LESSON F. RATIO AND PROPORTION LESSON F. RATIO AND PROPORTION 7
8 TOPIC F PROPORTIONAL REASONING II
Overview You have already studied fractions. Now you will use fractions as you study ratio and proportion. In this lesson, you will learn the definition of ratio. You will also learn how to set up and solve proportions. Then you will see how ratio and proportion apply to real life situations. Before you begin, you may find it helpful to review the following mathematical ideas which will be used in this lesson. To help you review, you may want to work out each example. To see these Review problems worked out, go to the Overview module of this lesson on the computer. Review Simplifying a fraction Simplify this fraction to lowest terms: Answer: 5 7 0 Review Recognizing equivalent fractions Is the fraction equivalent to the fraction? 9 5 Answer: No Review Finding a fraction equivalent to a given fraction Find a fraction with denominator 00 equivalent to the fraction. 5 60 Answer: 00 LESSON F. RATIO AND PROPORTION OVERVIEW 9
Explain In Concept : Ratios, you will find a section on each of the following: How to Use a Ratio to Compare Two Quantities The Definition of Equivalent Ratios How to Use a Ratio to Represent a Rate CONCEPT : RATIOS How to Use a Ratio to Compare Two Quantities A ratio is a way to compare two quantities using division. For example, a salsa recipe calls for onions and 5 peppers. The ratio of onions to 5 peppers is to 5. This ratio can also be written by using a colon, : 5, or by using a fraction,. 5 In general, the ratio of the number a to the number b can be written in the following ways: using words using a colon using a fraction a a to b a : b b Since division by zero is undefined, the second number, b, cannot be 0. Most of the time a ratio is written as a fraction. Since a fraction represents division, this is a reminder that a ratio compares two numbers using division. Remember, the fraction means. You may find these Examples useful while doing the homework for this section. Sometimes a ratio may compare more than two numbers. You usually use colons to represent these ratios. For example, suppose there are 5 apples, 6 oranges, and bananas in a bag. The ratio of apples to oranges to bananas is 5 : 6 :. Example The ratio could be written as. But here, the is left in the denominator of the ratio to represent the number of cups of cilantro, the second quantity in the ratio.. In her salsa, Maria uses cups of tomatoes for each cup of cilantro. Find the ratio of cups of tomatoes to cups of cilantro. Since there are cups of tomatoes for cup of cilantro, the ratio of cups of tomatoes to cups of cilantro is: The ratio is. number of cups of tomatoes number of cups of cilantro = Example. What is the ratio of.5 cups of flour to.5 cups of sugar? Since there are.5 cups of flour for.5 cups of sugar, the ratio of number of cups of flour to number of cups of sugar is: number of cups of flour number of cups of sugar Notice that the ratio contains decimal numbers. To clear the decimals: 0.5 0 5 Multiply the ratio by, written as. = 0.5 0 5 5 Simplify the ratio. =.5 5 5 So, the ratio can be expressed as,, or..5 5 =.5.5 0 TOPIC F PROPORTIONAL REASONING II
. What is the ratio of cars to boat to bicycles? Example Since more than quantities are being compared, use colons to write the ratio. The ratio of cars to boats to bicycles is : :.. There are 0 students in a class. students are boys and 8 students are girls. Write the ratio of the number of girls in the class to the total number of students. Example The ratio of the number of girls in the class to the total number of students is: There are 8 girls in the class, and there are 0 students in the class. So the ratio of the number of girls to the total number of students is: the number of girls the total number of students 8 0 The Definition of Equivalent Ratios A salsa recipe calls for onions and 5 peppers per batch. The table below compares the number of onions to the number of peppers for different size batches. SALSA Batch Batches Batch Number of Onions 6.5 Number of Peppers 5 0.5 Ratio of Onions to Peppers 5 6 0.5.5 6.5 6 6 The ratios and can be simplified to : = = 0.5 5 0 0 5.5.5 = =.5.5 5 6.5 So, the fractions,, and are equivalent fractions. 5 0.5 They are also called equivalent ratios. No matter the size of the batch, the ratio of onions to peppers is always to 5. In general, to find a ratio equivalent to a given ratio: Multiply or divide the numerator and denominator of the given ratio by the same number. LESSON F. RATIO AND PROPORTION EXPLAIN
Ratios can be used to compare quantities. For example, ratios can be used to compare amounts of money, quantities of time, lengths, or weights. In order to use a ratio to do such comparisons, the quantities being compared need to have the same units of measurement. For example, to write a ratio comparing dollars to cent, you must first write both quantities using the same unit. Since dollar = 00 cents, dollars = 00 cents. 00 cents 00 So, the ratio of dollars to cent becomes the ratio or. cent In order to write this ratio, dollars was changed to 00 cents. But it would also have been correct to change cents into dollars. Here s how: You can write cent as part of a dollar, like this: $0.0. Now you are working with decimals instead of whole numbers. dollars The ratio of dollars to cent becomes the ratio or. 0.0 dollars 0.0 To get rid of the decimal, multiply the numerator and denominator by 00: 00 00 = = 0.0 0.0 00 So, the ratio of dollars to cent is still 00 to. In general, if you d like to work with a ratio of whole numbers instead of decimals, write both quantities using the smaller unit. (For example, use cents instead of dollars, minutes instead of hours, inches instead of feet, etc.) You may find these Examples useful while doing the homework for this section. Example 5 5. A recipe calls for onions for every 5 peppers. If you want to make the receipe using 8 onions, how many peppers do you need? To find how many peppers you need: Write the ratio of onions to peppers. 5 8 Find an equivalent fraction with 8 as the numerator. = 5? 6 8 Multiply the numerator and denominator by 6. = = 5 5 6 0 The denominator of the equivalent fraction is the number of peppers you need. 0 So, you need 0 peppers. TOPIC F PROPORTIONAL REASONING II
