Free Pre-Algebra Lesson 39! page 1. d = rt. second seconds. gas mileage = miles gallon

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1 Free Pre-Algebra Lesson 39! page 1 Lesson 39 Units in Ratios and Rates For the last several lessons, we ve been exploring ratios, rates, and proportions as ways of comparing data using division. The many examples show how thinking proportionally is a very important part of how we understand data in everyday life. We ve seen that there are two types of comparisons with division, ratios and rates. In this lesson we compare and contrast those concepts so that you can use them more confidently in your own life and work. Rates We ve been using rates in this text since the distance-rate-time formula was introduced in Lesson 6. As you ve seen in many examples and homework problems, the formula really just states the relationship between the units in the problem. In the formula summary in Lesson 34, we saw: Rates Rate (Speed) is Distance over Time r = d t The related multiplication is Distance = Rate Time d = rt Examples with units: miles = miles hour hours feet feet = second seconds actions = actions minute minutes Other rates: items = items box boxes gas mileage = miles gallon servings per container = amount per container amount per serving Since units in the numerator cancel with units in the denominator, the multiplicative version d = rt could be translated numerator units numerator units = denominator units denominator units which seems almost too obvious to count as a formula. When you read a rate in words, you say the word per in place of the fraction bar. So we have miles per hour, feet per second, actions per minute, items per box, miles per gallon. The last example in the formula list, servings per container is made up of two other rates, amount per container and amount per serving. Watch how the units work in this especially complicated example: servings per container = amount container = amount amount container amount serving = amount container serving amount = servings container serving If we have any two of the numbers in a rate, we can find the third by writing a proportion and cross-multiplying. Example: Find the missing quantity in each problem. distance 50 miles time 2 hours rate r miles per hour distance 50 miles time t hours rate 25 miles per hour distance d miles time 2 hours rate 25 miles per hour 50 miles 2 hours = x miles 1 hour 50 = 2x 50 miles t hours = 25 miles 1 hour 50 = 25t t = 2 hours d miles 2 hours = 25 miles 1 hour d = 25 2 d = 50 miles x = 25 miles per hour

2 rate 25 miles per hour weight 20 g Free Pre-Algebra volume 5 Lesson cm 3 39! page 2 density d g / cm 3 Compare the distance-rate-time example to the density example below: Example: Find the missing quantity in each problem. weight 20 g volume v cm 3 density 4 g / cm 3 weight 20 g volume 5 cm 3 density d g / cm 3 20 g 5 cm = d g 3 1 cm 3 20 = 5d d = 4 g/cm 3 weight 20 g volume v cm 3 density 4 g / cm 3 20 g v cm = 4 g 3 1 cm 3 20 = 4v v = 5 cm 3 weight w g volume 5 cm 3 density 4 g / cm 3 w g 5 cm = 4 g 3 1 cm 3 w = 20 g Because the units are part of the rate, the number that we get for the rate is dependent on the units used. If you change the units of speed from miles per hour to feet per second, the speed of the car is the same but the number given in the rate is different because the units are different. If you measure density in pounds per cubic inch instead of grams per cubic centimeter the numbers you get will be different although the material is the same. If you calculate the unit price in dollars per ounce you will get a different number than euros per gram. The Body Mass Index in Lesson 38 is a rate of weight in kilograms to height in meters. The number for the BMI will change if we use weight in pounds and height in inches, which is why we used the multiplier 703 in the formula with pounds and inches. A change in unit in a rate can be used to bring large numbers down to size so we can understand them better. For example, if you know that there were 4.3 million babies born in the U.S. in 2007, you could write this fact as a rate of 4,300,000 babies per year. But you could help people understand it better if you used a smaller unit of time in the denominator. Since 1 year = 365 days, you could write 4,300,000 babies 4,300,000 babies =! 11,781 babies per day 1 year 365 days This is still to large a number for easy comprehension. Using 24 hours = 1 day, and then 60 minutes = 1 hour, we get 11,781 babies 11,781 babies =! 491 babies per hour 1 day 24 hours 491 babies 491 babies =! 8 babies per minute 1 hour 60 minutes Babies are not really born at this exact rate per minute, as if they were cars on an assembly line. Probably minutes pass in which no babies are born, and then 20 or so are born all at once. But the rate gives us a clearer feeling for what the birthrate of 4.3 million per year means. A rate like this is called an average rate because the occurrences are not really perfectly regular.

