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Fractions, Decimals and Percents Alexander Grothendieck 3.3 Adding and Subtracting Decimals Alexander Grothendieck Add 5.763 + 3.84

Use front estimation to estimate the sum: 5 + 3 = 8

Add . Write each number with the same decimal places, using zeros as place holders .Record the number without the decimal point EXAMPLE 1 Write each fraction as a decimal.. Identify each decimal as terminating or repeating

1/2=

0.5 so this would be a terminating decimal

0.83333333... since this is a repeating decimal we will put a line over it like said in the explanation 0.83 3.4 Multiplying Decimals 3.1

Fractions to decimals Numbers can be written in both fraction and decimal form. For example , 4 can be written as 3/1 and 3.0.

Decimals such as 0.1 and 0.25, are terminating decimals each has a definite numbers of decimal places.

Decimals such as 0.333 333 333.......or 0.4545 4545 4545 are repeating decimals.Some digits in repeating decimals repeat forever. So we put a bar over the digits that repeat .

EG: 3/66=0.45454545 4545..which is written 0.45 EXAMPLE 1 EXAMPLE 2 Multiply . EXAMPLE 1 Jan bought 18.9 m of framing to make picture frames.

Each picture needs 1.8 m of frames.

How much frames can Jan make.

How much framing material is left over. EXAMPLE 2 A 0.4-kg bag of oranges costs $1.34

a) What is the cost of 1 kg of oranges ?

How do you know your answer is reasonable ?

b) Suppose you spent $10 on oranges.

What mass of orange did you buy EXAMPLE 3 Alex finds a remnant of landscaping fabric at a garden store. The fabric is standard width, with length 9.88 m.

Alex needs fourteen 0.8 pieces for a garden patio. a) How many 0.8 m pieces can Alex cut from the remnant ?

b) Will Alex have all the fabric he needs ? Why or why not ?

c) If your answers in part b is no how much more fabric does Alex need ?

d) Alex redesigns his patio so that he need fourteen 0.7 m pieces of fabric. Will the remnant be enough fabric ? Explain. 3.6 Order of operations with decimals Here is how you do the order of operation.

To make sure everyone has the same answers for the given expression we add, subtract multiply and divide in the order:

Do the operation in brackets first.

Then divide and multiply in the order from left to right

Then add and subtract from left to right

B brackets

E equation

D division

M multiplication

A addition

S subtraction EXAMPLE 1 Evaluate

a) 4.8 + 2 x 8.7

2x8.7= 17.4 + 4.8

= 22.2

b) 14-9 + 5.8

14-9 = 5 + 5.8

= 10.8

c) 8 - 3.6 x 2

3.6 x 2 = 7.2

= 0.8

d) 16+2.4 - 7

16+2.4 = 18.4 - 7

= 11.4 EXAMPLE 2 Evaluate

a) 164.5/7 x 10 + 7.2 b) 73.8 x (3.2 + 6.8)-14.1/0.2

164.5/7 = 23.5 x 10 (3.2 + 6.8)= 10 x 73.8

= 235 + 7.2 = 738/0.2

= 242.2 = 3690 - 14.1

= 3675.9

c) (4.7 - 3.1) x 5 - 7.5/2.5

(4.7 - 3.1) = 1.6

= 1.6 x 5

= 8/2.5

= 3.2 - 5

= 1.8 EXAMPLE 3 Solve the problem

A) The cross country teamed members ran timed circuits. Here are their times: 15.8min,12.5 min, 18.0 min, 14.2 min,13.9 min, 16.0 min, 16.2 min,

17.5 min, 16.3 min, 15.6 min

Find the mean time.

The mean time is the sum 15.8+12.5+18.0+14.2+13.9+16.0+16.2+17.5+16.3+15.6/10

of the times divided by 15.8+12.5+18.0+14.2+13.9+16.0+16.2+17.5+16.3+15.6 = 156

the number of times. = 156/10

= 15.6

The mean time they was 15.6 minutes.

B) 0.38 + 16.2 x (2.1 + 4)+ 21/3.5

(2.1 + 4) = 6.1 x 16.2

= 98.82/3.5 3.5 Dividing Decimals EXAMPLE 1 Evaluate To divide: 18.9/1.8 ignore the decimal points.

