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# Techniques of Estimation: Mental Arithmetic- Using the Distibutive Property

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses using the distributive property. By the end of the module students should understand the distributive property and be able to obtain the exact result of a multiplication using the distributive property.

## Section Overview

• The Distributive Property
• Estimation Using the Distributive Property

## The Distributive Property

### Distributive Property

The distributive property is a characteristic of numbers that involves both addition and multiplication. It is used often in algebra, and we can use it now to obtain exact results for a multiplication.

Suppose we wish to compute 3(2 + 5)3(2 + 5) size 12{3 $$"2 "+" 5"$$ } {}. We can proceed in either of two ways, one way which is known to us already (the order of operations), and a new way (the distributive property).

1. Compute 3(2 + 5)3(2 + 5) size 12{3 $$"2 "+" 5"$$ } {} using the order of operations.

3 ( 2 + 5 ) 3 ( 2 + 5 ) size 12{3 $$"2 "+" 5"$$ } {}

Operate inside the parentheses first: 2 + 5 = 72 + 5 = 7 size 12{"2 "+" 5 "=" 7"} {}.

3 ( 2 + 5 ) = 3 7 3 ( 2 + 5 ) = 3 7 size 12{3 $$"2 "+" 5"$$ =" 3 " cdot " 7"} {}

Now multiply 3 and 7.

3 ( 2 + 5 ) = 3 7 = 21 3 ( 2 + 5 ) = 3 7 = 21 size 12{3 $$"2 "+" 5"$$ =" 3 " cdot " 7 "=" 21"} {}

Thus, 3(2 + 5)= 213(2 + 5)= 21 size 12{3 $$"2 "+" 5"$$ =" 21"} {}.

2. Compute 3(2 + 5)3(2 + 5) size 12{3 $$"2 "+" 5"$$ } {} using the distributive property.

We know that multiplication describes repeated addition. Thus,

3(2+5) = 2+5+2+5+2+5 2 + 5 appears 3 times = 2+2+2+5+5+5 (by the commutative property of addition) = 32+35 (since multiplication describes repeated addition) = 6+15 = 21 3(2+5) = 2+5+2+5+2+5 2 + 5 appears 3 times = 2+2+2+5+5+5 (by the commutative property of addition) = 32+35 (since multiplication describes repeated addition) = 6+15 = 21

Thus, 3(2 + 5)= 213(2 + 5)= 21 size 12{3 $$"2 "+" 5"$$ =" 21"} {}.

Let's look again at this use of the distributive property.

3(2+5) = 2+5+2+5+2+5 2 + 5 appears 3 times 3(2+5) = 2+5+2+5+2+5 2 + 5 appears 3 times

3(2+5) = 2+2+2 2 appears 3 times + 5+5+5 5 appears 3 times 3(2+5) = 2+2+2 2 appears 3 times + 5+5+5 5 appears 3 times

The 3 has been distributed to the 2 and 5.

This is the distributive property. We distribute the factor to each addend in the parentheses. The distributive property works for both sums and differences.

### Sample Set A

#### Example 1

Using the order of operations, we get

4(6+2) = 48 = 32 4(6+2) = 48 = 32

#### Example 2

Using the order of operations, we get

8(9+6) = 815 = 120 8(9+6) = 815 = 120

### Practice Set A

Use the distributive property to compute each value.