3 Free Pre-Algebra Lesson 39! page 3 Ratios A ratio is different from a rate because it compares quantities of the same kind, so that it is possible to cancel units. Instead to the word per, we usually use to as below: L = 3 inches W= 1 inch The ratio of length to width is 3 inches 1 inch = 3 1 = 3 The ratio of length to width is 3 to 1. (or 3:1) Because the units cancel in the ratio, it doesn t matter what units we use to measure the rectangle. If the same rectangle were measured in centimeters, the ratio of length to width would still be 3 to 1. L = 7.62 cm W= 2.54 cm 7.62 cm The ratio of length to width is = = cm In this case we supply a denominator of 1 to write the two parts of the ratio. It s still a ratio of 3 to 1. Because the units cancel, the ratio directly compares the length and width. The division in the ratio has a related multiplication that is helpful. length width = ratio length = ratio width The related multiplication gives us an important interpretation of the ratio: The length is 3 times the width. Because the units are the same for the numerator and denominator they don t give us a clue about which numbers go where, so it s especially important to be careful in arranging ratios so that corresponding quantities are aligned. Putting the numbers in a table is helpful. Remember Thales in Lesson 2? Thales measured the height of the pyramid by measuring the length of the pyramid s shadow. In that lesson we had to wait for the moment that shadows were the same length as the objects that cast them. Now that we know ratios, we can use shadows at any time of day. At any particular time the angle of the sun makes the ratio of the height of an object to the length of the shadow the same for all the objects in the vicinity. Example: The length of the shadow is proportional to the height of the animal casting the shadow. Duck is 3 feet tall, and right now her shadow measures 2 feet. At the same time, Raging Dino s shadow measures 2 meters long. Find Raging Dino s height. Note that this is a ratio since we are comparing the length of the shadow and the height of the animal, measured in the same units. It doesn t matter that Dino and Duck are using different units, so long as the units within the ratio are the same. Here written words really help to organize the information. The shadow measurement can go on either the top or bottom of the fraction, as long as it goes in the same place for both animals. height 3 feet shadow 2 feet = x meters 2 meters The units will cancel from the ratios. x has to be 3 to make the ratios equal. Raging Dino is 3 meters tall.

4 Free Pre-Algebra Lesson 39! page 4 If we put the information about Dino and Duck into a table, it would look like Duck Dino height 3 ft 3 m shadow 2 ft 2 m Remember that the row ratios should also be proportional? That gives us 3 feet 3 meters = 2 feet, which is true, but are these ratios? 2 meters Recall the map and model scales that worked more practically with a change of unit? The dollhouse scale is a true ratio, 1:12, and the dollhouse dimensions are 1/12 of a real house dimensions. But it s far more practical to just change the name of the unit involved, because 12 inches is 1 foot. The ratio is 1:12, but practically, we just change 5 inches to 5 feet to make a conversion. 3 feet So technically, the division is a ratio. To write the true ratio without units, we d have to convert either feet to 3 meters meters or meters to feet. As long as the conversion is possible (both units measure length), we have a (potential) ratio. If the conversion is not possible (miles are not the same kind of unit as hours, since one measures length and the other time), we have a rate. Example: Write the true ratio of 3 feet to 3 meters. (1 foot = meters) First we convert 3 feet to meters. 3 feet Then we re-write the ratio in terms of meters: 3 feet meters = meters 1 foot 3 meters = meters 3 meters = You can see that the ratio is just the conversion factor originally given. Conversion factors for changing one unit to another have a true ratio of meter 60 minutes 5280 feet = 1 = 1 = 1 1 foot 1 hour 1 mile The conversion factors can be built into a ratio formula so that you can put units in conveniently. For example, Widmark s formula for blood alcohol concentration yields a true ratio of ounces of alcohol in the blood to ounces of blood. But the multiplier (7 for men, 7.8 for women) allows us to enter weight in pounds instead of ounces of blood into the formula, because our weight in pounds is easier to find and use. Ratios of Part to Part or Part to Whole Suppose you make lemonade from a can of concentrate. The instructions tell you to add 4 cans of water to the concentrate. In the end you will have a whole pitcher of lemonade, which contains 5 total cans of liquid. You can write several different ratios for this situation. concentrate water 1 can 4 cans or concentrate lemonade 1 can 5 cans The first gives a ratio of part to part, the second a ratio of part to whole. We would say The concentrate is 1/5 of the lemonade, or There is 4 times as much water as concentrate, depending on what was important to convey in the situation.

