Math 6 Unit 7 Notes: Proportional relationships

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1 Math 6 Unit 7 Notes: Proportional relationships Objectives: (3.2) The student will translate written forms of fractions, decimals, and percents to numerical form. (5.1) The student will apply ratios in mathematical and practical situations involving monetary and measurement conversions. (5.2) The student will use unit cost comparisons in practical situations. (6.17) The student will determine actual measurements based upon a scale drawing. A RATIO is a comparison between two quantities. Ratios are used every day; one Pepsi costs 50 cents describes a ratio. Another common example of a ratio is seen on a map. The legend might indicate that one inch is equivalent to 50 miles. The statement one hand has five fingers is another example of a ratio. All theses are examples of comparisons-ratios. 3 Ways to Write a Ratio A ratio can be written three different ways. For example, to show the comparison of one inch representing 50 miles on a map the following notations can be used: Method 1: 1 to 50 (Use to to separate the numbers being compared) Method 2: 1:50 (Use a : to separate the numbers being compared) Method 3: 1 (Use a fraction bar to separate the numbers being compared) 50 Most maps use Method 2 for the legend. Because ratios will be used to solve problems, it is easier to write the ratios using fractional notation (Method 3). Looking at the ratio of one inch representing 50 miles ( 1 ), 2 inches 50 2 represents miles ( ), and 3 inches represents 150 miles ( 3 ). These ratios can be 150 related to equivalent fractions. CRT Example: Math 6 Notes Unit 7: Proportional Relationships Page 1 of 15

2 Simplifying Ratios Because fractional notations are commonly used to describe ratios, it makes it simple to simplify ratios. To simplify a ratio, simply write it in fractional notation and reduce as you would a normal fraction. Example In the statement above, 3 inches represents150 miles. Simplify this ratio. 3 1 Using method 3, the ratio can be written as a fraction:. Reduce this fraction to The simplified ratios shows that 1 inch represents 50 miles. Applications of Ratios Rate is a ratio that compares two quantities measured in different units. Example Person A travels 300 miles in 5 hours, find A s rate. A s rate = 300 miles 5 hours Unit rate is a rate whose denominator is 1. To convert a rate to a unit rate, divide both the numerator and denominator by the denominator. Example Find the unit rate of person A. 300 miles 5 5 hours 5 = 60 miles 1 hour read 60 miles per hour Example Bob s hearts beats 520 times every four minutes, find Bob s heartbeat per minute. In other words, find his unit heart rate. 520 beats 4 4 minute 4 = 130 beats 1 minute read 130 beats per minute Math 6 Notes Unit 7: Proportional Relationships Page 2 of 15

3 A Proportion is a statement of equality between 2 ratios. Looking at a proportion like 1 = 3, notice the relationships that exist For instance, what would happen if both ratios were tipped up-side down? and, 1 3 notice they are also equal, so = 1 3 How about writing the original proportion sideways: and, notice they are equal also, so = Another relationship occurs when cross multiplying the original proportion. 1 3 = Notice that the cross products are equal. In general, whether tipping proportions up-side down, writing them sideways or cross multiplying, the proportions remain equal. If a = c, then b d 1. b d = (up-side-down) 2. a c a b = (Sideways) 3. ad bc c d = (cross multiply) Those 3 observations are referred to as Properties of Proportions. Those properties can be used to help solve problems. To solve problems involving proportions, most people use either equivalent fractions or cross multiplying. Generally you use equivalent fractions when either the numerator or denominator of a fraction is a multiple of the numerator or denominator of the other fraction. If that is not immediately obvious, then cross multiply. Math 6 Notes Unit 7: Proportional Relationships Page 3 of 15

4 Example Find the value of x = 36 x This problem can be done by equivalent fractions or by cross multiplying. By cross multiplying, we have 6x = 360. x = 60. Example If a turtle travels 5 inches every 10 seconds, how far will it travel in 50 seconds? The first step is to set up a proportion. Identify the comparison to be made; in this case 5 inches every 10 seconds. Notice the order of the comparison, inches to seconds. 5 On one side of the proportion describes inches to seconds. On the other side of the 10 proportion use the same comparison, inches to seconds. Because inches is not known, us a variable n. Where will the 50 go in the ratio, top or bottom? Bottom, because it describes seconds = n 50 Solve for n by equivalent fractions or use property 3 and cross multiply = n 50 10n = n = 250 n = 25 The turtle will travel 25 inches in 50 seconds It is very important to write the same comparisons on both sides of the equal signs. In other words, if one side is a ratio comparing inches to seconds, then the other ratio must also compare inches to seconds. If the number of boys to girls were compared on one side of a proportion, the ratios on the other side of the proportion must also compare boys to girls. If it was changed to girls to boys on one side then the ratio on the other side must also be changed to girls to boys. The first Property of Proportion, tipping the ratios upside down, permits this to happen. In the above examples, the fraction could have been reduced before cross multiplying. By reducing first, the numbers are kept smaller. The answer is the same. Math 6 Notes Unit 7: Proportional Relationships Page 4 of 15

