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Ratios! What is a Ratio? How to Use Ratios? How to Simplify? Proportions! What is a proportion? Properties of proportions? How to use proportions?

Ratios! What is a Ratio? How to Use Ratios? How to Simplify? Proportions! What is a proportion? Properties of proportions? How to use proportions?

1.
Ratios and Proportions

2.
• Ratios!

What is a Ratio?

How to Use Ratios?

How to Simplify?

Proportions!

What is a proportion?

Properties of proportions?

How to use proportions?

• Mysterious Problems…

What is a Ratio?

How to Use Ratios?

How to Simplify?

Proportions!

What is a proportion?

Properties of proportions?

How to use proportions?

• Mysterious Problems…

3.
What is a Ratio?

• A ratio is a comparison of two numbers.

• Ratios can be written in three different ways:

a to b

a:b

a

Because a ratio is a fraction, b can not be zero

b

Ratios are expressed in simplest form

• A ratio is a comparison of two numbers.

• Ratios can be written in three different ways:

a to b

a:b

a

Because a ratio is a fraction, b can not be zero

b

Ratios are expressed in simplest form

4.
How to Use Ratios?

• The ratio of boys and girls in the class is

12 to11. This means, for every 12 boys

you can find 11 girls to match.

Howcould

manybe dogs

justand cats do11I

• The ratio of length and width of this rectangle

• There 12 boys,

girls.have? We don’t know, all we

is 4 to 1. knowcould

• There is if they’d

girls.each dog

start a22

be 24 boys, fight,

4cmhas to fight 2 cats.

• There could be 120 boys, 110

girls…a huge class 1cm

What is the ratio if the

rectangle is 8cm long and

.• The ratio of cats and dogs at my home is 2 to 1

2cm wide?

Still 4 to 1, because for

every 4cm, you can find 1cm

to match

• The ratio of boys and girls in the class is

12 to11. This means, for every 12 boys

you can find 11 girls to match.

Howcould

manybe dogs

justand cats do11I

• The ratio of length and width of this rectangle

• There 12 boys,

girls.have? We don’t know, all we

is 4 to 1. knowcould

• There is if they’d

girls.each dog

start a22

be 24 boys, fight,

4cmhas to fight 2 cats.

• There could be 120 boys, 110

girls…a huge class 1cm

What is the ratio if the

rectangle is 8cm long and

.• The ratio of cats and dogs at my home is 2 to 1

2cm wide?

Still 4 to 1, because for

every 4cm, you can find 1cm

to match

5.
How to simplify ratios?

• The ratios we saw on last The ratio of boys and girls in the

slide were all simplified. class is 12

How was it done? 11

Ratios can be expressed

a

4

The ratio of the rectangle is

a 1

b

in fraction form…

b

This allows us to do math The ratio of cats and dogs in my

house is 2

on them.

1

• The ratios we saw on last The ratio of boys and girls in the

slide were all simplified. class is 12

How was it done? 11

Ratios can be expressed

a

4

The ratio of the rectangle is

a 1

b

in fraction form…

b

This allows us to do math The ratio of cats and dogs in my

house is 2

on them.

1

6.
How to simplify ratios?

• Now I tell you I have 12 cats and 6 dogs. Can you simplify the ratio of cats and dogs

to 2 to 1?

12 12 / 6 2

= =

6 6/6 1

Divide both numerator and

denominator by their

Greatest Common Factor 6.

• Now I tell you I have 12 cats and 6 dogs. Can you simplify the ratio of cats and dogs

to 2 to 1?

12 12 / 6 2

= =

6 6/6 1

Divide both numerator and

denominator by their

Greatest Common Factor 6.

7.
How to simplify ratios?

A person’s arm is 80cm, he is 2m tall.

Find the ratio of the length of his arm to his total height

To compare them, we need to convert both

numbers into the same unit …either cm or m.

• Let’s try cm

arm 80cm 80cm Once we have the

height

same units, we can

2m 200cm simplify them.

80 2

200 5

A person’s arm is 80cm, he is 2m tall.

Find the ratio of the length of his arm to his total height

To compare them, we need to convert both

numbers into the same unit …either cm or m.

• Let’s try cm

arm 80cm 80cm Once we have the

height

same units, we can

2m 200cm simplify them.

80 2

200 5

8.
How to simplify ratios?

• Let’s try m now!

arm 80cm 0.8m Once we have the

height

same units, they

2m 2m simplify to 1.

8

2

20 5

To make both numbers

integers, we multiplied both

numerator and denominator by

10

• Let’s try m now!

arm 80cm 0.8m Once we have the

height

same units, they

2m 2m simplify to 1.

8

2

20 5

To make both numbers

integers, we multiplied both

numerator and denominator by

10

9.
How to simplify ratios?

• If the numerator and denominator do not

have the same units it may be easier to

convert to the smaller unit so we don’t

have to work with decimals…

3cm/12m = 3cm/1200cm = 1/400

2kg/15g = 2000g/15g = 400/3

5ft/70in = (5*12)in / 70 in = 60in/70in = 6/7

2g/8g = 1/4 Of course, if they are already in the same units, we

don’t have to worry about converting. Good deal

• If the numerator and denominator do not

have the same units it may be easier to

convert to the smaller unit so we don’t

have to work with decimals…

3cm/12m = 3cm/1200cm = 1/400

2kg/15g = 2000g/15g = 400/3

5ft/70in = (5*12)in / 70 in = 60in/70in = 6/7

2g/8g = 1/4 Of course, if they are already in the same units, we

don’t have to worry about converting. Good deal

10.
More examples…

8 1 12 6

24 = 50 =

3 25

40 27 3

1 =

= 18 2

200 5

27 3

=

9 1

8 1 12 6

24 = 50 =

3 25

40 27 3

1 =

= 18 2

200 5

27 3

=

9 1

11.
Now, on to proportions!

What is a proportion?