#### Exercise 1

6(8 + 4)6(8 + 4) size 12{6 $$"8 "+" 4"$$ } {}

##### Solution

6 8 + 6 4 = 48 + 24 = 72 68+64=48+24=72

#### Exercise 2

4(4 + 7)4(4 + 7) size 12{4 $$"4 "+" 7"$$ } {}

##### Solution

4 4 + 4 7 = 16 + 28 = 44 44+47=16+28=44

#### Exercise 3

8(2 + 9)8(2 + 9) size 12{8 $$"2 "+" 9"$$ } {}

##### Solution

8 2 + 8 9 = 16 + 72 = 88 82+89=16+72=88

#### Exercise 4

12(10 + 3)12(10 + 3) size 12{"12" $$"10 "+" 3"$$ } {}

##### Solution

12 10 + 12 3 = 120 + 36 = 156 1210+123=120+36=156

#### Exercise 5

6(11 - 3)6(11 - 3) size 12{6 $$"11 - 3"$$ } {}

##### Solution

6 11 - 6 3 = 66 - 18 = 48 611-63=66-18=48

#### Exercise 6

8(9 - 7)8(9 - 7) size 12{8 $$"9 - 7"$$ } {}

##### Solution

8 9 8 7 = 72 56 = 16 8987=7256=16

#### Exercise 7

15(30 - 8)15(30 - 8) size 12{"15" $$"30 - 8"$$ } {}

##### Solution

15 30 - 15 8 = 450 - 120 = 330 1530-158=450-120=330

## Estimation Using the Distributive Property

We can use the distributive property to obtain exact results for products such as 25 2325 23 size 12{"25 " cdot " 23"} {}. The distributive property works best for products when one of the factors ends in 0 or 5. We shall restrict our attention to only such products.

### Sample Set B

Use the distributive property to compute each value.

#### Example 5

25 2325 23 size 12{"25 " cdot " 23"} {}

Notice that 23 = 20 + 323 = 20 + 3 size 12{"23 "=" 20 "+" 3"} {}. We now write

Thus, 25 23 = 57525 23 = 575 size 12{"25 " cdot " 23 "=" 575"} {}

We could have proceeded by writing 23 as 30 - 730 - 7 size 12{"30 - 7"} {}.

#### Example 6

15 3715 37 size 12{"15 " cdot " 37"} {}

Notice that 37 = 30 + 737 = 30 + 7 size 12{"37 "=" 30 "+" 7"} {}. We now write

Thus, 15 37 = 55515 37 = 555 size 12{"15 " cdot " 37 "=" 555"} {}

We could have proceeded by writing 37 as 40 - 340 - 3 size 12{"40 - 3"} {}.

#### Example 7

15 8615 86 size 12{"15 " cdot " 86"} {}

Notice that 86 = 80 + 686 = 80 + 6 size 12{"86 "=" 80 "+" 6"} {}. We now write

We could have proceeded by writing 86 as 90 - 490 - 4 size 12{"90 - 4"} {}.

### Practice Set B

Use the distributive property to compute each value.

#### Exercise 8

25 1225 12 size 12{"25 " cdot " 12"} {}

##### Solution

2510+2=2510+252=250+50=3002510+2=2510+252=250+50=300 size 12{"25" left ("10"+2 right )="25" cdot "10"+"25" cdot 2="250"+"50"="300"} {}

#### Exercise 9

35 1435 14 size 12{"35 " cdot " 14"} {}

##### Solution

3510+4=3510+354=350+140=4903510+4=3510+354=350+140=490 size 12{"35" left ("10"+4 right )="35" cdot "10"+"35" cdot 4="350"+"140"="490"} {}

#### Exercise 10

80 5880 58 size 12{"80 " cdot " 58"} {}

##### Solution

8050+8=8050+808=4,000+640=4,6408050+8=8050+808=4,000+640=4,640 size 12{"80" left ("50"+8 right )="80" cdot "50"+"80" cdot 8=4,"000"+"640"=4,"640"} {}

#### Exercise 11

65 6265 62 size 12{"65 " cdot " 62"} {}

##### Solution

6560+2=6560+652=3,900+130=4,0306560+2=6560+652=3,900+130=4,030 size 12{"65" left ("60"+2 right )="65" cdot "60"+"65" cdot 2=3,"900"+"130"=4,"030"} {}

## Exercises

Use the distributive property to compute each product.

### Exercise 12

15131513 size 12{"15" cdot "13"} {}

#### Solution

15(10+3)=150+45=19515(10+3)=150+45=195 size 12{"15" $$"10"+3$$ ="150"+"45"="195"} {}

### Exercise 13

15141514 size 12{"15" cdot "14"} {}

### Exercise 14

25112511 size 12{"25" cdot "11"} {}

#### Solution

25(10+1)=250+25=27525(10+1)=250+25=275 size 12{"25" $$"10"+1$$ ="250"+"25"="275"} {}

### Exercise 15

25162516 size 12{"25" cdot "16"} {}

### Exercise 16

15161516 size 12{"15" cdot "16"} {}

#### Solution

15204=30060=24015204=30060=240 size 12{"15" left ("20" - 4 right )="300" - "60"="240"} {}

