Multiplying decimals by whole numbers is very much like multiplying whole numbers except there is a decimal to deal with. Although students might initially he trouble with it, through the power of rounding and estimating, they can generally get it quite quickly. Many teachers will tell students to ignore the decimal and multiply the numbers just like they would whole numbers. This is a good strategy to use. Figuring out where the decimal goes at the end can be accomplished by counting how many decimal places were in the original question and giving the answer that many decimal places. To better understand this method, students can round the two factors and multiply in their head to get an estimate then place the decimal based on their estimate. For example, multiplying 9.84 × 91, students could first round the numbers to 10 and 91 (keep 91 since multiplying by 10 is easy) then get an estimate of 910. Actually multiplying (ignoring the decimal) gets you 89544. To get that number close to 910, the decimal needs to go between the 5 and the 4, thus 895.44. Note that there are two decimal places in the factors and two decimal places in the answer, but estimating made it more understandable rather than just a method.
(7.1) Multiplying Decimals by 1-Digit Whole Numbers Multiply 2-digit tenths by 1-digit whole numbers Multiply 2-digit hundredths by 1-digit whole numbers Multiply 2-digit thousandths by 1-digit whole numbers Multiply 3-digit tenths by 1-digit whole numbers Multiply 3-digit hundredths by 1-digit whole numbers Multiply 3-digit thousandths by 1-digit whole numbers Multiply various decimals by 1-digit whole numbers (7.2) Multiplying Decimals by 2-Digit Whole Numbers Multiplying 2-digit tenths by 2-digit whole numbers Multiplying 2-digit hundredths by 2-digit whole numbers Multiplying 3-digit tenths by 2-digit whole numbers Multiplying 3-digit hundredths by 2-digit whole numbers Multiplying 3-digit thousandths by 2-digit whole numbers Multiplying various decimals by 2-digit whole numbers (7.3) Multiplying Decimals by Tenths Multiplying 2-digit whole by 2-digit tenths Multiplying 2-digit tenths by 2-digit tenths Multiplying 2-digit hundredths by 2-digit tenths Multiplying 3-digit whole by 2-digit tenths Multiplying 3-digit tenths by 2-digit tenths Multiplying 3-digit hundredths by 2-digit tenths Multiplying 3-digit thousandths by 2-digit tenths Multiplying various decimals by 2-digit tenths (7.4) Multiplying Decimals by Hundredths Multiplying 2-digit whole by 2-digit hundredths Multiplying 2-digit tenths by 2-digit hundredths Multiplying 2-digit hundredths by 2-digit hundredths Multiplying 3-digit whole by 2-digit hundredths Multiplying 3-digit tenths by 2-digit hundredths Multiplying 3-digit hundredths by 2-digit hundredths Multiplying 3-digit thousandths by 2-digit hundredths Multiplying various decimals by 2-digit hundredths (7.5) Multiplying Decimals by Various Decimal Places Multiplying 2-digit by 2-digit numbers with various decimal places Multiplying 3-digit by 2-digit numbers with various decimal places (7.6) Decimal Long Multiplication in Various Ranges Decimal Multiplication (range 0.1 to 0.9) Decimal Multiplication (range 1.1 to 9.9) Decimal Multiplication (range 10.1 to 99.9) Decimal Multiplication (range 0.01 to 0.99) Decimal Multiplication (range 1.01 to 9.99) Decimal Multiplication (range 10.01 to 99.99) Random # Digits Random # Places (7.7) European Format Multiplying Decimals by 2-Digit Whole Numbers European Format 2-digit whole × 2-digit hundredths European Format 2-digit tenths × 2-digit whole European Format 2-digit hundredths × 2-digit whole European Format 3-digit tenths × 2-digit whole European Format 3-digit hundredths × 2-digit whole European Format 3-digit thousandths × 2-digit whole (7.8) European Format Multiplying Decimals by 2-Digit Tenths European Format 2-digit whole × 2-digit tenths European Format 2-digit tenths × 2-digit tenths European Format 2-digit hundredths × 2-digit tenths European Format 3-digit whole × 2-digit tenths European Format 3-digit tenths × 2-digit tenths European Format 3-digit hundredths × 2-digit tenths European Format 3-digit thousandths × 2-digit tenths (7.9) European Format Multiplying Decimals by 2-Digit Hundredths European Format 2-digit tenths × 2-digit hundredths European Format 2-digit hundredths × 2-digit hundredths European Format 3-digit whole × 2-digit hundredths European Format 3-digit tenths × 2-digit hundredths European Format 3-digit hundredths × 2-digit hundredths European Format 3-digit thousandths × 2-digit hundredths (7.10) European Format Multiplying Decimals by Various Decimal Places European Format 2-digit × 2-digit with various decimal places European Format 3-digit × 2-digit with various decimal places (7.11) Dividing Decimals by Whole Numbers Divide Tenths by a Whole Number Divide Hundredths by a Whole Number Divide Thousandths by a Whole Number Divide Ten Thousandths by a Whole Number Divide Various Decimals by a Whole NumberIn case you aren't familiar with dividing with a decimal divisor, the general method for completing questions is by getting rid of the decimal in the divisor. This is done by multiplying the divisor and the dividend by the same amount, usually a power of ten such as 10, 100 or 1000. For example, if the division question is 5.32/5.6, you would multiply the divisor and dividend by 10 to get the equivalent division problem, 53.2/56. Completing this division will result in the exact same quotient as the original (try it on your calculator if you don't believe us). The main reason for completing decimal division in this way is to get the decimal in the correct location when using the U.S. long division algorithm.
