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CONCEPTUAL ARITHMETIC METHODS WITH DECIMALS
Multiplication
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 tenths
4 5 4 ones and 5 tenths
Step 2: Find the partial products.tens ones tenths hundredth
sreasoning
2 3 2 ones and 3 tenths
4 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 tenths
4 5 4 ones and 5 tenths
1 5
1 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 tenths
4 5 4 ones and 5 tenths
1 5
1 0
1 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 tenths
4 5 4 ones and 5 tenths
1 5
1 0
1 2
8
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 tenths
4 5 4 ones and 5 tenths
1 5
1 0
1 2
8
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 8
9 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 8
9 0 7
5 6
ones tenthshundredth
sthousandt
hs
tenthousandt
hs
hundredthousandt
hs
0 0 8
9 0 7
5 6
7 2
ones tenthshundredth
sthousandt
hs
tenthousandt
hs
hundredthousandt
hs
0 0 8
9 0 7
5 6
7 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
dths
7 0 8 9
2 3 0 5
4 5
hundreds tens ones tenths
hundredths
thousandths
tenthousan
dths
hundredthousan
dths
7 0 8 9
2 3 0 5
4 5
4 0
hundreds tens ones tenths
hundredths
thousandths
tenthousan
dths
hundredthousan
dths
7 0 8 9
2 3 0 5
4 5
4 0
3 5
hundreds tens ones tenths
hundredths
thousandths
tenthousan
dths
hundredthousan
dths
7 0 8 9
2 3 0 5
4 5
4 0
3 5
2 7
hundreds tens ones tenths
hundredths
thousandths
tenthousan
dths
hundredthousan
dths
7 0 8 9
2 3 0 5
4 5
4 0
3 5
2 7
2 4
hundreds tens ones tenths
hundredths
thousandths
tenthousan
dths
hundredthousan
dths
7 0 8 9
2 3 0 5
4 5
4 0
3 5
2 7
2 4
2 1
hundreds tens ones tenths
hundredths
thousandths
tenthousan
dths
hundredthousan
dths
7 0 8 9
2 3 0 5
4 5
4 0
3 5
2 7
2 4
2 1
1 8
hundreds tens ones tenths
hundredths
thousandths
tenthousan
dths
hundredthousan
dths
7 0 8 9
2 3 0 5
4 5
4 0
3 5
2 7
2 4
2 1
1 8
1 6
hundreds tens ones tenths
hundredths
thousandths
tenthousan
dths
hundredthousan
dths
7 0 8 9
2 3 0 5
4 5
4 0
3 5
2 7
2 4
2 1
1 8
1 6
1 4
hundreds tens ones tenths
hundredths
thousandths
tenthousan
dths
hundredthousan
dths
7 0 8 9
2 3 0 5
4 5
4 0
3 5
2 7
2 4
2 1
1 8
1 6
1 4
1 6 3 4 0 1 4 5
70.89 x 2.305 = 163.40145
