Ratios and Rates Lesson 4.2.1 Lesson 4.2.1 Ratios and Rates California Standard: What it means for you: Measurement and Geometry 3.2 Use measures expressed as rates (e.g., speed,
Download ReportTranscript Ratios and Rates Lesson 4.2.1 Lesson 4.2.1 Ratios and Rates California Standard: What it means for you: Measurement and Geometry 3.2 Use measures expressed as rates (e.g., speed,
Slide 1
Ratios and Rates
Lesson 4.2.1
1
Slide 2
Lesson
4.2.1
Ratios and Rates
California Standard:
What it means for you:
Measurement and Geometry 3.2
Use measures expressed as rates
(e.g., speed, density) and measures
expressed as products (e.g., persondays) to solve problems; check the
units of the solutions; and use
dimensional analysis to check the
reasonableness of the answer.
You’ll learn what rates are
and how you can use them to
compare things — such as which
size product is better value.
Key words:
•
•
•
•
•
rate
ratio
fraction
denominator
unit rate
2
Slide 3
Lesson
4.2.1
Ratios and Rates
Rates are used a lot in daily life. You often hear people talk
about speed in miles per hour, or the cost of groceries in
dollars per pound.
A rate tells you how much one thing changes when
something else changes by a certain amount.
Imagine buying apples for $2 per
pound — the cost will increase by
$2 for every pound you buy.
Best apples
$2 per pound
3
Slide 4
Lesson
4.2.1
Ratios and Rates
Ratios are Used to Compare Two Numbers
You might remember ratios from grade 6.
Ratios compare two numbers, and don’t have any units.
For example, the ratio of boys
to girls in a class might be 5 : 6.
There are three ways of expressing a ratio.
The ratio 5 : 6 could also be expressed as “5 to 6”
or as the fraction 5 .
6
4
Slide 5
Lesson
4.2.1
Example
Ratios and Rates
1
There are four nuts between three squirrels.
What is the ratio of nuts to squirrels?
Solution
There are 4 nuts to 3 squirrels so the ratio
of nuts to squirrels is 4 : 3.
4
This could also be written “4 to 3” or .
3
5
Solution follows…
Slide 6
Lesson
4.2.1
Ratios and Rates
Ratios Compare Quantities With Different Units
A rate is a special kind of ratio, because it compares
two quantities that have different units.
For example, if you travel 60 miles in 3 hours
you would be traveling at a rate of 60 miles .
3 hours
You’d normally write this as a unit rate.
That’s one with a denominator of 1.
60 miles 20 miles
So
=
, or 20 miles per hour.
3 hours
1 hour
6
Slide 7
Lesson
4.2.1
Example
Ratios and Rates
2
John takes 110 steps in 2 minutes.
What is his unit rate in steps per minute?
Solution
110 steps in 2 minutes means a rate of:
110 steps 55 steps
=
= 55 steps per minute
2 minutes 1 minute
7
Solution follows…
Slide 8
Lesson
4.2.1
Ratios and Rates
Numerator ÷ Denominator Gives a Unit Rate
Dividing the numerator by the denominator of a rate
gives the unit rate.
So, if it costs 2 dollars for 3 apples, the unit rate is the
2 dollars
price per apple, which is
= 2 dollars ÷ 3 apples
3 apples
= 0.67 dollars per apple
8
Slide 9
Lesson
4.2.1
Example
Ratios and Rates
3
A car goes 54 miles in 3 hours.
Write this as a unit rate in miles per hour.
Solution
Divide the top by the bottom of the rate.
54 miles
= (54 ÷ 3) miles per hour = 18 miles per hour.
3 hours
This is a unit rate because the denominator is now 1
18
(it’s equivalent to
mi/h).
1
9
Solution follows…
Slide 10
Lesson
4.2.1
Example
Ratios and Rates
4
If a wheel spins 420 times in 7 minutes,
what is its unit rate in revolutions per minute?
Solution
420
The rate is
revolutions per minute.
7
Divide the top by the bottom of the rate.
(420 ÷ 7) revolutions per minute = 60 revolutions per minute.
This is a unit rate because 60 revolutions per minute has a
60
denominator of 1 (60 =
).
1
10
Solution follows…
Slide 11
Lesson
4.2.1
Ratios and Rates
Guided Practice
In Exercises 1–3, find the unit rates.
1. $3.60 for 3 pounds of tomatoes.
$1.20 per pound of tomatoes
2. $25 for 500 cell phone minutes.
$0.05 per minute
3. 648 words typed in 8 minutes.
81 words per minute
4. Joaquin buys 2 meters of fabric, which costs him $9.50.
$4.75 per meter
What was the price per meter?
5. Mischa buys a $19.98 ticket for unlimited rides
at a fairground. She goes on six rides.
How much did she pay per ride? $3.33 per ride
11
Solution follows…
Slide 12
Lesson
4.2.1
Ratios and Rates
Use “Unit Rates” to Find the Better Buy
Stores often sell different sizes of the same thing,
such as clothes detergent or fruit juice.
A bigger size is often a better buy — meaning that you
get more product for the same amount of money.