A much simpler strategy, in our opinion, is to initially ignore the decimals all together and use estimation to place the decimal in the quotient. In the same example as above, you would complete 532/56 = 95. If you "flexibly" round the original, you will get about 5/5 which is about 1, so the decimal in 95 must be placed to make 95 close to 1. In this case, you would place it just before the 9 to get 0.95. Combining this strategy with the one above can also help a great deal with more difficult questions. For example, 4.584184 梅 0.461 can first be converted the to equivalent: 4584.184 梅 461 (you can estimate the quotient to be around 10). Complete the division question without decimals: 4584184 梅 461 = 9944 then place the decimal, so that 9944 is about 10. This results in 9.944.
Dividing decimal numbers doesn't he to be too difficult, especially with the worksheets below where the decimals work out nicely. To make these worksheets, we randomly generated a divisor and a quotient first, then multiplied them together to get the dividend. Of course, you will see the quotients only on the answer page, but generating questions in this way makes every decimal division problem work out nicely.
(7.12) Decimal Long Division with Quotients That Work Out Nicely Dividing Decimals by Various Decimals with Various Sizes of Quotients Dividing Decimals by 1-Digit Tenths (e.g. 0.72 梅 0.8 = 0.9) Dividing Decimals by 1-Digit Tenths with Larger Quotients (e.g. 3.2 梅 0.5 = 6.4) Dividing Decimals by 2-Digit Tenths (e.g. 10.75 梅 2.5 = 4.3) Dividing Decimals by 2-Digit Tenths with Larger Quotients (e.g. 387.75 梅 4.7 = 82.5) Dividing Decimals by 3-Digit Tenths (e.g. 1349.46 梅 23.8 = 56.7) Dividing Decimals by 2-Digit Hundredths (e.g. 0.4368 梅 0.56 = 0.78) Dividing Decimals by 2-Digit Hundredths with Larger Quotients (e.g. 1.7277 梅 0.39 = 4.43) Dividing Decimals by 3-Digit Hundredths (e.g. 31.4863 梅 4.61 = 6.83) Dividing Decimals by 4-Digit Hundredths (e.g. 7628.1285 梅 99.91 = 76.35) Dividing Decimals by 3-Digit Thousandths (e.g. 0.076504 梅 0.292 = 0.262) Dividing Decimals by 3-Digit Thousandths with Larger Quotients (e.g. 2.875669 梅 0.551 = 5.219)These worksheets would probably be used for estimating and calculator work.
(7.13) Horizontally Arranged Decimal Division Random # Digits Random # Places (7.14) European Format Dividing Decimals with Quotients That Work Out Nicely European Format Divide Tenths by a Whole Number European Format Divide Hundredths by a Whole Number European Format Divide Thousandths by a Whole Number European Format Divide Ten Thousandths by a Whole Number European Format Divide Various Decimals by a Whole NumberIn the next set of questions, the quotient does not always work out well and may he repeating decimals. The answer key shows a rounded quotient in these cases.
(7.15) European Format Dividing Decimals by Whole Numbers European Format Divide Tenths by a Whole Number European Format Divide Hundredths by a Whole Number European Format Divide Thousandths by a Whole Number European Format Divide Ten Thousandths by a Whole Number European Format Divide Various Decimals by a Whole Number (7.16) European Format Dividing Decimals by Decimals European Format Decimal Tenth (0,1 to 9,9)Divided by Decimal Tenth (1,1 to 9,9) European Format Decimal Hundredth (0,01 to 9,99)Divided by Decimal Tenth (1,1 to 9,9) European Format Decimal Thousandth (0,001 to 9,999)Divided by Decimal Tenth (1,1 to 9,9) European Format Decimal Ten Thousandth (0,0001 to 9,9999)Divided by Decimal Tenth (1,1 to 9,9) European Format Various Decimal Places (0,1 to 9,9999)Divided by Decimal Tenth (1,1 to 9,9) European Format Various Decimal Places (0,1 to 9,9999)Divided by Various Decimal Places (1,1 to 9,9999)