For the Birds
Rates and Proportions
Learning Goals
Key Term
In this lesson, you will:
 convert
 Write ratios and rates.
 Write proportions.
 Scale up and scale down proportions.
W
hich bird lays the largest egg for its size? That honor goes to the little
spotted kiwi—a native of New Zealand with no tail, a long ivory beak, and poor
eyesight. The little spotted kiwi lays an egg that is more than one quarter its own
body weight.
By contrast, the bird that lays the smallest egg for its size is—you guessed it—the
ostrich! (Perhaps you didn’t guess that.) Although ostriches lay the largest eggs,
15
a typical ostrich egg weighs less than _____ of its mother’s weight.
1000
Why do you think that the ostrich egg is both the largest egg that any bird lays,
© Carnegie Learning
but is also the smallest egg in comparison to the mother’s weight?
1.3
Rates and Proportions
•
23
Problem 1
Eggsactly!
The table shows the weights of four different adult birds and the weights of their eggs.
Mother’s Weight
(oz)
Egg Weight
(oz)
Pigeon
10
0.75
Chicken
80
2
Swan
352
11
Robin
2.5
0.1
1. Compare the weights of the eggs. List the birds in order from the bird with the largest
egg to the bird with the smallest egg.
2. Determine the ratio of egg weight to mother’s weight for each
bird. Use your calculator to help you. Write the ratios as decimals.
Remember to
carefully read which
quantity should come
first in the ratio!
3. Use your decimal representations 2 to answer each question. Explain your reasoning.
b. Which of the birds listed lays the smallest egg for its size?
c. Compare the ratios of egg weight to mother’s weight. List the birds in order from
greatest ratio to least ratio.
24
•
Chapter 1
Ratios and Rates
© Carnegie Learning
a. Which of the birds listed lays the largest egg for its size?
Problem 2
The Coyote and the . . . Ostrich?
Although the ostrich is the largest living bird, it is also the fastest runner. The table shows
distances that four birds ran, and the amount of time it took each bird to run that distance.
Bird
Distance Covered
Time
Ostrich
22 miles
30 minutes
Great Roadrunner
300 yards
30 seconds
Quail
20 yards
2.5 seconds
Pheasant
200 yards
50 seconds
Each row in the table shows a rate. The rate for each bird in this situation is the distance
covered per the amount of time.
The rate, or running speed, for the ostrich is 22 miles per
22 mi .
30 minutes, or _______
30 min
1. Write the rates for the other three birds.
© Carnegie Learning
a. Great roadrunner:
b. Quail:
Remember,
a rate is a ratio that
compares two quantities
that are measured in
different units.
c. Pheasant:
1.3
Rates and Proportions
•
25
There are many situations in which you need to convert measurements to different units.
To convert a measurement means to change it to an equivalent measurement in
different units. Converting measurements can help you compare rates. When the units of
measure are the same, you can more easily compare the rates.
The table shows some common measurement conversions.
Length
Weight
Capacity
Time
12 in. 5 1 ft
16 oz 5 1 lb
8 l oz 5 1 c
60 sec 5 1 min
36 in. 5 1 yd
2000 lb 5 1 t
2 c 5 1 pt
60 min 5 1 hr
3 ft 5 1 yd
4 c 5 1 qt
3600 sec 5 1 hr
5280 ft 5 1 mi
2 pt 5 1 qt
24 hrs 5 1 day
4 qt 5 1 gal
You can use the table of common measurements as rates to change one measurement to
an equivalent measurement in different units.
26
•
3. Write each amount of time in the table
a rate.
as a rate.
a. 12 in. 5 1 ft
a. 60 sec 5 1 min
b. 36 in. 5 1 yd
b. 60 min 5 1 hr
c. 3 ft 5 1 yd
c. 3600 sec 5 1 hr
d. 5280 ft 5 1 mi
d. 24 hrs 5 1 day
Chapter 1
Ratios and Rates
© Carnegie Learning
2. Write each length in the table as
You can convert the running speed of the ostrich from being represented in miles per
minute to show the units in miles per hour.
You know that the ostrich ran 22 miles in 30 minutes. You can use
a proportion to describe the ostrich’s speed in miles per hour.
32
distance
________
time
22 mi
_______
30 min
44 mi
5 _______
60 min
32
60 min 5 1 hour
5
44 mi
______
1h
The ostrich’s speed is 44 miles per hour.
60 min , to convert the ostrich’s speed
You can also use the unit rate, _______
1 hr
from miles per minute to miles per hour.
You can
represent
multiplication by
using ? or by using
parenthesis like
(22)(2).
2
60 min
60 min 5 _______
22 mi ? _______
22 mi ? _______
_______
30 min
1 hr
30 min
1
1 hr
22 ? 2 mi
5 ________
1 hr
© Carnegie Learning
44 mi
5 ______
1 hr
The ostrich’s speed is 44 miles per hour.
1.3
Rates and Proportions
•
27
You can scale up the rate for the roadrunner to describe its speed in miles per hour.
There are
3600 seconds
in 1 hour. There
are 1760 yards
in 1 mile.
3 120
distance
________
time
300 yd
_______
30 s
36,000 yd
5 _________
3600 s
3 120
1 mile < 20.5 miles
36,000 yd 3 ________
1760 yd
The roadrunner’s speed is 20.5 miles per hour.
You can use a unit rate to convert the roadrunners speed to
miles per hour.
10
yd _________
300
yd
300
3600
sec
_______ ? _________ 5 _______
? 3600 sec
30 sec
30 sec
1 hr
1 hr
1
36,000 yd
1 mi
5 _________ ? ________
1 hr
1760 yd
36,000 mi 20.5 mi
5 _________ < ________
1 hr
1760 hr
4. Write a proportion or use rates to determine the quail’s and pheasant’s speeds in
miles per hour. Use your calculator to help you.
b. Pheasant’s speed:
28
•
Chapter 1
Ratios and Rates
© Carnegie Learning
a. Quail’s speed:
5. Write the birds in order from the fastest run to the slowest run.
You can scale down the ratio for the ostrich to describe its speed in
miles per minute.
4 30
distance
________
time
22 mi
_______
30 min
__
0.73 min
5 ________
1 min
4 30
The ostrich’s speed was about 0.73 mile per minute.
Problem 3
Up and Down
© Carnegie Learning
1. Scale each common measurement up or down to determine the unknown quantity.
48 in.
12 in. 5 ______
a. ______
1 ft
?
?
3 ft 5 ____
b. ____
1 yd 4 yd
360 min 5 ____
?
c. ________
1 hr
6 hrs
300 cm 5 _______
100 cm
d. _______
3m
?
?
64 l oz 5 ______
e. _______
8 cups
1 cup
?
16 c 5 ____
f. ____
8 pt
1 pt
16 oz
32 oz 5 ______
g. ______
2 lb
?
5 km
1 km 5 _____
h. ______
0.6 mi
?
?
5280 ft 5 ____
i. _______
1 mi
2 mi
?
72 hours 5 ______
j. ________
3 days
1 day
1.3
Rates and Proportions
•
29
2. Use a rate and multiply to determine each measurement conversion.
a. How many quarts in 12 cups?
b. How many gallons in 16 quarts?
c. How many pounds in 2 tons?
d. How many ounces in 4 pounds?
e. How many seconds in 1 day?
© Carnegie Learning
Be prepared to share your solutions and methods.
30
•
Chapter 1
Ratios and Rates