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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?
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…
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
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
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
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.
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
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
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
10.
More examples…
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
12.
Properties of a proportion?
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
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
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
16.
How about an example?
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?
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
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
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
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
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
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