6. There are a total of apples and oranges in a bowl. The ratio of apples to oranges is to. How many apples and how many oranges are in the bowl? Example 6 To find how many apples and how many oranges are in the bowl: Start with 7 pieces of fruit since the ratio of apples to oranges is to. Add fruit 7 pieces at a time until you have pieces of fruit. Count the number of apples and oranges: 9 apples and oranges. So, there are 9 apples and oranges in the bowl. 7. Write a ratio to compare hours to 5 minutes. Example 7 To find the ratio of hours to 5 minutes: Write both quantities using the same unit. Here, write hours using minutes. hour = 60 minutes. So: hours = 60 minutes = 0 minutes hour = 60 minutes = 0 minutes hours = 50 minutes hours 50 minutes 0 Write the ratio = = 5 minutes 5 minutes 0 So, the ratio of hours to 5 minutes is. 8. Write a ratio to compare ounces to pounds. 0 Here, can be written as 0. But the is left in the denominator of the ratio to represent the second quantity in the ratio. Example 8 To find the ratio of ounces to pounds: Write both quantities using the same unit. Here, write pounds using ounces. pound = 6 ounces. So: pounds = 6 ounces = ounces ounces ounces Write the ratio. = = pounds ounces So, the ratio of ounces to pounds is. LESSON F. RATIO AND PROPORTION EXPLAIN
How to Use a Ratio to Represent a Rate Often a ratio is used to compare quantities that have very different units. These ratios are sometimes called rates. Here are two examples. Suppose you travel 60 miles for each hour you drive. You can use a ratio to compare the distance, 60 miles, to the time, hour. The word per tells you that you use division to compare miles to hours. Notice, in each example, the rate was written as a ratio with denominator. 60 miles The ratio of 60 miles to hour is. This rate is usually read 60 miles per hour. hour Suppose you buy a 5 pound bag of apples for $.00. You can use a ratio to compare your cost, $.00, to the weight, 5 pounds. $.00 $.00 5 $0.60 The ratio of dollars to 5 pounds is = =. 5 pounds 5 pounds 5 pound By writing the ratio with denominator, you see that the apples cost $0.60 for pound. That is, the apples cost $0.60 per pound. These examples suggest a way to use a ratio to find a rate. Here s how: Write the ratio. Find an equivalent ratio with in the denominator. Read the rate. You may find these Examples useful while doing the homework for this section. Example 9 9. Suppose daisies cost $ per dozen. What is the cost per daisy? Here s one way to find the cost per daisy: the cost of the daisies Write the ratio. = the number of daisies Find an equivalent ratio with in the denominator. = = $.00 daisies $.00 daisies $0.5 daisy Read the rate. = $0.5 per daisy So, the cost is $0.5 per daisy. Example 0 0. Suppose you earn $5 for working 0 hours. Find your pay rate in dollars per hour. Here s one way to find the pay rate: the pay for 0 hours Write the ratio. = 0 hours Find an equivalent ratio with in the denominator. = = $5 0 hours $5 0 0 hours 0 $6.5 hour Read the rate. $6.5 per hour So, the pay rate is $6.5 per hour. TOPIC F PROPORTIONAL REASONING II
Explain In Concept : Proportions, you will find a section on each of the following: How to Solve a Proportion How to Set Up a Proportion How to Set Up and Solve a Proportion with Similar Triangles CONCEPT : PROPORTIONS How to Solve a Proportion A proportion is a statement that shows one ratio equal to another ratio. For example, a menu is planned so there are grams of fat for every 9 grams of carbohydrates. That is, the ratio of grams of fat to grams of carbohydrates is. 9 If the entire meal ends up having grams of fat, then it has 8 grams of carbohydrates. Here, the ratio of grams of fat to grams of carbohydrates is. 8 The ratios and are equivalent fractions. = 8 9 8 9 Here s another way to write the proportion =. 8 9 Use colons to write each ratio. : 8 = : 9 Replace the equals sign with two colons. : 8 :: : 9 The equation = is an example of a proportion. 8 9 A proportion is made up of four numbers. If you know three of the numbers in the proportion then you can find the fourth number. Here s one way to find a missing value in a proportion if the other three values are known: Find the cross products of the proportion and set them equal to each other. (Two ratios are equal if their cross products are equal. Thus, in a proportion, the cross products are equal.) Solve the resulting equation for the missing value. That is, get the missing value by itself on one side of the equation. You may find these Examples useful while doing the homework for this section. Example x. Find the missing number, x, that makes this proportion true. = 9 Here s one way to find the missing number, x, that makes this proportion true: ` Find the cross products and set them equal to each other. x 9 = Solve the equation for x. 9x = 9x To get x by itself, divide both sides of the equation by 9. = 9 9 A dot,, is used to represent multiplication. x= 9 Rewrite as a mixed numeral. x = 9 So, x =. 9 LESSON F. RATIO AND PROPORTION EXPLAIN 5
