Download 14 proportions

RATIOS, RATES, &
PROPORTIONS
MSJC ~ San Jacinto Campus
Math Center Workshop Series
Janice Levasseur
RATIOS
• A ratio is the comparison of two quantities
with the same unit.
• A ratio can be written in three ways:
– As a quotient (fraction in simplest form)
– As two numbers separated by a colon (:)
– As two numbers separated by the word “to”
• Note: ratios are “unitless” (no units)
Ex: Write the ratio of 25 miles to 40 miles in
simplest form.
What are we comparing?
miles
25 miles to 40 miles
25 miles 25 5


40 miles 40 8
Units, like factors, simplify (divide common units out)
Simplify
The ratio is 5/8 or 5:8 or 5 to 8.
Ex: Write the ratio of 12 feet to 20 feet in
simplest form.
What are we comparing?
feet
12 feet to 20 feet
12 feet
12 3


20 feet
20 5
Units, like factors, simplify (divide common units out)
Simplify
The ratio is 3/5 or 3:5 or 3 to 5.
Ex: Write the ratio of 21 pounds to 7
pounds in simplest form.
What are we comparing?
pounds
21 lbs
7 lbs
21 pounds to 7 pounds
21 3


1
7
Units, like factors, simplify (divide common units out)
Simplify
The ratio is 3/1 or 3:1 or 3 to 1.
Your Turn to try a few
RATES
• A rate is the comparison of two quantities
with different units.
• A rate is written as a quotient (fraction) in
simplest form.
• Note: rates have units.
Ex: Write the rate of 25 yards to 30
seconds in simplest form.
What are we comparing?
yards & seconds
25 yards to 30 seconds
25 yards 5 yards

30 sec
6 sec
Units can’t simplify since they are different.
Simplify
The rate is 5 yards/6 seconds.
Ex: Write the rate of 140 miles in 2 hours in
simplest form.
What are we comparing?
miles & hours
140 miles to 2 hours
140 miles
70 miles

2 hours
1 hour
Units can’t simplify since they are different.
Simplify
The rate is 70 miles/1 hour (70 miles per hour, mph).
Notice the denominator is 1 after simplifying.
Your Turn to try a few
UNIT RATES
• A unit rate is a rate in which the
denominator number is 1.
• The 1 in the denominator is dropped and
• often the word “per” is used to make the
comparison.
Ex: miles per hour  mph
miles per gallon  mpg
Ex: Write as a unit rate
20 patients in 5 rooms
What are we comparing?
patients & rooms
20 patients in 5 rooms
20 patients 4 patients

5 rooms
1 room
Units can’t simplify since they are different.
Simplify
The rate is 4 patients/1room 
Four patients per room
Ex: Write as a unit rate
8 children in 3 families
What are we comparing?
Children& families
8 children in 3 families
8 children 8 children  3 8 / 3 children 2 2 3 children



3 families 3 families  3
1 family
1 family
Units can’t simplify since they are different.
How do we write the rate with a denominator of 1?
Divide top and bottom by 3
The rate is 2 2/3 children/1 family 
2 2/3 children per family
Your Turn to try a few
PROPORTIONS
• A proportion is the equality of two ratios or
rates.
• If a/b and c/d are equal ratios or rates,
then a/b = c/d is a proportion.
• In any true proportion the cross products
are equal:
Why?
Multiply thru by the LCM
(bd)
a c (bd)

 ad = bc
b d
Simplify
 Cross products are equal!
• We will use the property that the cross
products are equal for true proportions to
x6
solve proportions.
Ex:
7 42

Solve the proportion
12 x
x 6  72
If the proportion is to be true, the cross products
must be equal  find the cross product equation:
7 42

12 x
 7x = (12)(42)
 7x = 504
 x = 72
4 n2

Ex: Solve the proportion
3
6
If the proportion is to be true, the cross products
must be equal  find the cross product equation:
4 n2

3
6
 24 = 3(n – 2)
 24 = 3n – 6
 30 = 3n
 10 = n
Check:
x2
4 8
4 10  2
 

3
6
3 6
x2
5
7

Ex: Solve the proportion
n 1 3
If the proportion is to be true, the cross products
must be equal  find the cross product equation:
5
7

n 1 3
 (5)(3) = 7(n + 1)
 15 = 7n + 7
 8 = 7n
 8/7 = n
Check:
5
8
   1
7

7
3
5
7

15 3
7
 15 
 5 3   7   7 
 
Your Turn to try a few
Ex: The dosage of a certain medication is 2 mg for
every 80 lbs of body weight. How many milligrams
of this medication are required for a person who
weighs 220 lbs?
What is the rate at which this medication is given?
2 mg
2 mg for every 80 lbs 
80 lbs
Use this rate to determine the dosage for 220-lbs by
setting up a proportion (match units) 
Let x = required dosage
2 mg
= x mg

2(220)
=
80x
80 lbs
220 lbs
 440 = 80x  x = 5.5 mg
Ex: To determine the number of deer in a game
preserve, a forest ranger catches 318 deer, tags
them, and release them. Later, 168 deer are
caught, and it is found that 56 of them are tagged.
Estimate how many deer are in the game preserve.
What do we need to find? Let d = deer population size
In the original population,
how many deer were tagged? 318
From the later catch, what is the tag rate?
56 tagged out of 168 deer
We will assume that the initial tag rate and
the later catch tag rate are the same
Set up a proportion comparing the initial tag rate to
the later catch tag rate
Initial tag rate = later catch tag rate
# initially tagged later catch # tagged

population size
later catch size
318 tagged 56 tagged

d deer
168 deer
 (318)(168) = 56d
 53,424 = 56d
56
56
 d = 954 deer in the reserve
Ex: An investment of $1500 earns $120 each
year. At the same rate, how much additional
money must be invested to earn $300 each year?
What do we need to find?
Let m = additional money to be invested
What is the annual return rate of the investment?
$120 for $1500 investment
What is the desired return?
$300
Set up a proportion comparing the current return rate
and the desired return rate
Initial return rate = desired return rate
initial return
desired return

initial investment new investment
$120 return
$300 desired return

$1500 invested ($1500  m) invested
 120(1500 + m) = (1500)(300)
 180,000 + 120m = 450,000
 120m = 270,000
Divide by 120
 m = $2250 additional needs to be invest
 new investment = $1500 + $2250 = $3750
Ex: A nurse is to transfuse 900 cc of blood over a
period of 6 hours. What rate would the nurse
infuse 300 cc of blood?
What do we need to find?
The rate of infusion for 300 cc of blood
What is the rate of transfusion?
900 cc of blood in 6 hours
Set up a proportion comparing the rate of tranfusion
to the desired rate of infusion 
But to set up the proportion we need to
know how long it takes to insfuse 300 cc of
Let h = hours required
blood 
proportion comparing the rate of tranfusion to the
desired rate of infusion 
900 cc 300 cc

6 hours h hours
 900h = (6)(300)
 900h = 1800
 h = 2 hours
Therefore, it will take 2 hours to insfuse 300 cc of
blood 
New insfusion rate = 300 cc / 2 hours 
300 cc
150 cc

2 hours 1 hours
150 cc/hour is the insfusion rate