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Lesson 38 ! page 1

Lesson 38

Ratios and Rates with Weight The use of ratios, rates, and proportions is so prevalent in everyday life that it almost passes unnoticed. We’re used to working with rates of speed, which measure distance (or other accomplishment) per unit time. In this section we work with some common rates that use weight in calculation. A rate is like a ratio, except that it compares different kinds of units and so the units do not cancel from the numerator and denominator. The idea is to see some of the mathematics that surrounds us, rather than to learn these specific concepts and formulas. Weighing Things In the U.S., weight is measured in pounds (lb.) and ounces (oz.), where 16 ounces = 1 pound. In the metric system weight is measured in milligrams (mg) , grams (g.) and kilograms (kg), where 1000 mg = 1 g and 1000 g = 1 kg. A paperclip and a dime are common objects that weigh about a gram. A kilogram is a little more than 2 pounds. (1 kilogram = 2.20462262 pounds) Comparing some measurement to weight is important in a variety of applications. Notice the use of the word “per” in rates. It is generally used where the fraction bar would go in when the information is written as a rate with numbers and units. Unit Prices (Rate) Food prices are often expressed as a rate of dollars per pound or dollars per ounce. This is called the unit price, and it gives the price for one unit of weight. Example: Ground beef costs $2.99 per pound. a. How much will 2.2 pounds cost?

$2.99 2.2 lb • = $6.58 1 1 lb

b. If a package costs $3.18, how many pounds does it contain? Round to the nearest tenth.

$2.99 x lb • = $3.18 1 1 lb 2.99x = 3.18 x = 1.1 lb

2.99x / 2.99 = 3.18 / 2.99

When you buy food products that come in different size boxes, you can often check the shelf for the best buy, because many supermarkets show the unit price for each size. The unit price makes it makes it easy to compare prices on boxes of different sizes. If you need to figure out the unit price, just divide the price by the unit of weight. Example: Find the unit price for each item. a. A box of laundry detergent that weighs 11.73 pounds costs $13.12. Find the cost per pound, rounded to the nearest cent.

$13.12 = $1.12 per pound 11.73 lb

© 2010 Cheryl Wilcox

b. The same brand costs $19.07 in a 22 pound size. Find the cost per pound, rounded to the nearest cent.

$19.07 = $0.87 per pound 22 lb

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Lesson 38 ! page 2

Medicine Dosage (Ratio) Doctors consider a variety of factors in determining medicine dosages, but an important one is the patient’s size. In general the dosage is proportional to the patient’s body weight. Both measurements are weights, but since the scale is so different, we include the appropriate units for each as we did with map scales. Example: The medication label shows that 75 to 150 milligram of medicine per kilogram of the child’s body weight per day is the appropriate dosage range for infants. If an infant weighs 7 kg (15.4 lb), what is the maximum dose per day that should be given? The rate is set up in a table so you can see the cross products, but you can probably see that you should just multiply the dose for 1 kg by 7 since the infant’s weight is 7 kg. MG KG

PACKAGE INFANT 150 mg x mg 1 kg 7 kg

x = 150 • 7 = 1050 mg

The daily dosage is usually divided into a number of smaller doses given every 6 or 8 hours. If the infant were to have 3 doses per day, each would be 1050 mg / 3 = 350 mg. Example: Pediatricians prescribe acetomenophen given every 4-6 hours at the rate of 6 mg per pound of the child’s weight. How many chewable tablets (80 mg) should be given to a child weighing 50 pounds? First compute the proper dosage: SCALE DOSAGE Acetomenophen in mg

6

d

Child’s weight in lbs

1

50

1d = 6 • 50 d = 300 mg

Then use proportions again to find the number of tablets:

SCALE DOSAGE TABLETS

1

d

Acetomenophen mg.