Example 5. Find the missing number, x, that makes this proportion true. = x Here is one way to find the missing number, x, that makes this proportion true: Find the cross products and set them equal to each other. 5 x = Solve the equation for x. 5x = To get x by itself, divide both sides of 5x the equation by 5. = 5 5 x= 5 Rewrite as a mixed numeral. x = 8 5 So, x = 8. 5 Example 6. 6. Find the missing number, x, that makes this proportion true. = 0 x Here s one way to find the missing number, x, that makes this proportion true: Find the cross products and set them equal to each other. 6 0 = 6. x Solve the equation for x. 60 = 6.x 60 6.x To get x by itself, divide both sides of the equation by 6.. = 6. 6. 60 = x 6. Simplify. 5 = x So, x = 5. Example 7 x. Find the missing number, x, that makes this proportion true. = 0 5 Here s one way to find the missing number, x, that makes this proportion true: Find the cross products and set them equal to each other. x 0 = 5 7 x 0 = 0 Solve the equation for x. 0x = 7 To get x by itself, multiply both sides of the 0 0x = 0 0 7 equation by. 0 Simplify. x = So, x =. 7 x = 7 7 0 70 5 6 TOPIC F PROPORTIONAL REASONING II
How to Set Up a Proportion Here s an example of an application that can be solved using a proportion. The nutritional label on a certain can of soda says there are 50 calories in ounces of the soda. How many calories are there in 7 ounces of the soda? ounces This proportion can be used to answer the question: = 50 calories Here are some other ways to set up a proportion to answer the question: ounces 50 calories 7 ounces = = 7 ounces x calories ounces Here s the result of cross multiplying each of these proportions: 50 7 = x x = 7 50 050 = x x = 050 And here s the result of cross multiplying the first proportion: ounces 50 calories = 7 ounces x calories 7 50 = x 050 = x In each case you get the same result. So, you can use any of the proportions to answer the question. In general, given a proportion with one missing number, say x, and another proportion also involving x, you can determine if the two proportions give the same value for x. Here s how: In the given proportion, cross multiply. In the other proportion, cross multiply. Compare the equations. 7 ounces x calories x calories 50 calories x 5. Which proportion below will not give the same value for x as the proportion =? 9 9 9 9 = = = x x x To determine which proportion will not give the same value for x as to the given proportion: x In the given proportion, cross multiply. = 9 Example 5 You may find these Examples useful while doing the homework for this section. x 9 = 9x = In the other proportions, cross multiply. 9 9 9 = = = x x x 9 x = 9 x = 9 = x 9x = 9x = 99 = x Compare the equations. The given proportion and the first two choices each give the same equation. But the equation 99 = x is different from the other equations. 9 So = will not give the same value for x as the given proportion. x LESSON F. RATIO AND PROPORTION EXPLAIN 7
Example 6 0 6. Which proportion below will not give the same value for x as the proportion =? 7 x 7 0 7 x = = = x 0 x 7 0 To determine which proportion will not give the 0 same value for x as = : 7 x 0 In the given proportion, cross multiply. = 7 x 7 = 0 x In the other proportions, cross multiply. = 0x 7 0 7 x = = = x 0 x 7 0 0 x = 7 7 0 = x 7 = x 0 0x = 70 = x = 0x Compare the equations. The given proportion and the first and third choices each give the same equation. But the equation 70 = x is different from the other equations. 7 So the proportion = will not give the same value for x 0 x 0 as the proportion =. 7 x Example 7 7. Suppose there are 6 calories in 8 ounces of juice. Find the number of calories, x, in.5 ounces of juice. One way to find the number of calories in.5 ounces of juice is to use a proportion: Write the ratio of 8 ounces of juice to 6 calories. Write the ratio of.5 ounces of juice to x calories. 8 ounces Set the two ratios equal to each other. = 6 calories Solve the proportion. Cross multiply..5 6 = 8 x 50 = 8x 50 8x 8 8 8.5 = x Solve for x. Divide both sides of the equation by 8. = So, there are 8.5 calories in.5 ounces of juice..5 ounces x calories 8 TOPIC F PROPORTIONAL REASONING II
8. On a map, of an inch represents 00 yards. Find the actual distance, x, from point A 8 Example 8 to point B if the distance on the map between these two points is inches. One way to find the distance from point A to point B is to use a proportion: Write the ratio of inches to 00 yards. 8 Write the ratio of inches to x yards. inch 8 inches Set the two ratios equal to each other. = 00 yards x yards Solve the proportion. Cross multiply. 00 = x 8 00 = x 8 8 8 Solve for x. Multiply both sides of the equation 00 = x 8 8 by, the reciprocal of. 8 800 = x So, the actual distance from point A to point B is 800 yards. How to Set Up and Solve a Proportion with Similar Triangles Look at the triangles below. side: inch longest side: inches side:.5 inches side: inch longest side: 6 inches side:.5 inches These ratios compare the lengths of the corresponding sides of the triangles. length of side of small triangle length of side of large triangle length of side of small triangle length of side of large triangle length of longest side of small triangle length of longest side of large triangle inch = = inches.5 inches = =.5 inches inches = = 6 inches The lengths of the corresponding sides of the two triangles are in the same ratio. Two such triangles are called similar triangles. Given similar triangles, if you are missing the length of the side of one of the triangles, here s a way to find that length: Write a proportion comparing the lengths of corresponding sides. Solve the proportion for the missing length. LESSON F. RATIO AND PROPORTION EXPLAIN 9