But this isn’t always the case, so it’s useful to be able to
work out which is the better buy.
You can do this by finding the price for a single unit of
each product. The units can be ounces, liters, meters, or
whatever is most sensible.
12
Slide 13
Lesson
4.2.1
Example
Ratios and Rates
5
A store sells two sizes of cereal.
Which is the better buy?
Solution
CEREAL
$3.20 for 16 ounce box
$4.32 for 24 ounce box
CEREAL
3.20 dollars
16 ounce box: Rate is
.
16 ounces
Unit rate = (3.20 ÷ 16) dollars per ounce = $0.20 per ounce
4.32 dollars
24 ounce box: Rate is
.
24 ounces
Unit rate = (4.32 ÷ 24) dollars per ounce = $0.18 per ounce
The 24 ounce box is the better buy
— the price per ounce is lower.
13
Solution follows…
Slide 14
Lesson
4.2.1
Ratios and Rates
Guided Practice
6. Determine which phone company offers the better deal:
Phone Company A: $40 for 800 minutes.
Phone Company B: $26 for 650 minutes.
Unit rate from Company A = $40 ÷ 800 minutes = 5 ¢ per minute
Unit rate from Company B = $26 ÷ 650 minutes = 4 ¢ per minute
So, phone Company B offers the best deal.
7. Determine which is the better deal on carrots:
$1.20 for 2 lb or $2.30 for 5 lb.
Unit rate deal 1 = $1.20 ÷ 2 lb = 60 ¢ per lb
Unit rate deal 2 = $2.30 ÷ 5 lb = 46 ¢ per lb
So, deal 2 — $2.30 for 5 lb is best.
14
Solution follows…
Slide 15
Lesson
4.2.1
Ratios and Rates
Independent Practice
In Exercises 1–6, write each as a unit rate.
1. $4.50 for 6 pens
$0.75 per pen
3. 200 pages in 5 days
40 pages per day
5. $400 for 10 items
$40 per item
2. 100 miles in 8 h
12.5 miles per hour
4. 120 miles in 2 h
60 miles per hour
6. $36 in 6 hours
$6 per hour
15
Solution follows…
Slide 16
Lesson
4.2.1
Ratios and Rates
Independent Practice
7. Peanuts are either $1.70 per pound or $8 for 5 pounds.
Which is the better buy? $8 for 5 pounds
8. Lemons sell for $4.50 for 6, or $10.50 for 15.
Which is the better buy? $10.50 for 15
9. “$40 for 500 pins or $60 for 800 pins.”
Which is the better buy? $60 for 800 pins
16
Solution follows…
Slide 17
Lesson
4.2.1
Ratios and Rates
Round Up
Rates compare one thing to another and always
have units.
A unit rate is a rate that has a denominator of one.
In the next Lesson you’ll see how rate is related to
the slope of a graph.
17
Ratios and Rates
Lesson 4.2.1
1
Slide 2
Lesson
4.2.1
Ratios and Rates
California Standard:
What it means for you:
Measurement and Geometry 3.2
Use measures expressed as rates
(e.g., speed, density) and measures
expressed as products (e.g., persondays) to solve problems; check the
units of the solutions; and use
dimensional analysis to check the
reasonableness of the answer.
You’ll learn what rates are
and how you can use them to
compare things — such as which
size product is better value.
Key words:
•
•
•
•
•
rate
ratio
fraction
denominator
unit rate
2
Slide 3
Lesson
4.2.1
Ratios and Rates
Rates are used a lot in daily life. You often hear people talk
about speed in miles per hour, or the cost of groceries in
dollars per pound.
A rate tells you how much one thing changes when
something else changes by a certain amount.
Imagine buying apples for $2 per
pound — the cost will increase by
$2 for every pound you buy.
Best apples
$2 per pound
3
Slide 4
Lesson
4.2.1
Ratios and Rates
Ratios are Used to Compare Two Numbers
You might remember ratios from grade 6.
Ratios compare two numbers, and don’t have any units.
For example, the ratio of boys
to girls in a class might be 5 : 6.
There are three ways of expressing a ratio.
The ratio 5 : 6 could also be expressed as “5 to 6”
or as the fraction 5 .
6
4
Slide 5
Lesson
4.2.1
Example
Ratios and Rates
1
There are four nuts between three squirrels.
What is the ratio of nuts to squirrels?
Solution
There are 4 nuts to 3 squirrels so the ratio
of nuts to squirrels is 4 : 3.
4
This could also be written “4 to 3” or .
3
5
Solution follows…
Slide 6
Lesson
4.2.1
Ratios and Rates
Ratios Compare Quantities With Different Units
A rate is a special kind of ratio, because it compares
two quantities that have different units.
For example, if you travel 60 miles in 3 hours
you would be traveling at a rate of 60 miles .
3 hours
You’d normally write this as a unit rate.
That’s one with a denominator of 1.
60 miles 20 miles
So
=
, or 20 miles per hour.
3 hours
1 hour
6
Slide 7
Lesson
4.2.1
Example
Ratios and Rates
2
John takes 110 steps in 2 minutes.
What is his unit rate in steps per minute?
