Business Math ppt

1. RATIOS, RATES, &
PROPORTIONS
MSJC ~ San Jacinto Campus
Math Center Workshop Series
Janice Levasseur

2. 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)

3. Ex: Write the ratio of 25 miles to 40 miles in
simplest form.
What are we comparing?
miles 25 miles to 40 miles
miles
40
miles
25
Units, like factors, simplify (divide common units out)
40
25

Simplify
8
5

The ratio is 5/8 or 5:8 or 5 to 8.

4. Ex: Write the ratio of 12 feet to 20 feet in
simplest form.
What are we comparing?
feet 12 feet to 20 feet
feet
20
feet
12
Units, like factors, simplify (divide common units out)
20
12

Simplify
5
3

The ratio is 3/5 or 3:5 or 3 to 5.

5. Ex: Write the ratio of 21 pounds to 7 pounds in
simplest form.
What are we comparing?
pounds 21 pounds to 7 pounds
lbs
7
lbs
21
Units, like factors, simplify (divide common units out)
7
21

Simplify
1
3

The ratio is 3/1 or 3:1 or 3 to 1.

6. Your Turn to try a few

7. 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.

8. Ex: Write the rate of 25 yards to 30 seconds in
simplest form.
What are we comparing?
yards & seconds 25 yards to 30 seconds
sec
30
yards
25
Units can’t simplify since they are different.
Simplify
The rate is 5 yards/6 seconds.
sec
6
yards
5


9. Ex: Write the rate of 140 miles in 2 hours in
simplest form.
What are we comparing?
miles & hours 140 miles to 2 hours
hours
2
miles
140
Units can’t simplify since they are different.
Simplify
The rate is 70 miles/1 hour (70 miles per hour, mph).
hour
1
miles
70

Notice the denominator is 1 after simplifying.

10. Your Turn to try a few

11. 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

12. Ex: Write as a unit rate
20 patients in 5 rooms
What are we comparing?
patients & rooms 20 patients in 5 rooms
rooms
5
patients
20
Units can’t simplify since they are different.
Simplify
The rate is 4 patients/1room 
room
1
patients
4

Four patients per room

13. Ex: Write as a unit rate
8 children in 3 families
What are we comparing?
Children& families 8 children in 3 families
families
3
children
8
Units can’t simplify since they are different.
How do we write the rate with a denominator of 1?
The rate is 2 2/3 children/1 family 
2 2/3 children per family
Divide top and bottom by 3
3
families
3
3
children
8



family
1
children
3
/
8

family
1
children
3
2
2


14. Your Turn to try a few

15. 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:
d
c
b
a

(bd) (bd)
Multiply thru by the LCM
Simplify
 ad = bc
 Cross products are equal!
Why?

16. • We will use the property that the cross
products are equal for true proportions to
solve proportions.
Ex: Solve the proportion
x
42
12
7

x
42
12
7

If the proportion is to be true, the cross products must
be equal  find the cross product equation:
 7x = (12)(42)  7x = 504
 x = 72
x 6
x 6  72

17. Ex: Solve the proportion
6
2
n
3
4 

If the proportion is to be true, the cross products must
be equal  find the cross product equation:
6
2
n
3
4 
  24 = 3n – 6
 24 = 3(n – 2)
 30 = 3n
 10 = n
Check:
6
2
10
3
4 

6
8
3
4


x 2
x 2

18. Ex: Solve the proportion
3
7
1
n
5


If the proportion is to be true, the cross products must
be equal  find the cross product equation:
3
7
1
n
5

  15 = 7n + 7
 (5)(3) = 7(n + 1)
 8 = 7n
 8/7 = n
Check:
5 7
3
8
1
7

 

 
 
 
5 7
15 3
7
     
15
5 3 7
7
 
  
 
 

19. Your Turn to try a few

20. 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 for every 80 lbs
lbs
80
mg
2

Use this rate to determine the dosage for 220-lbs by
setting up a proportion (match units) 
lbs
80
mg
2
Let x = required dosage
=
220 lbs
x mg
 2(220) = 80x
 440 = 80x  x = 5.5 mg

21. 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

22. Set up a proportion comparing the initial tag rate to the
later catch tag rate
Initial tag rate = later catch tag rate
size
catch
later
tagged
#
catch
later
size
population
tagged
initially
#

deer
168
tagged
56
deer
d
tagged
318

 (318)(168) = 56d
 53,424 = 56d
56 56
 d = 954 deer in the reserve

23. 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

24. Set up a proportion comparing the current return rate and
the desired return rate
Initial return rate = desired return rate
investment
new
return
desired
investment
initial
return
initial

invested
)
m
1500
($
return
desired
300
$
invested
1500
$
return
120
$


 120(1500 + m) = (1500)(300)
 180,000 + 120m = 450,000
 120m = 270,000
 m = $2250 additional needs to be invest
 new investment = $1500 + $2250 = $3750
Divide by 120

25. 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 blood 
Let h = hours required

26. hours
h
cc
300
hours
6
cc
900

proportion comparing the rate of tranfusion to the
desired rate of infusion 
 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 
hours
2
cc
300
hours
1
cc
150

150 cc/hour is the insfusion rate