You may find these Examples useful while doing the homework for this section. Example 9 9. The triangles below are similar triangles. Which proportion below will help you find x, the length of the longest side of the large triangle? x x = = =.5.75 side: side:.75 side: side:.5 longest side: longest side: x Here s how to find the proportion that will help find the length x: Notice that the first proportion compares to x, which are not lengths of corresponding sides. The same is true for the third proportion. The second proportion compares the lengths of corresponding sides. Here s how: length of longest side of large triangle length of longest side of small triangle x So, the second proportion will help you find the length x. = = length of side of large triangle length of side of small triangle Example 0 0. The triangles below are similar triangles. Find x, the length of the side of the large triangle. side: longest side: 7 side:.5 longest side: 0.5 side: 5 side: x Here s one way to find the length, x: length of longest side of large triangle Write a = length of longest side of small triangle proportion comparing the lengths of 0.5 x corresponding sides. = 7 5 Solve the proportion. Cross multiply. x 7 = 0.5 5 7x = 5.5 Solve for x. Divide both 7x 5.5 = sides of the equation by 7. 7 7 x = length of side of large triangle length of side of small triangle 5.5 7 So, the length x is 7.5. x = 7.5 50 TOPIC F PROPORTIONAL REASONING II
. A man, 6 feet tall, is standing feet from a street light. The length of his shadow produced by the street light is feet. Find the height, x, of the street light. Example x 6 feet feet feet You can use similar triangles to find the height of the street light. Here s how: Draw and label two similar triangles. x feet + feet = 8 feet 6 feet feet Notice that even though you aren t given the lengths of two of the sides, you can still find x. Write a length of vertical side of large triangle length of horizontal side of large triangle proportion = length of vertical side of small triangle length of horizontal side of small triangle comparing the x 8 lengths of corresponding sides. = 6 Solve the proportion. Cross multiply. 8 6 = x 68 = x Solve for x. Divide both sides 68 x of the equation by. = = x So, the height of the street light is feet. LESSON F. RATIO AND PROPORTION EXPLAIN 5
5 TOPIC F PROPORTIONAL REASONING II
Explore This Explore contains two investigations. Inverting a Ratio Similar Rectangles You have been introduced to these investigations in the Explore module of this lesson on the computer. You can complete them using the information given here. Investigation : Inverting a Ratio The areas in which people live can be catagorized in two different ways: metropolitan and rural. A metropolitan area consists of towns and/or cities where business or industry provides a majority of the jobs for the residents of the area. A rural area may consist of some small towns, and agriculture provides most of the jobs for area residents. acre In a certain region, the ratio of undeveloped land to developed land is or. 9 acres 9. Write some possible numbers of acres of undeveloped and developed land that satisfy this ratio: acres of undeveloped land acres of developed land = 9. Interpret the data in question. That is, describe the setting. Could the setting be a metropolitan area? Could it be a rural area?. What if the values were inverted? That is, consider this ratio: acres of undeveloped land acres of developed land What kind of setting would this ratio represent? =. Examine some ratios in your own community in this same way. Some possible quantities to compare are the number of bicycles on a school campus to the number of cars on a school campus, the number of homes for sale in a neighborhood to the number of homes not for sale, etc. Write your ratios. 5. Interpret the data. What do the ratios say about the situation? 6. Invert your ratios. 9 7. Interpret this new data. That is, what do the inverted ratios say about the situation? LESSON F. RATIO AND PROPORTION EXPLORE 5
Investigation : Similar Rectangles. A rectangular piece of land has a length of 6 meters and a width of meters. Let the length of each side of a square on a piece of graph paper represent meter. Draw a scale drawing of this piece of land. Label each side with the appropriate measurement.. The perimeter of a figure is the distance around the figure. For instance, if you wanted to put a fence around the piece of land in question, you would want to know its perimeter. Find the perimeter of the piece of land by adding the lengths of the four sides. Perimeter = meters. The area of a figure measures the space inside the boundary of the figure. For example, if you were going to cover the entire piece of land in the rectangle in question with cement, you would want to know its area. One way to find area is to draw a grid on the figure and count the number of by squares it takes to precisely fill the figure. So, find the area of the rectangular piece of land by counting the number of by squares inside the rectangle you drew in question. Area = square meters. Now you will use graph paper to draw some other scale drawings of the piece of land described in question. Each small square on the graph paper is unit long by unit wide. The area of each small square is square unit. If you let unit represent 6 meters, then each square on the graph paper represents a square 6 meters long by 6 meters wide. Figure shows a drawing of the piece of land using this scale. The first row in the chart below shows the length, width, and perimeter (in units), and area (in square units) of this scale drawing. Now draw three more scale drawings and label them Drawing, Drawing, and Drawing. In Drawing, let unit represent meters. In Drawing, let unit represent meters. In Drawing, let unit represent meter. Use your drawings to fill in the rest of this chart. Drawing Drawing Number Length Width Perimeter Area (in units) (in units) (in units) (in square units) 6 0 units 6 units Actual piece of land 6 meters meters 0 meters 86 square meters Figure 5 TOPIC F PROPORTIONAL REASONING II
