RATIOS, RATES, & PROPORTIONS. RATIOS A ratio is the comparison of two quantities with the same unit. A ratio can be written in three ways: –As a quotient.

RATIOS, RATES, & PROPORTIONS

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.


feet 12 feet to 20 feet

feet20feet12


2012

Simplify

53


Ex: Write the ratio of 21 pounds to 7 pounds in simplest form.


pounds 21 pounds to 7 pounds

lbs7lbs21


721

Simplify

13


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.


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.


miles & hours 140 miles to 2 hours

hours2miles140


Simplify

The rate is 70 miles/1 hour (70 miles per hour, mph).

hour1miles70

Notice the denominator is 1 after simplifying.


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


patients & rooms 20 patients in 5 rooms

rooms5patients20


Simplify

The rate is 4 patients/1room

room1patients4

Four patients per room

Ex: Write as a unit rate 8 children in 3 families


Children& families 8 children in 3 families

families3children8


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

3families3

3children8

family1children3/8

family1

children322


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


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


37

1n5 15 = 7n + 7

(5)(3) = 7(n + 1)

8 = 7n

8/7 = n

Check: 5 7

381

7

5 715 37

155 3 7

7


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 lbslbs80mg2

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 hoursSet 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

RATIOS, RATES, & PROPORTIONS. RATIOS A ratio is the comparison of two quantities with the same unit. A ratio can be written in three ways: –As a quotient.

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