5 Free Pre-Algebra Lesson 39! page 5 Because the two parts of a ratio must be distinguished by some criteria, it s sometimes difficult to tell if the units are the same or different. For example, if we are comparing students taking 9#,.!:0(-*!&-1033;#-*!)-!<)$*,-"#!&.(",* =,33!4#1;$!7AAB!*0!7AAC =,33!4#1;$ <)$*,-"# &.(",*)0- :033#># &-1033;#-* 7AAE 6'EG7 77'6EH 7AA8 6'GG6 76'667 7AAF 7'BAI 7A'HA8 7AAG E'BG7 7A'66H 7AAH 8'BF6 7A'FGG 7AAI F'H6F 76'GIF distance education courses to all students enrolled at the college with a quotient is that a ratio or a rate? It is a ratio, because the first quantity, students taking distance education courses is a PART of the second quantity, all students enrolled at the college. The unit is students enrolled at the college. When the first quantity in a ratio is part of the second, the ratio is essentially a fraction. It tells the fraction of all students that are taking distance education. We could also write a ratio of students taking a distance education course to students (enrolled but) not taking such a course. This would be a ratio of part to part. Ratios such as two girls for every boy in the song Surf City are ratios of part to part. Girls and boys are not different units for this ratio they are each part of the whole quantity young people in Surf City. Ratios of part to whole are often expressed as percents. That s the subject of the next lesson.!

6 Free Pre-Algebra Lesson 39! page 6 Lesson 39: Units in Ratios and Rates Worksheet Name This side of the page has RATES. Solve with proportions by cross-multiplying. 1. A gardener weeded 3 rows per hour. If she worked 6 hours, how many rows were weeded? 2. Another gardener finished 20 rows in 5 hours. What was her rate? 3. A pipe fills a tank at the rate of 19 gallons per minute. The tank holds 250 gallons. How long will it take to fill the tank? 4. The population density of a region is measured in people per unit area. Find the population density and write it with units for the city of Manila, in the Phillipines, with the data given. Population 1,660,714; Area square miles 5. In 2003, each person [in the U.S.] consumed about 142 pounds of sugar per year. U.S. News and World Report What is the amount of sugar the average person consumed per day? 6. If you look at the information in problem 5 a different way, there is a rate of 142 pounds of sugar per person. If the U.S. population in 2003 was approximately 294,043,000 people, what was the total amount of sugar consumed nationwide that year? Convert the answer to ounces (16 oz = 1 lb) per day.

7 Free Pre-Algebra Lesson 39! page 7 This side of the page has RATIOS. Solve with proportions by cross-multiplying. 4. The shadow of a tree is 40 feet at the same time and place that the shadow of a 5 foot tall woman is 6 feet long. What is the height of the tree? 5. The ratio of students taking distance ed to students enrolled is If 21,673 students are enrolled, how many are taking distance ed? 6. Fill in the blank to complete the sentence. a. The ratio of length to width of a rectangle is 1.5 to 1. The length of the rectangle is times the width. 7. Fill in the blank to complete the sentence. a. The length of the rectangle is 2 times the width. The ratio of length to width of the rectangle is to 1. b. The ratio of students who smoke to the number of students is The number of students who smoke is times the number of students. b. There are 4 times as many students enrolled as are taking distance education courses. The ratio of students enrolled to students taking distance education courses is to The length of a rectangle is 1.5 times the width of the rectangle. If the length is 4.8 meters, what is the width? 9. Find the true ratio of length to width if the length is 8.8 meters and the width is 4.4 cm. (100 cm = 1 m).