5 Applications Proportions Similar Polygons Similar Polygons have the same shape but not necessarily the same size. Corresponding parts of polygons are in the same relative position Similar Polygons - two polygons are similar if a) the measure of their corresponding angles are equal b) the ratio of the lengths of their corresponding sides are proportional Example Find the value of x given these two rectangles are similar. 4 in. 10 in. x 5 in. Since the two rectangles are similar, their sides must be in proportion. That is, the left side is to the bottom as the left side is to the bottom. Another way of saying that is the width is to the length as the width is to the length. 4 x = 10 5 (4)(5) = 10x 20 = 10x 2 = x Math 6 Notes Unit 7: Proportional Relationships Page 5 of 15

6 CRT Example: Percents Percents are special fractions whose denominators are. The number in front of the percent symbol (%) is the numerator. The denominator is not written, but understood to be. Examples 6% = 6 14% = 14 87% = 87 Because a percent is a special fraction, then, just like with decimals, all the rules for percents come from the rules for fractions. Adding & Subtracting Percents Adding or subtracting percents is exactly the same as adding fractions with like denominators. Add the numerators and keep the common denominator. Math 6 Notes Unit 7: Proportional Relationships Page 6 of 15

7 Example Add 34% + 15% This is the same as + = 49 can be written as 49%. Multiplying Percents To multiply percents, go back to the rules for multiplying fractions. To multiply fractions, multiply the numerators then multiply the denominators. To multiply percents, do the same. Example 5% 12% This example is equivalent to the following fraction multiplication problem: = 10, 000 Now, remember that a percent is a fraction with a denominator of. The product above does not have a denominator of. Its denominator is 10,000. To obtain a denominator of, divide both the numerator and denominator of 60 by (because 10,000 =). 10, This yields =. That means, 5% 12% =.6%. 10, 000 Converting Percents to Fractions and Decimals To convert a percent to a fraction, we just use the definition. The number in front of the percent symbol is the numerator, the denominator is, then simplify. Example Convert 53% to a fraction. 53 Math 6 Notes Unit 7: Proportional Relationships Page 7 of 15

8 Example Convert 53% to a decimal 53 53% =, but that s a fraction. There is one more step, change 53 to a decimal by dividing. To divide by move the decimal point 2 places to the left. 53%=.53. Converting % to fraction to decimal % to fraction--remove % symbol and put in the denominator. % to decimal--move decimal two places to the left. Decimal to %--Move decimal two places to the right and put a % symbol at the end. fractions to decimal--divide numerator by denominator. fraction to %--Convert fraction to a decimal then move decimal two places to the right. Example Convert 3% to a decimal. Remove the % symbol then move the decimal two places to the left. 3% =.03 Example Convert.34 to a percent. Move the decimal point 2 places to the right and put a percent symbol at the end..34 = 34% Hints To convert a decimal, the loop on the d in decimal curves to the left, so move the decimal point to the left 2 places. To convert to a percent, the loop on the p in percent curves to the right, so move the decimal point to the 2 places. Math 6 Notes Unit 7: Proportional Relationships Page 8 of 15

9 Example Convert 63% to a decimal. The loop on the d curves left, move the decimal point 2 places in that direction. The answer is.63. Example Convert.427 to a percent. The loop on the p curves to the right, move the decimal point 2 places in that direction. The answer is 42.7%. This is just a short cut, but here is why.427 = 42.7%..427 = To convert that to a percent, rewrite that fraction with a denominator of. Since the denominator is 0, both the numerator and denominator must be divided by 10 (because 0 10 = ) = 42.7% 0 10 Example Convert 1 % 4 to a fraction. Follow the same procedure as the first example. Remove the % symbol and write the number as a fraction using as the denominator %= 4 Simplifying that complex fraction, invert and multiply, then reduce = = = 4 = Math 6 Notes Unit 7: Proportional Relationships Page 9 of 15

10 Notice, the problems looked different, but the same strategies were used, put the numerator over then simplify. Convert to fractions % 2. 9% % 4. 30% 5. 45% 6. 2 % 7..4% % 3 Convert to decimals 9. 65% 10. 7% % % % % % % Convert to percents Percent Proportion A percent is nothing more than a way of interpreting information, writing a ratio, then rewriting the ratio so that the denominator is. For instance, let s say a student gets 8 correct out of 10 problems on a quiz. To determine the grade, the teacher would typically take that information and convert it to a percent. In other words, set up a proportion like this. # correct total =? Filling in the numbers: 8 n 8 80 = = 80% Getting 8 out of 10 is equivalent to 80%. Math 6 Notes Unit 7: Proportional Relationships Page 10 of 15