a c A proportion is an equation

that equates two ratios

b d

The ratio of dogs and cats was 3/2

The ratio of dogs and cats now is 6/4=3/2

3 6

So we have a proportion :

2 4

What is a proportion?

a c A proportion is an equation

that equates two ratios

b d

The ratio of dogs and cats was 3/2

The ratio of dogs and cats now is 6/4=3/2

3 6

So we have a proportion :

2 4

12.
Properties of a proportion?

3 6

2 4 Cross Product Property

2x6=12 3x4 = 12

3x4 = 2x6

3 6

2 4 Cross Product Property

2x6=12 3x4 = 12

3x4 = 2x6

13.
Properties of a proportion?

• Cross Product Property

a c

ad = bc

b d

means

extremes

• Cross Product Property

a c

ad = bc

b d

means

extremes

14.
Properties of a proportion?

Let’s make sense of the Cross Product Property…

For any numbers a, b, c, d:

a c a c

d d

b d b d

a a

d c d b b c

b b

ad bc

Let’s make sense of the Cross Product Property…

For any numbers a, b, c, d:

a c a c

d d

b d b d

a a

d c d b b c

b b

ad bc

15.
Properties of a proportion?

• Reciprocal Property

3 6

If

2 4 Can you see it?

If yes, can you think

Then 2 4 of why it works?

3 6

• Reciprocal Property

3 6

If

2 4 Can you see it?

If yes, can you think

Then 2 4 of why it works?

3 6

16.
How about an example?

7 x

Solve for x:

2 6

7(6) = 2x Cross Product Property

42 = 2x

21 = x

7 x

Solve for x:

2 6

7(6) = 2x Cross Product Property

42 = 2x

21 = x

17.
How about another example?

7 12

Solve for x:

2 x

7x = 2(12) Cross Product Property

7x = 24

24 Can you solve it

x=

7 using Reciprocal

Property? If yes,

would it be easier?

7 12

Solve for x:

2 x

7x = 2(12) Cross Product Property

7x = 24

24 Can you solve it

x=

7 using Reciprocal

Property? If yes,

would it be easier?

18.
Can you solve this one?

7 3

Solve for x:

x 1 x

7x = (x-1)3 Cross Product Property

7x = 3x – 3

Again, Reciprocal

4x = -3

3 Property?

x= 4

7 3

Solve for x:

x 1 x

7x = (x-1)3 Cross Product Property

7x = 3x – 3

Again, Reciprocal

4x = -3

3 Property?

x= 4

19.
Now you know enough about properties,

let’s solve the Mysterious problems!

If your car gets 30 miles/gallon, how many gallons

of gas do you need to commute to school

5 miles to home

5 miles to school

Let x be the number gallons we need for a day:

Can you solve

30miles (5 5)miles 30 10 it from here?

1gal x _ gal 1 x

1

x= 3 Gal

let’s solve the Mysterious problems!

If your car gets 30 miles/gallon, how many gallons

of gas do you need to commute to school

5 miles to home

5 miles to school

Let x be the number gallons we need for a day:

Can you solve

30miles (5 5)miles 30 10 it from here?

1gal x _ gal 1 x

1

x= 3 Gal

20.
5 miles to home

5 miles to school

So you use up 1/3 gallon a day. How many gallons would

you use for a week?

Let t be the number of gallons we need for a week:

1 / 3 gal t _ gal 1/ 3 t

1day 5days 1 5

What property

is this?

1 t 5

3 5 1(5) 3t t Gal

3

5 miles to school

So you use up 1/3 gallon a day. How many gallons would

you use for a week?

Let t be the number of gallons we need for a week:

1 / 3 gal t _ gal 1/ 3 t

1day 5days 1 5

What property

is this?

1 t 5

3 5 1(5) 3t t Gal

3

21.
So you use up 5/3 gallons a week (which is about 1.67

gallons). Consider if the price of gas is 3.69 dollars/gal,

how much would it cost for a week?

Let s be the sum of cost for a week:

3.69dollars s _ dollars

3.69 s

1gallon 1.67 gallons

1 1.67

3.69(1.67) = 1s s = 6.16 dollars

5 miles to home

5 miles to school

gallons). Consider if the price of gas is 3.69 dollars/gal,

how much would it cost for a week?

Let s be the sum of cost for a week:

3.69dollars s _ dollars

3.69 s

1gallon 1.67 gallons

1 1.67

3.69(1.67) = 1s s = 6.16 dollars

5 miles to home

5 miles to school

22.
So what do you think?

5 miles

10 miles

You pay about 6 bucks a week just to get to school!

What about weekends?

If you travel twice as much on weekends, say drive

10 miles to the Mall and 10 miles back, how many

gallons do you need now? How much would it cost

totally? How much would it cost for a month?

Think proportionally! . . . It’s all about

5 miles

10 miles

You pay about 6 bucks a week just to get to school!

What about weekends?

If you travel twice as much on weekends, say drive

10 miles to the Mall and 10 miles back, how many

gallons do you need now? How much would it cost

totally? How much would it cost for a month?

Think proportionally! . . . It’s all about

23.