ratios & proportions. ratio (ray-she-yo) a ratio is the comparison of two numbers by division. a...
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Ratios & Ratios & ProportionsProportions
Ratio (ray-she-yo)Ratio (ray-she-yo)A ratio is the comparison of two
numbers by division.
A classroom has 16 boys and 12 girls.
Also written as 16 boys, 16:12 or 16 to 12 12 girls
Generally, ratios are in lowest terms: 16 = 16/4 = 4
12 12/4 3
Ratio, continuedRatio, continuedRatios can compare two unlike
things:◦Joe earned $40 in five hours ◦The ratio is 40 dollars or 8 dollars
5 hours 1 hour
◦When the denominator is one, this is called a unit rate.
Ratio, continuedRatio, continued
Let’s look at a classroom:Ratios can be part-to-part
◦16 boys15 girls
Ratios can be part-to-whole◦16 boys
31 students
Now, on to proportions!Now, on to proportions!
d
c
b
a
What is a proportion?
A proportion is an equation that equates two ratios
So we have a proportion :4
6
2
3
Tell whether the ratios are proportional.
410
615
A.
Since the cross products are equal, the ratios are proportional.
60
=?
Using Cross Products to Identify Proportions
60 = 60
Find cross products.60410
615
A mixture of fuel for a certain small engine should be 4 parts gasoline to 1 part oil. If you combine 5 quarts of oil with 15 quarts of gasoline, will the mixture be correct?
4 parts gasoline1 part oil
=? 15 quarts gasoline5 quarts oil
4 • 5 = 20 1 • 15 = 15
20 ≠ 15
The ratios are not equal. The mixture will not be correct.
Set up ratios.
Find the cross products.
Using Cross Products to Identify Proportions
Tell whether the ratios are proportional.
Try This: Example 1A
Since the cross products are equal, the ratios are proportional.
20
20 = 20
Find cross products.2024
510
24
510
A. =?
A mixture for a certain brand of tea should be 3 parts tea to 1 part sugar. If you combine 4 tablespoons of sugar with 12 tablespoons of tea, will the mixture be correct?
Try This: Example 1B
3 parts tea 1 part sugar
=? 12 tablespoons tea4 tablespoons sugar
3 • 4 = 12 1 • 12 = 12
12 = 12
The ratios are equal. The mixture will be correct.
Set up ratios.
Find the cross products.
RATIOS & PROPORTIONSRATIOS & PROPORTIONSAre the following proportions?
7
3
9
4
20
9
9
4
27
12
9
4
18
8
9
4
3974 2728
FALSE – Not a proportion
99204 8180
FALSE – Not a proportion
129274 108108
TRUE – this is a proportion
89184 7272
TRUE – this is a proportion
Solving Proportions Solving Proportions When you do not know one of the
four numbers in a proportion, set the cross products equal to each other and solve.
Solve the proportion.
6p = 12 • 5
p = 10
6p = 60
Find the cross products.
Solve.
56
p12
=
; the proportion checks.56
1012
=
Solve the proportion.
14 • 3 = 2g
21 = g
42 = 2g
Find the cross products.
Solve.
23
14g
=
; the proportion checks.23
1421
=
RATIOS & PROPORTIONSRATIOS & PROPORTIONSFind the missing numbers to make the following proportions.
21
93
9213 x
x963
x79 9
10
5
4
5410 x2010 x
10 10 2x
9
4
1
941 x36x
ProportionsProportionsProportion is a statement that
says two ratios are equal.◦In an election, Damon got three votes for each two votes that Shannon got. Damon got 72 votes. How many votes did Shannon get?
◦Damon 3 = 72 ◦Shannon 2 n
n = 48, so Shannon got 48 votes.
Proportions, continuedProportions, continuedTires cost two for $75. How much
will four tires cost?
◦# of tires 2 = 4 cost 75 n
n = 150, so four tires cost $150
Proportion, continuedProportion, continuedThree cans of soup costs $5. How
much will 12 cans cost?
# of cans 3 = 12 cost 5 n
n = 20, so 12 cans cost $20
Now you know enough about Now you know enough about properties, let’s solve the Mysterious properties, let’s solve the Mysterious problems!problems!
galx
miles
gal
miles
_
)55(
1
30
x
10
1
30
3
1
If your car gets 30 miles/gallon, how many gallons of gas do you need to commute to school everyday?
5 miles to school
5 miles to home
Let x be the number gallons we need for a day:Can you
solve it from here?
x = Gal
So you use up 1/3 gallon a day. How many gallons would you use for a week?
5 miles to school
5 miles to home
Let t be the number of gallons we need for a week:
days
galt
day
gal
5
_
1
3/1
51
3/1 t
53
1 t t3)5(1 3
5t Gal
What property is
this?
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:
5 miles to school
5 miles to home
gallons
dollarss
gallon
dollars
67.1
_
1
69.3
67.11
69.3 s
3.69(1.67) = 1s s = 6.16 dollars
So what do you think?So what do you think?
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?
5 miles
Think proportionally! . . . It’s all about proportions!
Exit Ticket
Tell whether each pair of ratios is proportional.
4842 =? 16
141. 20
15 =? 34
2.
Solve each proportion.
3. 4.
yes no
n = 30 n = 164518
n12 = n
2469 =