8 Free Pre-Algebra Lesson 39! page 8 Lesson 39: Units in Ratios and Rates Homework 39A Name 1. Use the nutrition label to make the comparisons. a. About how many crackers are in a container? b. How many grams per container? c. If you eat 12 crackers, how many servings have you eaten? d. How many calories in 12 crackers? Figure Figure (a) (a) Data Data points points from fromu.s. U.S. Navy Navycruises used usedbybyrpw08, and andthethe data data release release area area (irregular polygon). (b) (b) Interannual changes in winter winter and and summer ice ice thickness from fromrpw08 and and K09 K09 centered on on the the ICESat ICESat campaigns. Blue Blue error error bars bars show showresiduals in in the the regression and and quality quality of of ICESat ICESat data. data. (c, (c, d) d) Spatial Spatial patterns patterns of of ice ice thickness in in winter winter (Feb Mar) and and fall fall (Oct Dec) of of (e) (e) Mean Mean sea sea ice ice concentration at at summer summer minimum minimum ( ). (f, (f, g) g) Spatial Spatial patterns patterns of of mean meanwinter winter (Feb Mar) (Feb Mar) and andfall fall (Oct Dec) (Oct Dec) ice ice thickness thickness from fromicesat ICESat ( ). (h) (h) Mean Meanseasea ice ice concentration concentration at summer summer minimum minimum ( ). 2008). Quantities Quantities in in Figures Figures 2c, 2c, 2d, 2d, 2f, 2f, and and 2g 2g are are mean mean and and standard standard deviation deviation of of ice ice thickness thickness within within the the DRA. DRA. 2. The table is from the article Decline in Arctic Sea Ice a. Compare the mean ice thickness in Nansen Basin in Period 3 to Thickness. Table 1. Mean Use Ice the Thickness table to answer at the End the of questions. Melt Season in the the Sixthickness Regions ofin the the Arctic same Ocean location Fromin Submarine Period 1 Cruises in one or in 1958 more 1976, Table 1. Mean Ice Thickness at the End of Melt Season Table , 1. Mean and IceICESat Thickness Acquisitions at the Endinof Melt Season a in the Six Regions of the Arctic Ocean From Submarine Cruises in , , and ICESat Acquisitions a sentences. Compare directly and using the difference. Period Change Period Change Period 1, Period 2, Period 3, (2) (1) (3) (1) (3) (2) Region Period , Period , Period , (2) (1) (3) (1) (3) (2) Thickness Percent Thickness Percent Thickness Percent Region Thickness Percent Thickness Percent Thickness Percent Chukchi Cap Chukchi Beaufort CapSea Beaufort CanadaSea Basin Canada NorthBasin Pole North Nansen PoleBasin Nansen Eastern Basin Arctic Eastern All Regions Arctic All Regions a Mean ice thickness 3.02 is shown in meters, 1.62 and changes 1.43 in thickness 1.40 are shown in 46 meters and percent a Mean ice thickness is shown in meters, and changes in thickness are shown in meters and percent. c. What is the ratio of mean ice thickness in All Regions during Period 3 to that during Period 2? Round to two decimal places. b. What is the ratio of ice thickness at the North Pole in Period 1 to ice 3thickness of 5 at the North Pole in Period 3? Round to two decimal 3places. of 5 Fill in the blank: The ice at the North Pole was 1.82 times as thick during Period 1 as it was during Period 3.

9 Free Pre-Algebra Lesson 39! page 9 3. The legend of a map shows 1 inch = 1 mile. What is the real scale (the true ratio)? (1 mile = 5280 feet, 1 foot = 12 inches) 4. The triangles are similar. Find side c of the larger triangle. 5. The shadow of a bell tower is 80 feet long at the same time a person 5.5 feet tall has a shadow of 4.8 feet. How tall is the bell tower? 6. Find the population density (rate of people per square mile) in San Francisco, California. Write the units with the rate. Population 776,733; Area square miles 7. An earring made of silver with density 10.5 g/cm 3 weighs 7 grams. What is the volume of the earring? 8. A man 6 feet 1 inch tall is aiming for a BMI of 24. What is his desired weight? W (lb) BMI = 703 H 2 (inches)