Find 189/18 and then estimate.

10.5 18.9/1.8 = 10.5

18 189.0 There will be 0.5 framing material

- 180 left.

90

- 90

0 When you divide 2 numbers, the number is the quotient. a) The cost of 1kg of orange is $ 3.35, to find the cost of 1kg you divide 1.34/0.4 and the total is $ 3.35.

b) About 3kg because you do $ 3.35 x 3 = $ 10.05 so there is 5 cents more,so it is little less than 3kg. a) He can cut 12.35 0.8 m of fabric.

b) No he will not have all the fabric he needs because he needs 14 0.8 fabric and he only has 12.35 0.8 fabric.

c) He needs 1.65 more remnant fabric.

d) If he redesigned his patio and uses 0.7 pieces there would be 14.1142857 so it would be enough remnant fabric for his patio. 3.7 You can divide decimals the same way as you divide a whole number.

Use front end estimation to place the decimals point.

For example, to divide 24.3/0.6 405

Estimate first: 24/1=24 6 2430

So, 24.3/0.6 is about 24. -24

Divide as you would whole numbers. 03

00

30

30 3.7 Relating Fractions, Decimals, and Percents You ca use Base Ten Blocks to multiply decimals.

To multiply 2.2 x 1.4 display base ten blocks as shown below.

2 0.2 1 0.4 0.8 So when you find out the answer you multiply 1 by 2, then 1 by 0.2,then 2 by 0.4 and then 0.4 by .0.2 and then 0.4 by 0.8 and then add all of the answers u 3.4 Multiplying Decimals You can also multiply decimals the same way you multiply whole numbers.

To multiply 2.3 x 1.8, multiply 24 x 18.

So you take out the decimal and when you find the answer you move the decimals in as the same number you moved the decimal out

24

x 18

432 So now you move the decimal in two spaces because for 1.8 you moved it out one and for 2.4 you moved it out one so the total is two spaces . The total is 4.32.

You can also use front-end estimation to place the decimal point : 2 x 1 = 2

So, 2.4 x 1.8 is about 2 decimals place.

Place the decimals point between the 4 and the 3.

The product is 4.32. Write a multiplication equation for each picture.

a) EXAMPLE 2 For each fraction write , write an equivalent fraction with denominator 10,100,1000. The write the fraction as a decimal.

2/5= 0.4 Since 4 in this rounds to 10(closest) it will go over 10. 4/10 0.4. Because it has one decimal place.

1/4: 0.25 25 will round up to 100. 25/100. This number has 2 decimals places it is in the 10's place.

37/500: 0.074 rounds to 1000. 74/1000. Thousands place. 1x1= 1, 1x0.2= 0.2, 1x0.2= 0.2, 0.2x0.2= 0.4, 0.2x0.4= 0.08 1+0.2+0.2+0.4+0.08= 1.88 a) 2.7x4.786= 12.9222

b) 12.52x13.923= 174.31596

c) 0.986x1.352= 1.333072 You can check your answer by using base ten blocks to see if a, b, and c is right. We can use number lines to show how percents relate to fractions and decimals

For example:

25%=25=0.25

100

Conversely, a decimal can be written as a percent:

0.15=15=15%

100

To write a fraction in a percent, write the equivalent fraction with denominator 100.

1 x50 50

2 x50 100 EXAMPLE 1 Write each percent as a fraction. a) 15% = 15 = 0.15

100

b) 40% = 40 = 0.40

100

c) 5% = 5 = 0.05

100 EXAMPLE 2 Write each percent as a fraction and as a decimal.

Sketch a number line to show how the numbers are related. a) 2 2 o.02

100

1 2 6 28 50

100 100 100

b) 6 6 0.06

100

c) 28 28 0.28

100 EXAMPLE 3 Divide the paper into these 4 sections EXAMPLE 3 Multiply, describe the pattern that you see a) 8.36x10 = 83.6

8.36x100 = 836

8.36x1000 =8360

Multiply by multiples of 10. The digits in the product move on place to the left each time. Or the decimal point moves one place right each time. EXAMPLE 3 write each fraction as decimal. What patterns do you see?