### Exercise 17

35123512 size 12{"35" cdot "12"} {}

### Exercise 18

45834583 size 12{"45" cdot "83"} {}

#### Solution

4580+3=3600+135=37354580+3=3600+135=3735 size 12{"45" left ("80"+3 right )="3600"+"135"="3735"} {}

### Exercise 19

45384538 size 12{"45" cdot "38"} {}

### Exercise 20

25382538 size 12{"25" cdot "38"} {}

#### Solution

25402=1,00050=95025402=1,00050=950 size 12{"25" left ("40" - 2 right )=1,"000" - "50"="950"} {}

### Exercise 21

25962596 size 12{"25" cdot "96"} {}

### Exercise 22

75147514 size 12{"75" cdot "14"} {}

#### Solution

7510+4=750+300=1,0507510+4=750+300=1,050 size 12{"75" left ("10"+4 right )="750"+"300"=1,"050"} {}

### Exercise 23

85348534 size 12{"85" cdot "34"} {}

### Exercise 24

65266526 size 12{"65" cdot "26"} {}

#### Solution

6520+6=1,300+390=1,6906520+6=1,300+390=1,690 size 12{"65" left ("20"+6 right )=1,"300"+"390"=1,"690"} {} or   65304=1,950260=1,69065304=1,950260=1,690 size 12{"65" left ("30" - 4 right )=1,"950" - "260"=1,"690"} {}

### Exercise 25

55515551 size 12{"55" cdot "51"} {}

### Exercise 26

1510715107 size 12{"15" cdot "107"} {}

#### Solution

15100+7=1,500+105=1,60515100+7=1,500+105=1,605 size 12{"15" left ("100"+7 right )=1,"500"+"105"=1,"605"} {}

### Exercise 27

2520825208 size 12{"25" cdot "208"} {}

### Exercise 28

3540235402 size 12{"35" cdot "402"} {}

#### Solution

35400+2=14,000+70=14,07035400+2=14,000+70=14,070 size 12{"35" left ("400"+2 right )="14","000"+"70"="14","070"} {}

### Exercise 29

8511085110 size 12{"85" cdot "110"} {}

### Exercise 30

95129512 size 12{"95" cdot "12"} {}

#### Solution

9510+2=950+190=1,1409510+2=950+190=1,140 size 12{"95" left ("10"+2 right )="950"+"190"=1,"140"} {}

### Exercise 31

65406540 size 12{"65" cdot "40"} {}

### Exercise 32

80328032 size 12{"80" cdot "32"} {}

#### Solution

8030+2=2,400+160=2,5608030+2=2,400+160=2,560 size 12{"80" left ("30"+2 right )=2,"400"+"160"=2,"560"} {}

### Exercise 33

30473047 size 12{"30" cdot "47"} {}

### Exercise 34

50635063 size 12{"50" cdot "63"} {}

#### Solution

5060+3=3,000+150=3,1505060+3=3,000+150=3,150 size 12{"50" left ("60"+3 right )=3,"000"+"150"=3,"150"} {}

### Exercise 35

90789078 size 12{"90" cdot "78"} {}

### Exercise 36

40894089 size 12{"40" cdot "89"} {}

#### Solution

40901=3,60040=3,56040901=3,60040=3,560 size 12{"40" left ("90" - 1 right )=3,"600" - "40"=3,"560"} {}

### Exercises for Review

#### Exercise 37

((Reference)) Find the greatest common factor of 360 and 3,780.

#### Exercise 38

((Reference)) Reduce 5945,1485945,148 size 12{ { {"594"} over {5,"148"} } } {} to lowest terms.

##### Solution

326326 size 12{ { {3} over {"26"} } } {}

#### Exercise 39

((Reference)) 159159 size 12{1 { {5} over {9} } } {} of 247247 size 12{2 { {4} over {7} } } {} is what number?

#### Exercise 40

((Reference)) Solve the proportion: 715=x90715=x90 size 12{ { {7} over {"15"} } = { {x} over {"90"} } } {}.

##### Solution

x=42x=42 size 12{x="42"} {}

#### Exercise 41

((Reference)) Use the clustering method to estimate the sum: 88 + 106 + 91 + 11488 + 106 + 91 + 114 size 12{"88 "+" 106 "+" 91 "+" 114"} {}.

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