80

300

80d = 300 80d / 80 = 300 / 80 d = 3.75 tablets

The child should be given 3 or 4 tablets, depending on other factors. Blood Alcohol Concentration (Ratio) Another common ratio that depends on body weight is the Blood Alcohol Concentration used to determine legal driving limits for drinkers. It is basically the ratio of alcohol (by weight) to blood in the body (by weight). In general, larger people have more blood, so they can consume proportionally more alcohol. The formula, called Widmark’s formula, uses constants to adjust units, transforming fluid ounces of alcoholic beverage to ounces by weight of alcohol, and body weight to blood weight. The formula also contains a differentiating factor for men and women since they have different proportions of body fat and blood by weight. If A is the amount of (pure) alcohol consumed, and W is the person’s weight in pounds, the basic formula for men is about

7A 7.8A , and for women is about . (The formula can be improved by taking into account the number of hours since W W

alcohol consumption began and subtracting the alcohol already metabolized at the rate of 0.015 ounces per hour.)

© 2010 Cheryl Wilcox

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Lesson 38 ! page 3

Example: Compare the blood alcohol concentration of a 120 pound woman to that of a 180 pound man after one drink (equivalent to about 0.6 liquid ounces of pure alcohol), using Widmark’s formula. For the woman, A = 0.6 and W = 120:

For the man, A = 0.6 and W = 180:

( )

( )

7A 7 0.6 = = 0.023 W 180

7.8A 7.8 0.6 = = 0.039 W 120

The same amount of alcohol consumed results in very different blood alcohol concentrations for the two people, with the woman’s level much higher.

Example: How many drinks (1 drink is equivalent to 0.6 oz alcohol) can a woman weighing 110 pounds have and remain under the California legal driving blood alcohol concentration (for persons over 21) of 0.08? Set up the formula for a woman with A as a variable, and solve the equation:

7.8A = 0.08 110 7.8A = 8.8

7.8A = 0.08 • 110 110 7.8A / 7.8 = 8.8 / 7.8 110 •

A ! 1.13 Stop and think about the units for A, ounces of pure alcohol. We need to translate that into a number of drinks, each containing 0.6 ounces. We’re asking “How many 0.6’s in 1.13?” so we divide 1.13 by 0.6. Another way to think about it is as a unit conversion, where 1 drink = 0.6 oz.

1.13 oz 1 drink • ! 1.9 drinks 1 0.6 oz If drinks come only in whole numbers, she can have one drink, but shouldn’t finish the second one, to stay under the legal limit. Body Mass Index (Rate) You’ve probably seen charts like that to the right for BMI (Body Mass Index) at your doctor’s office or in fitness literature. The body mass index compares a person’s weight and height to give a general indication of whether the person is at a healthy weight or is under- or over-weight. BMI is a ratio calculated most easily in metric units. In the metric system, weight is measured in kilograms, abreviated kg. (1 kg = 2.2 lbs.) Height is measured in meters, (1 m = 3.2 feet). The formula for BMI with W = weight in kg and H = height in meters, is

W . In general, a BMI between 18 and 25 is considered a H2

healthy weight, although BMI does not account for muscle mass, and muscular individuals may have a higher BMI without being unhealthy.

© 2010 Cheryl Wilcox

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Example: a. Calculate the BMI for a man weighing 96 kg who is 1.8 meters tall.

b. The man wants to reduce his BMI to 25. What is his desired weight?

W = 25 (1.8)2

W 96 = = 29.6 2 H (1.8)2 According to the BMI scale, this person’s weight is high for his height (BMI > 25).

W = 25 • (1.8)2 = 81 kg

To adjust the formula for U.S. standard units of pounds and inches, the ratio is multiplied by 703 so that the BMI range is the same for both systems. The formula where W = weight in pounds and H = height in inches, is 703

W . H2

Example: Calculate the BMI for a person weighing 114 lb who is 64 inches tall.

703

W 114 = 703 • ! 19.6 H2 64 2

According to the BMI scale, this person’s weight is in the healthy range. Density (Rate) Some materials are heavier than others – an iron cannonball weighs more than a wooden sphere of the same size. To talk about this scientists use a rate called density. The density of a material is defined as its mass per unit volume. Scientists differentiate between mass and weight, but in everyday usage you can think of density as weight divided by volume. Since density is a scientific concept it is most often measured in metric units. Example: A gold bar with volume 150 cubic centimeters weighs 2895 grams. A silver bar of the same size weighs 1575 grams. Find the density of gold and the density of silver. To find the density, divide weight by volume:

density of gold =

2895 g 3

150 cm

= 19.3 g per cm3 density of silver =

1575 g 3

150 cm

= 10.5 g per cm3

Example: You have commissioned a crown made of 200 cm3 pure gold. Upon receiving the crown, however, you are suspicious that perhaps the jeweler has mixed some lesser metal with the gold and taken the profit for himself. You have determined that the volume of the crown is indeed 200 cm3, and that it weighs 3500 g. Is it pure gold? If the crown is pure gold, its density will be 19.3 g per cm3. If the density is not 19.3 g per cm3, it has some other metal mixed in or it is hollow somewhere.