1/999: 0.010

2/999:0.0200

54/999:0.540

113/999: 0.113

The patterns that you see that they are all repeating decimals. Plus the number that is being divided by 999 comes out the same when you divide but its just a repeating number. 3.2 Comparing and Ordering Fractions and Decimals Any fraction greater than 1 can be written as a mixed number. The bench mark 0,1/2 and 1 can be used to compare the fraction parts of a mixed number.

We can use bench marks on a number line to order these numbers: 2/11,2 2 3/8, 1 1/16, 14/9, 14/15

2/11 is close to 0. Since 3/8 is close to 1/2 2 3/8 is close to 2 1/2, but less than 2 1/2, 1 1/16 is close to 2, but greater than 1. 0 2/11 1/2 14/15 1 1 1/16 1 1/2 14/9 2 2 3/8 2 1/2 3 EXAMPLE 1 Use bench marks and a number line to order the set of numbers from least to greatest:

7/6, 15/12, 1 2/9, 1

least to greatest: 1, 7/6, 1 2/9, 15/17

1 7/6 1 2/9 15/17 EXAMPLE 2 Amrita, Paul, Cory baked pizzas for the fund raiser sale. The students cut their pizzas into different sized slices.

Amrita sold 11/6 pizzas. Pauls sold 1.875. Cory sold 9/4 pizzas.

a) Order From least to greatest: 11 6, 1.875, 9 4

b) Who sold the most pizzas?

Corey sold the most pizzas and amrita sold the leastest EXAMPLE 3 Which number in the number line is placed incorrectly?

1 1/2 15/8 2 2 1/8 11/4 2 1/2 11/4; 2 1/2= 10/4 which is less than 11/4 3.8 Solving Percent Problems A papaerback novel usually costs $7.99. Its is on sale for 15% off. To find how much you saved, calculate 15%of $7.99. 15%=15/100 of %7.99

=0.15 x 7.99

Use a calculator:

0.15 x 7.99=1.1985; round to the nearest cent and you save $1.20 buying the book on black friday.

You can also show it on a number line:

0 $1.20 $7.99 EXAMPLE 1 Calculate:

a) 10% of 30: 10 x 30= 300(3) b) 20% of 50: 20 x 50 =100(10)

c) 18% of 36= 18 x 36= 6.48 d) 67% of 112: 67 x 112= 75.04 EXAMPLE 2 The Good and Service tax(GST) is currently 6%.

For each item below find the GST. Find the cost of the item including the GST.

A) Bicycle: 129.00

so what you do here is that you need to multiply the cost of the bike by the GST. 129.00 x 6%= 7.74. This is how mch GST will be added. Now if you want to add the GST in the price of the bike. ADD: 129.00+7.74:136.74 EXAMPLE 3 The regular price of a radio is $60.00. How much will the radio be for when its on sale.

sale for 25%: 45.00 30% off: 42.00 40% off: 35.00 EXAMPLE 1 The tallest building in the world is the Burj Khalifa. Its height is 0.800, in Dubai . The 2nd tallest building in the world is the Taipei in Taiwan. Its height is 0.509km What is the difference in the heights of the buildings.

So we would subtract the heights for the buildings.

0.800-0.509= 291km or you can do front estimation 8-5= 3 Example 2 The Robb family and the Khalifa family have similar homes. The Robb family sets there thermostats to to 20 degrees Celsius during the winter months. These were the monthly heating bills: $134.35 $171.23 and 123.21

The Khalifa family used programmable thermostat lower the temperature at night and during the day when the family was out. Their heating bills were: $134.25, 103.27, $98.66

a)How much did each family pay to heat its home during the winter months?

Robb Family: $134.35+171.23+123.21+=$428.79

Khlifa Family:$134.25+103.37+$98.66=$336.28

b)How much more did the Robb family pay: $428.79-336.28=92.51

The Robb family paid $92.51 dollars more

EXAMPLE 3 A student subtracted 0.373 from 4.81 and got the difference of 0.108.

a) What mistake did the student make?

