CONCEPTUAL ARITHMETIC METHODS WITH DECIMALSMultiplication

Multiplication with decimals The following three techniques will be

covered in this presentation:

Using upper and lower product bounds to correctly place the decimal point

Converting to fractions

Place value multiplication

Using upper and lower product bounds to correctly place the decimal point

Technique 1

Example 1: Find the product of 3.8 and 0.52

1. Find upper and lower bounds for the factors:

3 < 3.8 < 4 and 0.5 < 0.52 < 0.6

2. Find upper and lower bounds for the product:

Example 1: Find the product of 3.8 and 0.52

3. Multiply the factors as if they were whole numbers:

4. Use the upper and lower bounds for the product to correctly place the decimal point.

Answer:

Example 2: Find the product of 72.3 and 8.201

1. 70 < 72.3 < 80 and 8 < 8.201 < 92.

3. Multiply the factors as if they were whole numbers:

4. Correctly place the decimal point using the bounds.

Answer:

Lower bound for the product: 70×8=560

Upper bound for the product: 80×9=720

Convert to fractions

Technique 2

Example 3: Find the product of 1.2 and 0.03

1. Convert each decimal to fraction form:

2. Multiply the fractions:

3. Rewrite in decimal form: 1.2 x 0.03 = 0.036

If you have trouble seeing the decimal form, note that 36/1000 = 30/1000 + 6/1000 = 3/100 + 6/1000 = 0.03 + 0.006 = 0.036

Example 4: Find the product of 0.025 and 0.08

1. Convert each decimal to fraction form:

2. Multiply the fractions:

3. Rewrite in decimal form: 0.025 x 0.08 = 0.002

Example 5: Find the product of 34.23 and 0.011

1. Convert each decimal to fraction form:

2. Multiply the fractions:

3. Rewrite in decimal form: 34.23 x 0.011 = 0.37653

Note that the final digit of 3 in the numerator 37653 from step 2 must be in the 100,000ths (hundred thousandths) place.

Place Value Multiplication

Technique 3

Multiplication of decimals using place value

Use a place value chart to organize the factors and partial products. The number of columns depends on the

problems. Leave room to add more columns if necessary.

hundreds tens ones tenths hundredths

thousandths

Example 6: Find the product of 2.3 and 4.5

Step 1: Enter the factors into a place value chart.

tens ones tenths hundredths

reasoning

2 3 2 ones and 3 tenths4 5 4 ones and 5 tenths

Step 2: Find the partial products.tens ones tenths hundredth

sreasoning

2 3 2 ones and 3 tenths4 5 4 ones and 5 tenths

1 5

Example 6: Find the product of 2.3 and 4.5

Step 2: Find the partial products.tens ones tenths hundredth

sreasoning

2 3 2 ones and 3 tenths4 5 4 ones and 5 tenths

1 51 0

Example 6: Find the product of 2.3 and 4.5

Step 2: Find the partial products.tens ones tenths hundredth

sreasoning

2 3 2 ones and 3 tenths4 5 4 ones and 5 tenths

1 51 01 2

Example 6: Find the product of 2.3 and 4.5

Step 2: Find the partial products.tens ones tenths hundredth

sreasoning

2 3 2 ones and 3 tenths4 5 4 ones and 5 tenths

1 51 01 28

Example 6: Find the product of 2.3 and 4.5

Step 3: Sum the partial products to obtain the final product.tens ones tenths hundredth

sreasoning

2 3 2 ones and 3 tenths4 5 4 ones and 5 tenths

1 51 01 28

1 0 3 52.3 x 4.5 =

10.35

Example 6: Find the product of 2.3 and 4.5

Example: Find the product of .08 and .907

Estimate practice: The answer should lie between

ones tenthshundredth

sthousandt

hs

tenthousandt

hs

hundredthousandt

hs

0 0 89 0 7

8100

×910

=72

1000=0.072 and

8100

×1=8

100=0.08.

Example 7: Find the product of .08 and .907

ones tenthshundredth

sthousandt

hs

tenthousandt

hs

hundredthousandt

hs

0 0 89 0 7

5 6

ones tenthshundredth

sthousandt

hs

tenthousandt

hs

hundredthousandt

hs

0 0 89 0 7

5 67 2

ones tenthshundredth

sthousandt

hs

tenthousandt

hs

hundredthousandt

hs

0 0 89 0 7

5 67 2

0 0 7 2 5 6

0.08 x 0.907 = 0.07256

Find the product of 2.305 and 70.89.

Estimating, we see that our answer should be between2 x 70 = 140 and 3 x 71 = 213. We can use this as a check at the end.

Example 8: Find the product of 2.305 and 70.89

hundreds tens ones tenths

hundredths

thousandths

tenthousan

dths

hundredthousan

dths7 0 8 9

2 3 0 54 5

hundreds tens ones tenths

hundredths

thousandths

tenthousan

dths

hundredthousan

dths7 0 8 9

2 3 0 54 5

4 0

hundreds tens ones tenths

hundredths

thousandths

tenthousan

dths

hundredthousan

dths7 0 8 9

2 3 0 54 5

4 03 5

hundreds tens ones tenths

hundredths

thousandths

tenthousan

dths

hundredthousan

dths7 0 8 9

2 3 0 54 5

4 03 5

2 7

hundreds tens ones tenths

hundredths

thousandths

tenthousan

dths

hundredthousan

dths7 0 8 9

2 3 0 54 5

4 03 5

2 72 4

hundreds tens ones tenths

hundredths

thousandths

tenthousan

dths

hundredthousan

dths7 0 8 9

2 3 0 54 5

4 03 5

2 72 4

2 1

hundreds tens ones tenths

hundredths

thousandths

tenthousan

dths

hundredthousan

dths7 0 8 9

2 3 0 54 5

4 03 5

2 72 4

2 11 8

hundreds tens ones tenths

hundredths

thousandths

tenthousan

dths

hundredthousan

dths7 0 8 9

2 3 0 54 5

4 03 5

2 72 4

2 11 8

1 6

hundreds tens ones tenths

hundredths

thousandths

tenthousan

dths

hundredthousan

dths7 0 8 9

2 3 0 54 5

4 03 5

2 72 4

2 11 8

1 61 4

hundreds tens ones tenths

hundredths

thousandths

tenthousan

dths

hundredthousan

dths7 0 8 9

2 3 0 54 5

4 03 5

2 72 4

2 11 8

1 61 41 6 3 4 0 1 4 5

70.89 x 2.305 = 163.40145