11 Notice the right side of the proportion is a fraction whose denominator is because that is the definition of a percent. Example Let s say Ashton made 23 out of 25 free throws playing basketball. How many shots would you expect Ashton to make at this rate if he were to shoot free throws? Again, begin with a proportion: attempts total = 23 n = = The proportion can be solved by making equivalent fractions or by cross-multiplying. Either way, the missing numerator is 92. Ashton is expected to make 92 free throws out of tries. These problems are just like the ratio and proportion problems from previous examples. The only difference is the denominator on the right side is because we are working with percents. A proportion that always has the denominator of the right side as is called the Percent Proportion. Percent Proportion part total = % Remembering that you have to describe the ratios the same way on each side of a proportion, we might think this should read. part total = part total Well, the percent ratio actually does compare parts to total on both sides. For a percent, the total is always and the percent is always the part you get. Speaking mathematically, the always goes on the bottom right side. That s a constant. The only things that can change is the part, total or percent. This information is obtained by reading the problem and placing the numbers in the correct spot, then solve. Math 6 Notes Unit 7: Proportional Relationships Page 11 of 15

12 There are only 3 different problems, we can look for a part, a total or a percent. Example (Looking for a percent) Bob got 17 correct on his history exam that had 20 questions. What percent grade did he receive? part total = % filling in the numbers, 17 n = 20 Solving, either by equivalent fractions or by cross-multiplying, n = 85which equals 85%. CRT Example: Example (Looking for a total) A company bought a used typewriter for $350, which was 80% of the original cost. What was the original cost? Does the $350 represent the total or part? 350 n = 80 Cross multiply: 80n = 350. Solve: 80n = 35,000 n = The original cost of the typewriter is $ Math 6 Notes Unit 7: Proportional Relationships Page 12 of 15

13 Example (Looking for a part) If a real estate broker receives 4% commission on an $80,000 sale, how much would he receive? Does $80,000 represent the part or total? n 80,000 = 4 n = 4 80, 00 n = 320, 000 n = 3,200 He would receive $3,200 in commission. While the first three examples were all percent problems percent proportion were used to solve them. In each case the unknown was something different. That s the beauty of the percent proportion, it can be used for any situation. In this next example, everything stays the same, but there is a slight variation in how the problem is written. To do this problem, how proportion problems are set up must be understood. Example Dad purchased a radio that was marked down 20% for $ What was the original cost of the radio! Setting up the proportion, does $68 represent the part or total? Filling in the proportion, paid total = % If the $68 represents the part you paid, what does the 20% represent? It represents the percentage off the total price. A proportion with paid is to total as amount off is to total is not correct!! The same ratio must be on both sides. That is paid to total as paid to total. If dad got 20% off, what percent did he pay? 80% Now, filling in the numbers, we have 68 n = 80 Solve: The original cost of the radio is $85. 80n = 6,800 n = 85 Math 6 Notes Unit 7: Proportional Relationships Page 13 of 15

14 3 different type problems were solved using the Percent Proportion. Percent proportions can solve for the part, total, and percent by using ratios and proportions. Scale Drawing Using Proportions Another application of proportions is in the use of scale drawings. A scale drawing is a two dimensional drawing that is similar to the object it represents. A scale model is a three dimensional model that is similar to the object it represents. The scale of a scale drawing or scale model gives the relationship between the drawing or model s dimensions and the actual dimensions. For example, if a map shows a scale of 1 cm : 5 m, it means that 1 centimeter on the scale drawing represents an actual distance of 5 meters. The scale of a scale drawing or scale model can be written without units if the measurements have the same unit. To write the scale from our example without units, write 5 meters as 500 centimeters. 1 cm 1 cm 1 cm : 5 m 1: m 500 cm So, we can write the scale without units as 1 : 500. Example: Example: On a map, the distance from your house to school is 5 centimeters. The scale is 1 cm : 500 m. What is the actual distance from your house to school? map distance 1 cm 5 cm = actual distance 500 m d m 1d = The distance from your house to school is 2500 meters. d = 2500 You have a scale model of an airplane, scale of 1:90. The length of the model airplane from nose to tail is 1.8 feet. Determine the length (from nose to tail) of the actual airplane. model length = airplane length 90 x The length of the airplane 1x = 162 is 162 feet. The length of the airplane is 162 feet. Math 6 Notes Unit 7: Proportional Relationships Page 14 of 15

15 CRT Example: Math 6 Notes Unit 7: Proportional Relationships Page 15 of 15