Solution
110 steps in 2 minutes means a rate of:
110 steps 55 steps
=
= 55 steps per minute
2 minutes 1 minute
7
Solution follows…
Slide 8
Lesson
4.2.1
Ratios and Rates
Numerator ÷ Denominator Gives a Unit Rate
Dividing the numerator by the denominator of a rate
gives the unit rate.
So, if it costs 2 dollars for 3 apples, the unit rate is the
2 dollars
price per apple, which is
= 2 dollars ÷ 3 apples
3 apples
= 0.67 dollars per apple
8
Slide 9
Lesson
4.2.1
Example
Ratios and Rates
3
A car goes 54 miles in 3 hours.
Write this as a unit rate in miles per hour.
Solution
Divide the top by the bottom of the rate.
54 miles
= (54 ÷ 3) miles per hour = 18 miles per hour.
3 hours
This is a unit rate because the denominator is now 1
18
(it’s equivalent to
mi/h).
1
9
Solution follows…
Slide 10
Lesson
4.2.1
Example
Ratios and Rates
4
If a wheel spins 420 times in 7 minutes,
what is its unit rate in revolutions per minute?
Solution
420
The rate is
revolutions per minute.
7
Divide the top by the bottom of the rate.
(420 ÷ 7) revolutions per minute = 60 revolutions per minute.
This is a unit rate because 60 revolutions per minute has a
60
denominator of 1 (60 =
).
1
10
Solution follows…
Slide 11
Lesson
4.2.1
Ratios and Rates
Guided Practice
In Exercises 1–3, find the unit rates.
1. $3.60 for 3 pounds of tomatoes.
$1.20 per pound of tomatoes
2. $25 for 500 cell phone minutes.
$0.05 per minute
3. 648 words typed in 8 minutes.
81 words per minute
4. Joaquin buys 2 meters of fabric, which costs him $9.50.
$4.75 per meter
What was the price per meter?
5. Mischa buys a $19.98 ticket for unlimited rides
at a fairground. She goes on six rides.
How much did she pay per ride? $3.33 per ride
11
Solution follows…
Slide 12
Lesson
4.2.1
Ratios and Rates
Use “Unit Rates” to Find the Better Buy
Stores often sell different sizes of the same thing,
such as clothes detergent or fruit juice.
A bigger size is often a better buy — meaning that you
get more product for the same amount of money.
But this isn’t always the case, so it’s useful to be able to
work out which is the better buy.
You can do this by finding the price for a single unit of
each product. The units can be ounces, liters, meters, or
whatever is most sensible.
12
Slide 13
Lesson
4.2.1
Example
Ratios and Rates
5
A store sells two sizes of cereal.
Which is the better buy?
Solution
CEREAL
$3.20 for 16 ounce box
$4.32 for 24 ounce box
CEREAL
3.20 dollars
16 ounce box: Rate is
.
16 ounces
Unit rate = (3.20 ÷ 16) dollars per ounce = $0.20 per ounce
4.32 dollars
24 ounce box: Rate is
.
24 ounces
Unit rate = (4.32 ÷ 24) dollars per ounce = $0.18 per ounce
The 24 ounce box is the better buy
— the price per ounce is lower.
13
Solution follows…
Slide 14
Lesson
4.2.1
Ratios and Rates
Guided Practice
6. Determine which phone company offers the better deal:
Phone Company A: $40 for 800 minutes.
Phone Company B: $26 for 650 minutes.
Unit rate from Company A = $40 ÷ 800 minutes = 5 ¢ per minute
Unit rate from Company B = $26 ÷ 650 minutes = 4 ¢ per minute
So, phone Company B offers the best deal.
7. Determine which is the better deal on carrots:
$1.20 for 2 lb or $2.30 for 5 lb.
Unit rate deal 1 = $1.20 ÷ 2 lb = 60 ¢ per lb
Unit rate deal 2 = $2.30 ÷ 5 lb = 46 ¢ per lb
So, deal 2 — $2.30 for 5 lb is best.
14
Solution follows…
Slide 15
Lesson
4.2.1
Ratios and Rates
Independent Practice
In Exercises 1–6, write each as a unit rate.
1. $4.50 for 6 pens
$0.75 per pen
3. 200 pages in 5 days
40 pages per day
5. $400 for 10 items
$40 per item
2. 100 miles in 8 h
12.5 miles per hour
4. 120 miles in 2 h
60 miles per hour
6. $36 in 6 hours
$6 per hour
15
Solution follows…
Slide 16
Lesson
4.2.1
Ratios and Rates
Independent Practice
7. Peanuts are either $1.70 per pound or $8 for 5 pounds.
Which is the better buy? $8 for 5 pounds
8. Lemons sell for $4.50 for 6, or $10.50 for 15.
Which is the better buy? $10.50 for 15
9. “$40 for 500 pins or $60 for 800 pins.”
Which is the better buy? $60 for 800 pins
16
Solution follows…
Slide 17
Lesson
4.2.1
Ratios and Rates
Round Up
Rates compare one thing to another and always
have units.
A unit rate is a rate that has a denominator of one.
In the next Lesson you’ll see how rate is related to
the slope of a graph.
17