5. Find the ratio in lowest terms of length to width for each of the scale drawings and for the actual piece of land. Length (in units) 6 Drawing : = = Width (in units) Drawing : Length (in units) = Width (in units) = Length (in units) Drawing : = Width (in units) = Length (in units) Drawing : = Width (in units) = Actual piece of land: Length (in meters) Width (in meters) = = 6. For a scale drawing where the length of one square represents 0 meters, what is the ratio in lowest terms of length (in units) to width (in units)? 7. Now, find the ratio in lowest terms of the perimeter of each scale drawing to the perimeter of the actual piece of land. Perimeter of Drawing (in units) Perimeter of Actual Piece of Land (in meters) 0 = = 0 6 Perimeter of Drawing (in units) Perimeter of Actual Piece of Land (in meters) = = Perimeter of Drawing (in units) Perimeter of Actual Piece of Land (in meters) = = Perimeter of Drawing (in units) Perimeter of Actual Piece of Land (in meters) = = 8. For a scale drawing where the length of one square represents 0 meters, what is the ratio in lowest terms of the perimeter of the scale drawing (in units) to the perimeter of the actual piece of land (in meters)? LESSON F. RATIO AND PROPORTION EXPLORE 55
9. Now, find the ratio in lowest terms of the area of each scale drawing to the area of the actual piece of land. Area of Drawing (in square units) Area of Actual Piece of Land (in square meters) = = 86 6 Area of Drawing (in square units) Area of Actual Piece of Land (in square meters) Area of Drawing (in square units) Area of Actual Piece of Land (in square meters) Area of Drawing (in square units) Area of Actual Piece of Land (in square meters) = = = = = = Observe that the ratio of the area of each scale drawing to the area of the actual piece of land is: Perimeter of Scale Drawing (in units) Perimeter of Actual Piece of Land (in meters) Perimeter of Scale Drawing (in units) Perimeter of Actual Piece of Land (in meters) 0. For a scale drawing where the length of one square represents 0 meters, what is the ratio in lowest terms of the area of the scale drawing (in square units) to the area of the actual piece of land (in square meters)? 56 TOPIC F PROPORTIONAL REASONING II
Homework CONCEPT : RATIOS How to Use a Ratio to Compare Two Quantities For help working these types of problems, go back to Examples in the Explain section of this lesson.. Write a fraction that expresses the ratio of onions to 7 peppers.. Write a fraction that expresses the ratio of 6 squash to potatoes.. Write a fraction that expresses the ratio of 7 girls to boys.. Write a fraction that expresses the ratio of 7 girls to 0 students. 5. Write a fraction that expresses the ratio of 5 new cars to 9 used cars. 6. Write a fraction that expresses the ratio of 6 bicycles to 9 tricycles. 7. Write a fraction that expresses the ratio of 5 dogs to 0 cats. 8. Write a fraction that expresses the ratio of 5 cows to 5 horses. 9. Write a fraction that expresses the ratio of 0 full time instructors to 7 part-time instructors. 0. Write a fraction that expresses the ratio of inches to inches.. Write a fraction that expresses the ratio of 5 ounces to 8 ounces.. Write a fraction that expresses the ratio of 5 cups to cups.. Write a fraction that expresses the ratio of.5 quarts to 5 quarts.. Write a fraction that expresses the ratio of.8 ounces to ounces. 5. Use colons (:) to write the ratio of 5 red marbles to 7 white marbles to 7 blue marbles. 6. Use colons (:) to write the ratio of ducks to 5 chickens to geese. 7. There are oranges and 8 apples in a bowl. Write the ratio of the number of oranges to the number of apples. 8. There are oranges and 8 apples in a bowl. Write the ratio of the number of apples to the number of oranges. 9. In a class of 0 students, 9 are girls and are boys. Write the ratio of the number of girls to boys. 0. In a class of 0 students, 9 are girls and are boys. Write the ratio of the number of boys to girls.. In a bag of 8 marbles, there are 0 blue marbles, 7 green marbles, 8 black marbles, and red marbles. Write the ratio of the number of blue marbles to the number of red marbles.. In a bag of 8 marbles, there are 0 blue marbles, 7 green marbles, 8 black marbles, and red marbles. Write the ratio of the number of green marbles to the number of black marbles.. A recipe calls for cups of flour and cups of sugar. Write the ratio of the number of cups of flour to the number of cups of sugar.. A recipe calls for cups of flour and cups of sugar. Write the ratio of the number of cups of sugar to the number of cups of flour. LESSON F. RATIO AND PROPORTION HOMEWORK 57