The student didn't line up the digits in the correct decimal places.

b) What is the correct answer?

4.437

Full transcriptUse front estimation to estimate the sum: 5 + 3 = 8

Add . Write each number with the same decimal places, using zeros as place holders .Record the number without the decimal point EXAMPLE 1 Write each fraction as a decimal.. Identify each decimal as terminating or repeating

1/2=

0.5 so this would be a terminating decimal

0.83333333... since this is a repeating decimal we will put a line over it like said in the explanation 0.83 3.4 Multiplying Decimals 3.1

Fractions to decimals Numbers can be written in both fraction and decimal form. For example , 4 can be written as 3/1 and 3.0.

Decimals such as 0.1 and 0.25, are terminating decimals each has a definite numbers of decimal places.

Decimals such as 0.333 333 333.......or 0.4545 4545 4545 are repeating decimals.Some digits in repeating decimals repeat forever. So we put a bar over the digits that repeat .

EG: 3/66=0.45454545 4545..which is written 0.45 EXAMPLE 1 EXAMPLE 2 Multiply . EXAMPLE 1 Jan bought 18.9 m of framing to make picture frames.

Each picture needs 1.8 m of frames.

How much frames can Jan make.

How much framing material is left over. EXAMPLE 2 A 0.4-kg bag of oranges costs $1.34

a) What is the cost of 1 kg of oranges ?

How do you know your answer is reasonable ?

b) Suppose you spent $10 on oranges.

What mass of orange did you buy EXAMPLE 3 Alex finds a remnant of landscaping fabric at a garden store. The fabric is standard width, with length 9.88 m.

Alex needs fourteen 0.8 pieces for a garden patio. a) How many 0.8 m pieces can Alex cut from the remnant ?

b) Will Alex have all the fabric he needs ? Why or why not ?

c) If your answers in part b is no how much more fabric does Alex need ?

d) Alex redesigns his patio so that he need fourteen 0.7 m pieces of fabric. Will the remnant be enough fabric ? Explain. 3.6 Order of operations with decimals Here is how you do the order of operation.

To make sure everyone has the same answers for the given expression we add, subtract multiply and divide in the order:

Do the operation in brackets first.

Then divide and multiply in the order from left to right

Then add and subtract from left to right

B brackets

E equation

D division

M multiplication

A addition

S subtraction EXAMPLE 1 Evaluate

a) 4.8 + 2 x 8.7

2x8.7= 17.4 + 4.8

= 22.2

b) 14-9 + 5.8

14-9 = 5 + 5.8

= 10.8

c) 8 - 3.6 x 2

3.6 x 2 = 7.2

= 0.8

d) 16+2.4 - 7

16+2.4 = 18.4 - 7

= 11.4 EXAMPLE 2 Evaluate

a) 164.5/7 x 10 + 7.2 b) 73.8 x (3.2 + 6.8)-14.1/0.2

164.5/7 = 23.5 x 10 (3.2 + 6.8)= 10 x 73.8

= 235 + 7.2 = 738/0.2

= 242.2 = 3690 - 14.1

= 3675.9

c) (4.7 - 3.1) x 5 - 7.5/2.5

(4.7 - 3.1) = 1.6

= 1.6 x 5

= 8/2.5

= 3.2 - 5

= 1.8 EXAMPLE 3 Solve the problem

A) The cross country teamed members ran timed circuits. Here are their times: 15.8min,12.5 min, 18.0 min, 14.2 min,13.9 min, 16.0 min, 16.2 min,

17.5 min, 16.3 min, 15.6 min

Find the mean time.

The mean time is the sum 15.8+12.5+18.0+14.2+13.9+16.0+16.2+17.5+16.3+15.6/10

of the times divided by 15.8+12.5+18.0+14.2+13.9+16.0+16.2+17.5+16.3+15.6 = 156

the number of times. = 156/10

= 15.6

The mean time they was 15.6 minutes.

B) 0.38 + 16.2 x (2.1 + 4)+ 21/3.5

(2.1 + 4) = 6.1 x 16.2

= 98.82/3.5 3.5 Dividing Decimals EXAMPLE 1 Evaluate To divide: 18.9/1.8 ignore the decimal points.