10 Free Pre-Algebra Lesson 39! page 10 Lesson 39: Units in Ratios and Rates Homework 39A Answers 1. Use the nutrition label to make the comparisons. a. About how many crackers are in a container? 28 servings 1 container 5 crackers = 140 crackers per container 1 serving b. How many grams per container? 28 servings 1 container 16 grams 1 serving = 448 grams per container c. If you eat 12 crackers, how many servings have you d. How many calories in 12 crackers? eaten? 80 calories 5 crackers 12 crackers = 5 crackers = x calories 12 crackers 1 serving x servings Figure Figure (a) (a) Data Data points points from fromu.s. U.S. Navy Navycruises used usedbybyrpw08, and andthe 5x the data = data 80 release release 12 = area 960 area (irregular polygon). (b) (b) Interannual 5x changes = 12 in winter 5x winter / 5 and = and summer 12 / 5 ice ice thickness from fromrpw08 and 5x and = K K09 centered 5x on on the / 5 the ICESat = ICESat 960 campaigns. / 5 Blue Blue error error bars bars show show x = residuals residuals 2.4 in in the the regression and and quality quality of of ICESat ICESat data. data. (c, (c, d) d) Spatial Spatial patterns patterns of of ice ice thickness in in winter winter (Feb Mar) and and fall fall (Oct Dec) of of (e) (e) Mean Mean sea sea ice ice concentration atx at summer = summer 192minimum ( ). (f, (f, g) g) Spatial Spatial patterns patterns ofyou ve mean meanwinter eaten winter (Feb Mar) 2.4 (Feb Mar) servings. and andfall fall (Oct Dec) (Oct Dec) ice ice thickness thickness from fromicesat ICESat ( ). (h) (h) Mean Meanseasea ice ice concentration at summer minimum 2008). Quantities in Figures There 2c, are 2d, 192 2f, and calories 2g are mean in 12 and crackers. concentration summer minimum ( ). Quantities in Figures 2c, 2d, 2f, and 2g are mean and standard standard deviation deviation of of ice ice thickness thickness within within the the DRA. DRA. 2. The table is from the article Decline in Arctic Sea Ice a. Compare the mean ice thickness in Nansen Basin in Period 3 to Thickness. Table 1. Mean Use Ice the Thickness table to answer at the End the of questions. Melt Season in the the Sixthickness Regions ofin the the Arctic same Ocean location Fromin Submarine Period 1 Cruises in one or in 1958 more 1976, Table 1. Mean Ice Thickness at the End of Melt Season Table , 1. Mean and IceICESat Thickness Acquisitions at the Endinof Melt Season a in the Six Regions of the Arctic Ocean From Submarine Cruises in , sentences. Compare directly and using the difference , and ICESat Acquisitions in a Period Change Period The mean ice thickness Change in Nansen Basin during Period 1, Period 2, Period 3, (2) (1) (3) (1) (3) (2) Period 3 was less than the mean ice thickness Region Period , Period , Period , (2) (1) (3) (1) (3) (2) Thickness Percent Thickness Percent Thickness Percent Region Thickness during Percent Period Thickness 1. The thickness Percent declined Thickness by Percent 1.77 Chukchi Cap Chukchi Beaufort CapSea meters, from meters 64 50to meters. 290 Beaufort CanadaSea Basin Canada NorthBasin Pole North Nansen PoleBasin Nansen Eastern Basin Arctic Eastern All Regions Arctic All Regions a Mean ice thickness 3.02 is shown in meters, 1.62 and changes 1.43 in thickness 1.40 are shown in 46 meters and percent a Mean ice thickness is shown in meters, and changes in thickness are shown in meters and percent. c. What is the ratio of mean ice thickness in All Regions during Period 3 to that during Period 2? Round to two decimal places. Period 3 Period m 1.62 m! 0.88 b. What is the ratio of ice thickness at the North Pole in Period 1 to ice thickness at the North Pole in Period 3? Round to two decimal places. 3 of 5 3 of 5 Fill in the blank: Period 1 Period m 1.89 m! 1.82 The ice at the North Pole was 1.82 times as thick during Period 1 as it was during Period 3.

11 Free Pre-Algebra Lesson 39! page The legend of a map shows 1 inch = 1 mile. What is the real scale (the true ratio)? (1 mile = 5280 feet, 1 foot = 12 inches) 4. The triangles are similar. Find side c of the larger triangle. 1 mile feet 1 mile 12 inches = 63,360 inches 1 foot 1 inch 1 mile = 1 inch 63,360 inches = 1 63, The shadow of a bell tower is 80 feet long at the same time a person 5.5 feet tall has a shadow of 4.8 feet. How tall is the bell tower? height (ft) shadow (ft) 4.8h = = h 80 ( )( 80) = h = h / 4.8 = 440 / 4.8 h = 91.6 The tower is about 91.7 feet tall. 7. An earring made of silver with density 10.5 g/cm 3 weighs 7 grams. What is the volume of the earring? 10.5 g 1 cm = 7 g 3 x cm x = x / 10.5 = 7 / 10.5 x = 0.6 The volume is about 0.7 cm 3 a c 2.1 c = c = ( 2.1) ( 2.1) = c = c / 1.5 = 4.41/ 1.5 c = 2.94 Side c is 2.94 cm. 6. Find the population density (rate of people per square mile) in San Francisco, California. Write the units with the rate. Population 776,733; Area square miles 776,733 people = x people mi 2 1 mi x = 776, x / = 776,733 / x = 16, There are about 16,636 people per square mile in San Francisco. 8. A man 6 feet 1 inch tall is aiming for a BMI of 24. What is his desired weight? W (lb) BMI = 703 H 2 (inches) 6 feet 1 inch = 72 inches + 1 inch = 73 inches 24 = 703 W W 5329 = W 5329 = W = His desired weight is about 182 pounds.