W 3500 g = = 17.5 g per cm3 V 200 cm3 Looks like you caught that jeweler cheating.

© 2010 Cheryl Wilcox

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Lesson 38 ! page 5

Example: The density of copper is 8.9 g / cm3. If a piece of copper pipe weighs 12.3 g, what is the volume of the pipe? GRAMS CUBIC CM

DENSITY PIPE 8.9 12.3 1 x

8.9x = 12.3

8.9x / 8.9 = 12.3 / 8.9

x ! 1.4 cm

3

The volume is 1.4 cubic cm.

Example: The density of platinum is 21.45 g/cm3. If you have a bar of platinum that is 5 cm long, 3 cm wide, and 2 cm thick, how much does it weigh? Density is a rate of weight to volume, so if we know the density and volume of the bar, we can figure out the weight.

V = LWH = (5 cm)(3 cm)(2 cm) = 30 cm3 21.45 g 3

cm

=

(

xg 30 cm3

)( )

x = 21.45 30 = 643.5

The bar of platinum weighs 643.5 grams. Floating and Sinking The density of water is 1 g per cm3. Objects with a density greater than water will sink, and those with density less than water will float. You can see from the densities we calculated above that bars of copper, silver, gold, and platinum will all sink, since their densities are greater than 1 g/cm3. Example: Which of these objects will float? Each is a cylinder 200 cubic centimeters in volume. A piece of bamboo weighing 60 g.

A piece of ebony weighing 240 g.

A piece of plastic weighing 175 g.

Note that you do not need to actually divide to figure out these problems. The rate of weight per unit volume is being compared to 1. If the weight is less than 200 g (the given volume), we have a proper fraction less than 1, and the object will float. If the weight is over 200 g, the fraction is greater than 1, and the object will sink. 60 < 200, so the bamboo will float.

240 > 200, so the ebony will sink.

175 < 200, so the plastic will float.

60 1 200

175 ?3H:;8CCCCCCCCFEQI

b. Find the volume in cm3 of a pile of salt weighing 220 g.

& 0'//'*&/%!"$+ & "H64:;64CCCCCCCEEIEQF /8=26=NCCCCCCCCDUEJF #=>??CCCCCCCCCEEELEPF

*:2T8HCCCCCCCCCLELF

0S=34:64CCCCCCCEEQEDF

WH>7:;64CCCCCCCCIDEKF

03558=CCCCCCCCCLEJU

+:HX8=CCCCCCCCCEDFEPF

G3H9CCCCCCCCCCDOEUF

!:;CCCCCCCCCCEQEUF

,=3;CCCCCCCCCCEQELF 11. Density: There are two

solid Y=>;:64CCCCCCCCDLEQF metal cylinders. The small Z:;2CCCCCCCCCCJEOF 1120 g. The large cylinder is />TCCCCCCCCCCFELK

)36??&V:=CCCCCCEFEKJ

-8HH3[&W:;8CCCCCCEEEFEPJ

%@3;NCCCCCCCCEEDEIF

B>H;67CCCCCCCCEEFEJQ

$>=2S&17>4>=>2TACCCEEFEKD

© 2010 Cheryl Wilcox

12. What rates or ratios are you familiar with in everyday life? o o o o o o o o o o o o o

model building with scale map scales price per pound or ounce BMI blood alcohol concentration density of materials miles per gallon population per square mile ratio of fat calories to total calories wages in $ per hour medicine dosage __________________________ __________________________

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Lesson 38 ! page 8

Lesson 38: Ratios and Rates with Weight

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Homework 38A

Name_________________________________________

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