Find 189/18 and then estimate.

10.5 18.9/1.8 = 10.5

18 189.0 There will be 0.5 framing material

- 180 left.

90

- 90

0 When you divide 2 numbers, the number is the quotient. a) The cost of 1kg of orange is $ 3.35, to find the cost of 1kg you divide 1.34/0.4 and the total is $ 3.35.

b) About 3kg because you do $ 3.35 x 3 = $ 10.05 so there is 5 cents more,so it is little less than 3kg. a) He can cut 12.35 0.8 m of fabric.

b) No he will not have all the fabric he needs because he needs 14 0.8 fabric and he only has 12.35 0.8 fabric.

c) He needs 1.65 more remnant fabric.

d) If he redesigned his patio and uses 0.7 pieces there would be 14.1142857 so it would be enough remnant fabric for his patio. 3.7 You can divide decimals the same way as you divide a whole number.

Use front end estimation to place the decimals point.

For example, to divide 24.3/0.6 405

Estimate first: 24/1=24 6 2430

So, 24.3/0.6 is about 24. -24

Divide as you would whole numbers. 03

00

30

30 3.7 Relating Fractions, Decimals, and Percents You ca use Base Ten Blocks to multiply decimals.

To multiply 2.2 x 1.4 display base ten blocks as shown below.

2 0.2 1 0.4 0.8 So when you find out the answer you multiply 1 by 2, then 1 by 0.2,then 2 by 0.4 and then 0.4 by .0.2 and then 0.4 by 0.8 and then add all of the answers u 3.4 Multiplying Decimals You can also multiply decimals the same way you multiply whole numbers.

To multiply 2.3 x 1.8, multiply 24 x 18.

So you take out the decimal and when you find the answer you move the decimals in as the same number you moved the decimal out

24

x 18

432 So now you move the decimal in two spaces because for 1.8 you moved it out one and for 2.4 you moved it out one so the total is two spaces . The total is 4.32.

You can also use front-end estimation to place the decimal point : 2 x 1 = 2

So, 2.4 x 1.8 is about 2 decimals place.

Place the decimals point between the 4 and the 3.

The product is 4.32. Write a multiplication equation for each picture.

a) EXAMPLE 2 For each fraction write , write an equivalent fraction with denominator 10,100,1000. The write the fraction as a decimal.

2/5= 0.4 Since 4 in this rounds to 10(closest) it will go over 10. 4/10 0.4. Because it has one decimal place.

1/4: 0.25 25 will round up to 100. 25/100. This number has 2 decimals places it is in the 10's place.

37/500: 0.074 rounds to 1000. 74/1000. Thousands place. 1x1= 1, 1x0.2= 0.2, 1x0.2= 0.2, 0.2x0.2= 0.4, 0.2x0.4= 0.08 1+0.2+0.2+0.4+0.08= 1.88 a) 2.7x4.786= 12.9222

b) 12.52x13.923= 174.31596

c) 0.986x1.352= 1.333072 You can check your answer by using base ten blocks to see if a, b, and c is right. We can use number lines to show how percents relate to fractions and decimals

For example:

25%=25=0.25

100

Conversely, a decimal can be written as a percent:

0.15=15=15%

100

To write a fraction in a percent, write the equivalent fraction with denominator 100.

1 x50 50

2 x50 100 EXAMPLE 1 Write each percent as a fraction. a) 15% = 15 = 0.15

100

b) 40% = 40 = 0.40

100

c) 5% = 5 = 0.05

100 EXAMPLE 2 Write each percent as a fraction and as a decimal.

Sketch a number line to show how the numbers are related. a) 2 2 o.02

100

1 2 6 28 50

100 100 100

b) 6 6 0.06

100

c) 28 28 0.28

100 EXAMPLE 3 Divide the paper into these 4 sections EXAMPLE 3 Multiply, describe the pattern that you see a) 8.36x10 = 83.6

8.36x100 = 836

8.36x1000 =8360

Multiply by multiples of 10. The digits in the product move on place to the left each time. Or the decimal point moves one place right each time. EXAMPLE 3 write each fraction as decimal. What patterns do you see?