The Definition of Equivalent Ratios For help working these types of problems, go to Examples 5 8 in the Explain section of this lesson. 5. Write a ratio to compare inches to feet. (Hint: inches = foot) 6. Write a ratio to compare yards to feet. (Hint: feet = yard) 7. Write a ratio to compare.5 quarts to gallons. (Hint: quarts = gallon) 8. Write a ratio to compare 7 cups to quarts. (Hint: cups = quart) 9. Write a ratio to compare 5 ounces to pounds. (Hint: 6 ounces = pound) 0. Write a ratio to compare. pounds to 9 ounces. (Hint: 6 ounces = pound). Write a ratio to compare yards to 5 feet. (Hint: feet = yard). Write a ratio to compare 5 inches to feet. (Hint: inches = foot). Write a ratio to compare 6 minutes to hours. (Hint: 60 minutes = hour). Write a ratio to compare 8 hours to 8 minutes. (Hint: 60 minutes = hour) 5. Write a ratio to compare cup to pints. (Hint: cups = pint) 6. Write a ratio to compare cups to quarts. (Hint: cups = quart) 7. Write a ratio to compare seconds to minutes. (Hint: 60 seconds = minute) 8. Write a ratio to compare hours to 50 seconds. Hint 600 seconds = hour) 9. Write a ratio to compare.75 miles to 000 yards. (Hint: 760 yards = mile) 0. Write a ratio to compare.8 miles to 00 feet. (Hint: 580 feet = mile). A recipe calls for onions and 7 potatoes. How many onions are needed if potatoes are used?. A recipe calls for stalks of celery and 5 carrots. How many carrots are needed if 8 stalks of celery are used?. Monia is filling gift bags for a child s party. In each bag, she wants to include erasers for every pencils. How many erasers does she need if she has pencils?. Ken is packing boxes of fruit. In each box he packs, he wants to include oranges for every 5 apples. If he has 5 oranges, how many apples does he need to complete his task? 5. A bowl contains a total of 8 apples and oranges. The ratio of the number of apples to the number of oranges is to. How many apples and how many oranges are in the bowl? 6. A bowl contains a total of oranges and bananas. The ratio of the number of oranges to the number of bananas is to 5. How many oranges and how many bananas are in the bowl? 7. There are a total of 05 cows and horses in a pen. The ratio of the number of cows to the number of horses is to. How many cows and how many horses are in the pen? 8. There are a total of chickens and ducks in a pen. The ratio of the number of chickens to the number of ducks is 5 to. How many chickens and how many ducks are in the pen? 58 TOPIC F PROPORTIONAL REASONING II
How to Use a Ratio to Represent a Rate For help working these types of problems, go to Examples 9 0 in the Explain section of this lesson. 9. Ernie loses 8 pounds in weeks. Find his weight loss per week. 50. Gladys loses pounds in 6 months. Find her weight loss per month. 5. Rachel earns $5 in 6 hours. Find her rate of pay in dollars per hour. 5. Simon earns $8 in 0 hours. Find his rate of pay in dollars per hour. 5. A certain car uses gallons of gas to travel 9 miles. Find the miles traveled per gallon. 5. A certain van uses 5 gallons of gas to travel.5 miles. Find the miles traveled per gallon. 55. A 500 square foot house costs $,500 to build. Find the cost per square foot. 56. An 875 square foot house costs $78,5 to build. Find the cost per square foot. 57. Jody travels 60 miles in 8 hours. Find her rate in miles per hour. 58. Jaime travels 675 miles in 9 hours. Find his rate in miles per hour. 59. A fish swims 8 feet in 0 seconds. Find its rate in feet per second. 60. Jonlyn walks miles in 5 minutes. Find her rate in miles per minute. 6. 5 pounds of bananas cost $.67. Find the price per pound of bananas. 6. 7 pounds of oranges cost $.. Find the price per pound of oranges. 6. Kyle cleans the house where he lives. If he can clean 8 rooms in hours, what is his cleaning rate in rooms per hour? 6. Kelly cleans stalls for a horse ranch. If she can clean stalls in 8 hours, what is her cleaning rate in stalls per hour? 65. A dozen pens cost $.79. Find, to the nearest cent, the price per pen. 66. A dozen pencils cost $0.50. Find, to the nearest cent, the price per pencil. 67. A box of paper contains 5 reams of paper. If the box costs $, what is the price per ream of paper? 68. A box of paper contains 5 reams of paper. If the box costs $6, what is the price per ream of paper? 69. In a certain rain storm it rained 8 inches in 0 hours. What is the rate of rainfall in inches per hour? 70. In a certain snow storm it snowed 6 feet in 5 hours. What is the rate of snowfall in feet per hour? 7. There are problems on a test. If it takes Elizabeth 0 minutes to finish the test, what is her rate in problems per minute? 7. There are 5 questions on a test. If it takes Ed 6 minutes to finish the test, what is his rate in questions per minute? LESSON F. RATIO AND PROPORTION HOMEWORK 59
CONCEPT : PROPORTIONS How to Solve a Proportion For help working these types of problems, go to Examples in the Explain section of this lesson. 7. Find the missing number, x, that makes this proportion true: x = 7 7. Find the missing number, x, that makes this proportion true: x = 5 75 75. Find the missing number, x, that makes this proportion true: 9 = 8 x 76. Find the missing number, x, that makes this proportion true: 7 = 0 x 77. Find the missing number, x, that makes this proportion true: x = 5 5 78. Find the missing number, x, that makes this proportion true: x 9 = 6 79. Find the missing number, x, that makes this proportion true: 6 = x 5 80. Find the missing number, x, that makes this proportion true: 75 = x 8. Find the missing number, x, that makes this proportion true: 6 = x 8. Find the missing number, x, that makes this proportion true: 7 x = 5 8 8. Find the missing number, x, that makes this proportion true: = 5 x 6 x 8. Find the missing number, x, that makes this proportion true: = 6 x 5 85. Find the missing number, x, that makes this proportion true: = 5 0 x 86. Find the missing number, x, that makes this proportion true: = 5. 8 87. Find the missing number, x, that makes this proportion true: = 7 x.5 8 88. Find the missing number, x, that makes this proportion true: = x 5 89. On a map, inch represents 5 miles. Find the actual distance, x, from point A to point B if these two points are.5 inches apart on.5 the map. To answer the question, solve this proportion for x: = 5 x 90. On a map, inch represents 8 miles. Find the actual distance, x, from point A to point B if these two points are.5 inches apart on.5 the map. To answer the question, solve this proportion for x: = 8 x 9. Carl is placing cut-up turkey in freezer storage bags. For every drumsticks he puts in a bag, he puts in wings. If he has 6 6 drumsticks, how many wings does he have? To answer this question, solve this proportion for x: = x 9. Jane is making a nut mix for a backpacking trip. For every cups of peanuts in her mix, she includes cup of cashews. If she has cups of peanuts, how many cups of cashews does she have? To answer this question, solve this proportion for x: = x 60 TOPIC F PROPORTIONAL REASONING II
