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**Conceptual arithmetic methods with decimals**

Multiplication

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**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

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Technique 1 Using upper and lower product bounds to correctly place the decimal point

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**Example 1: Find the product of 3.8 and 0.52**

1. Find upper and lower bounds for the factors: 3 < 3.8 < and < 0.52 < 0.6 2. Find upper and lower bounds for the product:

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**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:

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**Example 2: Find the product of 72.3 and 8.201**

3. Multiply the factors as if they were whole numbers: 4. Correctly place the decimal point using the bounds. Answer:

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Technique 2 Convert to fractions

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**Example 3: Find the product of 1.2 and 0.03**

Convert each decimal to fraction form: Multiply the fractions: Rewrite in decimal form: x 0.03 = 0.036 If you have trouble seeing the decimal form, note that /1000 = 30/ / = 3/ / = = 0.036

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**Example 4: Find the product of 0.025 and 0.08**

Convert each decimal to fraction form: Multiply the fractions: Rewrite in decimal form: x 0.08 = 0.002

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**Example 5: Find the product of 34.23 and 0.011**

Convert each decimal to fraction form: Multiply the fractions: Rewrite in decimal form: x = Note that the final digit of 3 in the numerator from step 2 must be in the 100,000ths (hundred thousandths) place.

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Technique 3 Place Value Multiplication

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**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

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**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 tenths 4 5 4 ones and 5 tenths

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**Example 6: Find the product of 2.3 and 4.5**

Step 2: Find the partial products. tens ones tenths hundredths reasoning 2 3 2 ones and 3 tenths 4 5 4 ones and 5 tenths 1

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**Example 6: Find the product of 2.3 and 4.5**

Step 2: Find the partial products. tens ones tenths hundredths reasoning 2 3 2 ones and 3 tenths 4 5 4 ones and 5 tenths 1

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**Example 6: Find the product of 2.3 and 4.5**

Step 2: Find the partial products. tens ones tenths hundredths reasoning 2 3 2 ones and 3 tenths 4 5 4 ones and 5 tenths 1

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**Example 6: Find the product of 2.3 and 4.5**

Step 2: Find the partial products. tens ones tenths hundredths reasoning 2 3 2 ones and 3 tenths 4 5 4 ones and 5 tenths 1 8

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**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 hundredths reasoning 2 3 2 ones and 3 tenths 4 5 4 ones and 5 tenths 1 8 2.3 x 4.5 = 10.35

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

Estimate practice: The answer should lie between ones tenths hundredths thousandths ten thousandths hundred 8 9 7

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**8 9 7 5 6 8 9 7 5 6 2 ones tenths hundredths thousandths**

ten thousandths hundred 8 9 7 5 6 ones tenths hundredths thousandths ten thousandths hundred 8 9 7 5 6 2

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**0.08 x 0.907 = 0.07256 8 9 7 5 6 2 ones tenths hundredths thousandths**

ten thousandths hundred 8 9 7 5 6 2 0.08 x =

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**Example 8: Find the product of 2.305 and 70.89**

Estimating, we see that our answer should be between 2 x 70 = 140 and 3 x 71 = 213. We can use this as a check at the end.

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**7 8 9 2 3 5 4 hundreds tens ones tenths hundredths thousandths**

ten thousandths hundred 7 8 9 2 3 5 4

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**7 8 9 2 3 5 4 hundreds tens ones tenths hundredths thousandths**

ten thousandths hundred 7 8 9 2 3 5 4

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**7 8 9 2 3 5 4 hundreds tens ones tenths hundredths thousandths**

ten thousandths hundred 7 8 9 2 3 5 4

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**7 8 9 2 3 5 4 hundreds tens ones tenths hundredths thousandths**

ten thousandths hundred 7 8 9 2 3 5 4

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**7 8 9 2 3 5 4 hundreds tens ones tenths hundredths thousandths**

ten thousandths hundred 7 8 9 2 3 5 4

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**7 8 9 2 3 5 4 1 hundreds tens ones tenths hundredths thousandths**

ten thousandths hundred 7 8 9 2 3 5 4 1

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**7 8 9 2 3 5 4 1 hundreds tens ones tenths hundredths thousandths**

ten thousandths hundred 7 8 9 2 3 5 4 1

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**7 8 9 2 3 5 4 1 6 hundreds tens ones tenths hundredths thousandths**

ten thousandths hundred 7 8 9 2 3 5 4 1 6

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**7 8 9 2 3 5 4 1 6 hundreds tens ones tenths hundredths thousandths**

ten thousandths hundred 7 8 9 2 3 5 4 1 6

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**70.89 x 2.305 = 163.40145 7 8 9 2 3 5 4 1 6 hundreds tens ones tenths**

hundredths thousandths ten thousandths hundred 7 8 9 2 3 5 4 1 6 70.89 x =

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