12 Free Pre-Algebra Lesson 39! page 12 Lesson 39: Units in Ratios and Rates Homework 39B Name 1. Use the nutrition label to make the comparisons. a. How many brownies are in a container? b. How many grams per container? c. If you eat 2 brownies, how many servings have you eaten? d. How many calories in 2 brownies? Figure Figure (a) (a) Data Data points points from fromu.s. U.S. Navy Navycruises used usedbybyrpw08, and andthethe data data release release area area (irregular polygon). (b) (b) Interannual changes in winter winter and and summer ice ice thickness from fromrpw08 and and K09 K09 centered on on the the ICESat ICESat campaigns. Blue Blue error error bars bars show showresiduals in in the the regression and and quality quality of of ICESat ICESat data. data. (c, (c, d) d) Spatial Spatial patterns patterns of of ice ice thickness in in winter winter (Feb Mar) and and fall fall (Oct Dec) of of (e) (e) Mean Mean sea sea ice ice concentration at at summer summer minimum minimum ( ). (f, (f, g) g) Spatial Spatial patterns patterns of of mean meanwinter winter (Feb Mar) (Feb Mar) and andfall fall (Oct Dec) (Oct Dec) ice ice thickness thickness from fromicesat ICESat ( ). (h) (h) Mean Meanseasea ice ice concentration concentration at summer summer minimum minimum ( ). 2008). Quantities Quantities in in Figures Figures 2c, 2c, 2d, 2d, 2f, 2f, and and 2g 2g are are mean mean and and standard standard deviation deviation of of ice ice thickness thickness within within the the DRA. DRA. 2. The table is from the article Decline in Arctic Sea Ice a. Compare the mean ice thickness in the Chukchi Cap in Period 3 Thickness. Table 1. Mean Use Ice the Thickness table to answer at the End the of questions. Melt Season in theto Sixthe Regions thickness of thein Arctic the same Oceanlocation From Submarine Period Cruises 1 in one in or 1958 more 1976, Table 1. Mean Ice Thickness at the End of Melt Season Table , 1. Mean and IceICESat Thickness Acquisitions at the Endinof Melt Season a in the Six Regions of the Arctic Ocean From Submarine Cruises in , , and ICESat Acquisitions a sentences. Compare directly and using the difference. Period Change Period Change Period 1, Period 2, Period 3, (2) (1) (3) (1) (3) (2) Region Period , Period , Period , (2) (1) (3) (1) (3) (2) Thickness Percent Thickness Percent Thickness Percent Region Thickness Percent Thickness Percent Thickness Percent Chukchi Cap Chukchi Beaufort CapSea Beaufort CanadaSea Basin Canada NorthBasin Pole North Nansen PoleBasin Nansen Eastern Basin Arctic Eastern All Regions Arctic All Regions a Mean ice thickness 3.02 is shown in meters, 1.62 and changes 1.43 in thickness 1.40 are shown in 46 meters and percent a Mean ice thickness is shown in meters, and changes in thickness are shown in meters and percent. c. What is the ratio of mean ice thickness in All Regions during Period 2 to that during Period 1? Round to two decimal places. b. What is the ratio of ice thickness at the North Pole in Period 3 to ice thickness at the North Pole in Period 1? Round to two decimal places. 3 of 5 3 of 5 Fill in the blank: The ice at the North Pole was times as thick during Period 3 as it was during Period 1.

13 Free Pre-Algebra Lesson 39! page The legend of a map shows 1 centimeter = 0.5 kilometers. What is the real scale (the true ratio)? (1 km = 1000 m, 1 m = 100 cm) 4. The triangles are similar. Find side a of the smaller triangle. 5. The shadow of a bell tower is 90 feet long at the same time a woman 5 feet tall has a shadow of 5.5 feet. How tall is the bell tower? 6. Find the population density (rate of people per square mile) in Pleasant Hill, California. Write the units with the rate. Population 34,199; Area square miles 7. A bracelet made of silver with density 10.5 g/cm 3 weighs 80 grams. What is the volume of the bracelet? 8. A man 5 feet 4 inches tall is aiming for a BMI of 22. What is his desired weight? W (lb) BMI = 703 H 2 (inches)