9. Steven is on a diet that requires him to eat grams of protein for every grams of carbohydrates. If his lunch contains grams of carbohydrates, how many grams of protein should he include to maintain the proper ratio? To answer this question, solve this x proportion for x: = 9. Erica is on a diet that requires her to eat grams of protein for every 5 grams of carbohydrates. If her breakfast contains grams of protein, how many grams of carbohydrates should she include to maintain the proper ratio? To answer this question, solve this proportion for x: = 5 x 95. A soup recipe that feeds people calls for cups of broccoli. How many cups of broccoli will be needed to make enough soup to 9 feed 9 people? To answer this question, solve this proportion for x: = x 96. A soup recipe that feeds 6 people calls for 5 cups of potatoes. How many people can be fed with a soup that contains 5 cups of 6 x potatoes? To answer this question, solve this proportion for x: = 5 5 How to Set Up a Proportion For help working these types of problems, go to Examples 5 8 in the Explain section of this lesson. 7 97. Which proportion below will not give the same value for x as the proportion =? x 5 x 5 x 7 x 7 = = = 7 5 5 8 6 98. Which proportion below will not give the same value for x as the proportion =? x 0 x 8 x 8 6 0 = = = 6 0 0 6 8 x x 99. Which proportion below will not give the same value for x as the proportion =? 6 6 x = = = 6 x x 6 x 8 00. Which proportion below will not give the same value for x as the proportion =? 7 x 8 8 7 7 = = = 7 x x 8 0 x 0. Which proportion below will not give the same value for x as the proportion =?.5.7 0 x.5 x.7.5 = = =.7 0.5 x 0.7.8 x 0. Which proportion below will not give the same value for x as the proportion =? 0 5.6 x 0.8 0 x 5.6 = = =.8 5.6 x 5.6.8 0 7.. 0. Which proportion below will not give the same value for x as the proportion =? x 5 x 5 x 7. x 7. = = = 7... 5 5. x 9. 0. Which proportion below will not give the same value for x as the proportion =?.8 x.8 x 9. = = = 9. 9..8.8 x 5 05. Which proportion below will not give the same value for x as the proportion =? x 5 x 5 x 5 5 = = = 5 x 5 5 LESSON F. RATIO AND PROPORTION HOMEWORK 6
8 x 06. Which proportion below will not give the same value for x as the proportion =? x x = = = 8 8 8 x 5 07. Which proportion below will not give the same value for x as the proportion =? x 5 5 5 x x = = = x 5 5 5 x 08. Which proportion below will not give the same value for x as the proportion =? 6 8 7 8 9 7 7 6 x 7 9 7 7 9 6 9 = = = 6 x x 8 7 5 8 7 09. A scale drawing of a floor plan uses inch to represent 5 feet. What is the length of a dining room if it is inches long on the scale drawing? 0. A scale drawing of a floor plan uses inch to represent feet. What is the length of a living room if it is inches long on the scale drawing?. A certain car can travel 0 miles on 7 gallons of gasoline. At this rate, how far can the car travel on a full tank of gallons?. A certain van can travel 5 miles on gallons of gasoline. At this rate, how far can the van travel on a full tank of 0 gallons?. A recipe calls for onions and 5 peppers. How many onions are needed if 5 peppers are used?. A recipe calls for cups of tomatoes and cup of celery. How many cups of tomatoes are needed if cup of celery is used? 5. Suppose it costs $5 for 0 pounds of apples. At this rate, how much does it cost for 7 pounds of apples? 6. Suppose it costs $.07 for pounds of plums. At this rate, how many pounds of plums can you buy for $5? Round your answer to the nearest tenth of a pound. 7. Betty is taking a trip and has traveled 0 miles in hours. At this rate, how long will it take her to complete the remaining 00 miles of the trip? 8. Boris has been driving for 6 hours and has traveled 0 miles. At this rate, how far can Boris drive in another hours? 9. Rita earns $50 in a 0 hour pay period. At this rate, how much will Rita earn in 0 hours? 0. Brennan earns $60 in 0 hours. At this rate, how many hours will Brennan have to work to earn $0? 6 TOPIC F PROPORTIONAL REASONING II
How to Set Up and Solve a Proportion with Similar Triangles For help working these types of problems, go to Examples 9 in the Explain section of this lesson.. The triangles below are similar triangles. Find x, the length of the longest side of the large triangle. side: side: 5 side: 9 side: 5 longest side: 7 longest side: x. The triangles below are similar triangles. Find x, the length of the side of the large triangle. side: side: 5 side: x side: 0 longest side: 7 longest side:. The triangles below are similar triangles. Find x, the length of the side of the small triangle. side: x side: 5 side: 6 side: 0 longest side: 7 longest side: 8. The triangles below are similar triangles. Find x, the length of the side of the small triangle. side: side: x side: 5 side: 90 longest side: 7 longest side: 05 5. The triangles below are similar triangles. Find x, the length of the longest side of the large triangle. side:. side:.6 side: 6. side: 0.8 longest side:.6 longest side: x 6. The triangles below are similar triangles. Find x, the length of the longest side of the small triangle. side:.6 side: 5.8 side: side: 9 longest side: x longest side: LESSON F. RATIO AND PROPORTION HOMEWORK 6
