the nearest thousandths place. 12

(Note: Since we must round to the hundredths place, we will carry the answer out 1 place beyond the hundredths place.)

12 5.0

.4 12 5.0 The remainder is not zero, so add another zero to the dividend. - 48 2

.41 12 5.00 - 4 8 The remainder is not zero, so add another zero to the dividend. 2 0 - 1 2

.416 12 5.000 - 4 8 2 0 - 1 2 8 0 - 7 2 8

.4166 12 5.0000 - 4 8 2 0

- 1 2 8 0 - 7 2 8 0

The remainder is still not zero. The remainder continues to be 8. This is a repeating decimal. We need our answer rounded to the nearest thousandths place, so we are ready to round.

0.4166 rounded to the nearest thousandths place is 0.417

Final answer: 0.417.

II. Dividing Decimals when the Divisor Contains a Decimal Point.

To Divide when the Divisor Contains a Decimal Point:

1.) Move the decimal point in the divisor to the right end of the number.

2.) Move the decimal point in the dividend the same number of places to the right as the decimal point was moved in the divisor.

3.) Insert a decimal point in the answer (quotient) directly above the decimal point in the dividend.

4.) Divide similar to how whole numbers are divided. Add zeros if necessary in the dividend.

Examples:

1.) Divide 0.544 0.68

.68 .544 The decimal point must be moved to the end of the divisor. .68 .544 Move the decimal two places to the right in the divisor and the dividend.

0. 68 54.4 Place a decimal point in the answer (quotient) directly above the decimal point in the dividend.

0.8 68 5.44 Divide 68 into 54.4. - 5 44 0

The remainder is zero, so the problem is completed.

Answer: 0.8 or .8

2.) Find the quotient: 0.0558 0.006

0.006 0.0558 The decimal point must be moved to the end of the divisor.

0.006 0.0558 Move the decimal three places to the right in the divisor and the dividend.

Note: If there were not three decimal places in the dividend, we would add zeros as place holders.

6. 55.8 or The decimal has now been moved. 6 55.8

Now, divide just like whole numbers. 9 6 55.8 - 54 1 Place the decimal in the quotient (answer) directly above the decimal in the dividend.

9. 6 55.8 -54 1 8

9.3 6 55.8 -54 1 8 - 1 8 0 The final answer is 9.3.

3.) Divide: 12 0.03

0.03 12 The decimal point must be moved to the end of the divisor.

0.03 1200 Move the decimal two places to the right in the divisor and the dividend. Since there are not two decimal places in the dividend, we must add zeros as place holders.

3 1200 The decimal has now been moved. Now divide. Just like whole numbers.

4 3 12 00 - 12 0 0 4 0 3 12 00 - 12 00 - 0 0 0

40 0 3 12 00 - 12 00 - 0 0 0 - 0 0

The remainder is zero, so the problem is completed.

The final answer is 400

III. Notes for Solving Word Problems-Decimals

Step 1: Read the problem through completely. Look for key words. Step 2: Write the number sentence from the problem. Step 3: Solve the problem.

1. Elliot Jones has 103.4 pounds of candy to place in boxes. Each box holds 4.7 pounds of candy. How many boxes will Elliot need to hold the candy?

Here we need to divide. Elliot is dividing the candy into boxes.

Numeric Problem: 103.4 ÷ 4.7 Answer: 22 2. A bookstore ordered 175 copies of a textbook at a cost of $58.25 each. What is the total cost of the textbooks?

Numeric Problem: 175 × $58.25 Answer: $10,193.75

3. The total ticket receipts for a play were $2362.50. If the tickets were $4.50 each, how many tickets were sold?

Numeric Problem: $2362.50 ÷ $4.50 Answer: 525

Elliot Jones has 245 pounds of candy to place in boxes. Each box holds 17.5 pounds of candy. How many boxes will Elliot need to hold the candy?

Here we need to divide. Elliot is dividing up the candy into boxes.

17.5 245

175. 2450. Move the decimal one place to the right in the divisor and the dividend.

1 175. 2450. - 175 7 00 14 175. 2450. - 175 700 - 700 0 The remainder is now zero, so the problem is completed.

The answer: Elliot will need 14 boxes for the candy.

Practice: 1.) 125.8 13.6 2.) 106.8/17.56 (Round to the hundredths place)

3.) The Spanish club collected $13.45 from each of its nine members. How much did the Spanish club collect?

4.) Henry earned $596.45 for working 32 hours. What is Henry’s hourly wage? Round to the nearest cent.

Answers:

1.) 9.25 2.) 6.08 3.) $121.05 4.) $18.64