1/999: 0.010

2/999:0.0200

54/999:0.540

113/999: 0.113

The patterns that you see that they are all repeating decimals. Plus the number that is being divided by 999 comes out the same when you divide but its just a repeating number. 3.2 Comparing and Ordering Fractions and Decimals Any fraction greater than 1 can be written as a mixed number. The bench mark 0,1/2 and 1 can be used to compare the fraction parts of a mixed number.

We can use bench marks on a number line to order these numbers: 2/11,2 2 3/8, 1 1/16, 14/9, 14/15

2/11 is close to 0. Since 3/8 is close to 1/2 2 3/8 is close to 2 1/2, but less than 2 1/2, 1 1/16 is close to 2, but greater than 1. 0 2/11 1/2 14/15 1 1 1/16 1 1/2 14/9 2 2 3/8 2 1/2 3 EXAMPLE 1 Use bench marks and a number line to order the set of numbers from least to greatest:

7/6, 15/12, 1 2/9, 1

least to greatest: 1, 7/6, 1 2/9, 15/17

1 7/6 1 2/9 15/17 EXAMPLE 2 Amrita, Paul, Cory baked pizzas for the fund raiser sale. The students cut their pizzas into different sized slices.

Amrita sold 11/6 pizzas. Pauls sold 1.875. Cory sold 9/4 pizzas.

a) Order From least to greatest: 11 6, 1.875, 9 4

b) Who sold the most pizzas?

Corey sold the most pizzas and amrita sold the leastest EXAMPLE 3 Which number in the number line is placed incorrectly?

1 1/2 15/8 2 2 1/8 11/4 2 1/2 11/4; 2 1/2= 10/4 which is less than 11/4 3.8 Solving Percent Problems A papaerback novel usually costs $7.99. Its is on sale for 15% off. To find how much you saved, calculate 15%of $7.99. 15%=15/100 of %7.99

=0.15 x 7.99

Use a calculator:

0.15 x 7.99=1.1985; round to the nearest cent and you save $1.20 buying the book on black friday.

You can also show it on a number line:

0 $1.20 $7.99 EXAMPLE 1 Calculate:

a) 10% of 30: 10 x 30= 300(3) b) 20% of 50: 20 x 50 =100(10)

c) 18% of 36= 18 x 36= 6.48 d) 67% of 112: 67 x 112= 75.04 EXAMPLE 2 The Good and Service tax(GST) is currently 6%.

For each item below find the GST. Find the cost of the item including the GST.

A) Bicycle: 129.00

so what you do here is that you need to multiply the cost of the bike by the GST. 129.00 x 6%= 7.74. This is how mch GST will be added. Now if you want to add the GST in the price of the bike. ADD: 129.00+7.74:136.74 EXAMPLE 3 The regular price of a radio is $60.00. How much will the radio be for when its on sale.

sale for 25%: 45.00 30% off: 42.00 40% off: 35.00 EXAMPLE 1 The tallest building in the world is the Burj Khalifa. Its height is 0.800, in Dubai . The 2nd tallest building in the world is the Taipei in Taiwan. Its height is 0.509km What is the difference in the heights of the buildings.

So we would subtract the heights for the buildings.

0.800-0.509= 291km or you can do front estimation 8-5= 3 Example 2 The Robb family and the Khalifa family have similar homes. The Robb family sets there thermostats to to 20 degrees Celsius during the winter months. These were the monthly heating bills: $134.35 $171.23 and 123.21

The Khalifa family used programmable thermostat lower the temperature at night and during the day when the family was out. Their heating bills were: $134.25, 103.27, $98.66

a)How much did each family pay to heat its home during the winter months?

Robb Family: $134.35+171.23+123.21+=$428.79

Khlifa Family:$134.25+103.37+$98.66=$336.28

b)How much more did the Robb family pay: $428.79-336.28=92.51

The Robb family paid $92.51 dollars more

EXAMPLE 3 A student subtracted 0.373 from 4.81 and got the difference of 0.108.

a) What mistake did the student make?

The student didn't line up the digits in the correct decimal places.

b) What is the correct answer?

4.437