ratios, rates, & proportions
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RATIOS, RATES, & PROPORTIONS
MSJC ~ San Jacinto CampusMath 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
miles40miles25
Units, like factors, simplify (divide common units out)
4025
Simplify
85
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
feet20feet12
Units, like factors, simplify (divide common units out)
2012
Simplify
53
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 pounds to 7 pounds
lbs7lbs21
Units, like factors, simplify (divide common units out)
721
Simplify
13
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
sec30yards25
Units can’t simplify since they are different.
Simplify
The rate is 5 yards/6 seconds.
sec6yards5
Ex: Write the rate of 140 miles in 2 hours in simplest form.
What are we comparing?
miles & hours 140 miles to 2 hours
hours2miles140
Units can’t simplify since they are different.
Simplify
The rate is 70 miles/1 hour (70 miles per hour, mph).
hour1miles70
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
rooms5patients20
Units can’t simplify since they are different.
Simplify
The rate is 4 patients/1room
room1patients4
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
families3children8
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
3families33children8
family1children3/8
family1
children322
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:
dc
ba (bd) (bd)
Multiply thru by the LCM
Simplify ad = bc
Cross products are equal!
Why?
• We will use the property that the cross products are equal for true proportions to solve proportions.
Ex: Solve the proportion x42
127
x42
127
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
Ex: Solve the proportion 62n
34
If the proportion is to be true, the cross products must be equal find the cross product equation:
62n
34 24 = 3n – 6
24 = 3(n – 2)
30 = 3n 10 = n
Check: 6210
34
68
34
x 2
x 2
Ex: Solve the proportion 37
1n5
If the proportion is to be true, the cross products must be equal find the cross product equation:
37
1n5 15 = 7n + 7
(5)(3) = 7(n + 1)
8 = 7n 8/7 = n
Check: 5 7
38 17
5 715 37
155 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 for every 80 lbs
lbs80mg2
Use this rate to determine the dosage for 220-lbs by setting up a proportion (match units)
lbs80mg2
Let x = required dosage
=220 lbs x mg 2(220) = 80x
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
sizecatchlatertagged#catchlater
sizepopulationtaggedinitially#
deer168tagged56
deerdtagged318
(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?$300
Set up a proportion comparing the current return rate and the desired return rate
Initial return rate = desired return rate
investmentnewreturndesired
investmentinitialreturninitial
invested)m1500($returndesired300$
invested1500$return120$
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
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
hourshcc300
hours6cc900
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
hours2cc300
hours1cc150
150 cc/hour is the insfusion rate
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