7. The triangles below are similar triangles. Find x, the length of the side of the small triangle. side: x longest side: 8. side: 0.5 side: 9 side:. longest side: 8. The triangles below are similar triangles. Find x, the length of the longest side of the large triangle. side: 5 _ longest side: side: side: _ side: 7 _ longest side: x 9. The triangles below are similar triangles. Find x, the length of the side of the small triangle. side: 5 _ longest side: 6 side: side: x side: 7 _ longest side: 6 0. The triangles below are similar triangles. Find x, the length of the longest side of the large triangle. side: 5 longest side: 7 side: 8. side:.5 side: longest side: x. The triangles below are similar triangles. Find x, the length of the side of the large triangle. side: longest side: 6 side: side: x longest side:. side:.8. The triangles below are similar triangles. Find x, the length of the side of the large triangle. side: longest side: 6 side: side: 0. longest side: 0.6 side: x 6 TOPIC F PROPORTIONAL REASONING II
. The triangles below are similar triangles. Find x, the length of the side of the small triangle. side: x side: longest side: 9 side: 9 side: 6 longest side: 8. The triangles below are similar triangles. Find x, the length of the longest side of the small triangle. side: side: 6 longest side: x side: 8 longest side: 6 side: 8 5. The triangles below are similar triangles. Find x, the length of the side of the large triangle. side: side: side: 60 side: x longest side: 5 longest side: 00 6. The triangles below are similar triangles. Find x, the length of the side of the large triangle. side: 5 side: side: x side: 8 longest side: longest side: 5 7. A man, 6 feet tall, is standing 5 feet from a street light. The length of his shadow produced by the street light is 5 feet. Find the height, x, of the street light. Use the similar triangles below to help you. x 6 feet 5 feet + 5 feet = 0 feet 5 feet 8. A woman is standing 7.5 feet from a street light that is feet tall. The length of her shadow created by the street light is.5 feet. Find the height, x, of the woman. Use the similar triangles below to help you. feet x 7.5 feet +.5 feet = 0.0 feet.5 feet LESSON F. RATIO AND PROPORTION HOMEWORK 65
9. Ariana, who is feet tall, is standing by a tree. The tree is 5 feet tall. How long is the shadow cast by the tree if Ariana s shadow is 6 feet long? Use the similar triangles below to help you. 5 feet feet x 6 feet 0. Gabe is standing by a tree. The tree is 0 feet tall and casts a shadow of feet. How tall is Gabe if his shadow is feet long? Use the similar triangles below to help you. 0 feet x feet feet. Lisa is flying a kite. When 8 feet of string is out, the kite is 0 feet off the ground. Lisa pulls in the string until there is only 0.5 feet of string out. How high is the kite from the ground now? (Assume that the angle the string makes with the ground does not change.) Use the similar triangles below to help you. 0 feet 8 feet x 0.5 feet. Dan is flying a kite. When 96 feet of string is out, the kite is feet off the ground. Dan pulls in the string until the kite is 8 feet off the ground. How much string is out now? (Assume that the angle the string makes with the ground does not change.) Use the similar triangles below to help you. feet 96 feet 8 feet x. Sean has drawn a scale drawing of his backyard showing the location of of his hiding places. His sister, Arlene, knows that the actual distance from the first hiding place to the second is 0 feet. How far is it from the second hiding place to the third? Use the similar triangles below to help you. # 0 feet # # 5 # # Scale Drawing Actual Distance # 66 TOPIC F PROPORTIONAL REASONING II
. Liza has drawn a scale drawing of her backyard showing the location of of her hiding places. Her brother, Arty, knows that the actual distance from the first hiding place to the second is feet. How far is it from the second hiding place to the third? Use the similar triangles below to help you. # feet # # 5 # # Scale Drawing Actual Distance # LESSON F. RATIO AND PROPORTION HOMEWORK 67
68 TOPIC F PROPORTIONAL REASONING II
Evaluate Take this Practice Test to prepare for the final quiz in the Evaluate module of the computer. Practice Test. In a choir consisting of sopranos, altos, tenors, and basses, there are 9 singers. Of this number, 5 are sopranos and 6 are tenors. a. What is the ratio of the number of sopranos to the number of tenors? b. What is the ratio of the number of sopranos to the number of singers?. In a fruit and nut mix, the ratio of the number of fruits to the number of nuts is 5 to 9. Select all the choices below that will keep the mix at this same ratio. a. Add 5 fruits and 9 nuts to the mix. b. Add 5 fruits and 5 nuts to the mix. c. Add 9 fruits and 5 nuts to the mix. d. Add 0 fruits and 8 nuts to the mix.. Write a ratio to compare 7 cents to dollars.. Nancy drove 60 miles in 8 hours. Find the rate that she drove in miles per hour. 5. Choose the ratio below that forms a proportion with the ratio. 8 9 8 a. b. c. d. 7 7 7 0 0 5 6. Solve this proportion for x: = x 7. After hiking 5.6 miles, Sharon found that she was of the way along the trail. 5 5.6 Use this proportion to find x, the length of the trail in miles: = x 5 8. The two triangles shown below are similar triangles. That is, the lengths of their corresponding sides are in the same ratio. 70 Use this proportion to find x, the missing length: = x 6 side: 56 side: longest side: 70 side: 8 side: 6 longest side: x LESSON F. RATIO AND PROPORTION EVALUATE 69
70 TOPIC F PROPORTIONAL REASONING II