RATIO & PROPORTION. 3 Ways to Write a Ratio

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1 Gr 7 Ch 5 RATIO & PROPORTION A RATIO is a comparison between two quantities. We use ratios everyday; one Pepsi costs 50 cents describes a ratio. On a map, the legend might tell us one inch is equivalent to 50 miles or we might notice one hand has five fingers. Those are all examples of comparisons ratios. 3 Ways to Write a Ratio A ratio can be written three different ways. If we wanted to show the comparison of one inch representing 50 miles on a map, we could write that as; 1 to 50 or Using a colon 1:50 or Using a fraction 1 50 Because we are going to learn to solve problems, it s easier to write the ratios using fractional notation. If we looked at the ratio of one inch representing 50 miles, 1/50, we might determine 2 inches represents 100 miles, 3 inches represents 150 miles by using equivalent fractions. Simplifying Ratios That just seems to make sense. Look at that from a mathematical standpoint, it appears that we might also be able to reduce 3/150 to 1/50. Does 3/150 represent the same comparison as 1/50? The answer is yes and if we looked at other ratios, we would see that reducing ratios does not affect those comparisons. We noticed that 3/150, 3 inches represents 150 miles, could be reduced to 1/50 meaning 1 inch represents 50 miles. Mathematically, by setting the ratios equal, we could write 1 50 = Applications of Ratios Rate is a ratio that compares two quantities measured in different units.

2 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. Find the unit rate of person A. 300 miles 5 5 hours 5 = 60 miles 1 hour ; read 60 miles per hour Bob s hearts beats 520 times every four minutes, find Bob s heartbeat per minute. 520 beats 4 4 minute 4 = 130 beats 1 minute ; read 130 beats per minute Slope is the ratio of rise to run. slope = rise run Find the slope of the line that connects the points in the graph.

3 The first point is (2, 1), the second point is (6, 5). How many units did I go up? Count them, up 4. Then we went over 4. So the slope is 4/4 = 1. Find the slope of the line that connects the points on the graph. In this case, I will start at the point furthest to the left in Quadrant II, ( 4, 5). To get to the point on the right in Quadrant IV (3, 2), I need to go down 7, ( 7) and go over 7, so the slope is 7/7 = 1 A PROPORTION is a statement of equality between 2 ratios. Looking at a proportion like 1 2 = 3, we might see some relationships that exist if we take 6 time and manipulate the numbers. For instance, what would happen if we tipped both ratios up-side down? 2 1 and 6 3, notice they are also equal, so 2 1 = 6 3 How about writing the original proportion sideways, will we get another equality? 1 3 and 2 6, notice they are equal also, so 1 3 = 2 6

4 If we continued looking at the original proportion, we might also notice we could cross multiply and retain an equality. In other words 1x6 = 2x3. Makes you wonder whether tipping ratios up-side down, writing them sideways or cross multiplying only works for our original proportion? Well, to make that determination, we would have to play with some more proportions. Try some, if our observation holds up, we ll be able to generalize what we saw. Let s try these observations with the proportion = 4 6 Can I tip them upside down and still retain an equality? In other words, does 3 2 = 6 4? How about writing them sideways, does 2 4 = 3 6? How about cross multiplying in the original proportion, does 2x6 = 3x4? The answer to all three questions is yes. Since everything seems to be working, we will generalize our observations using letters instead of numbers. If a b = c d, then 1. b a = d c 2. a c = b d 3. ad = bc Those 3 observations are referred to as Properties of Proportions. Those properties can be used to help us solve problems. To solve problems, most people use either equivalent fractions or cross multiplying to solve proportions. 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.

5 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 or x = 60. If a turtle travels 5 inches every 10 seconds, how far will it travel in 50 seconds? What we are going to do is set up a proportion. How surprising? The way we ll do this is to identify the comparison we are making. In this case we are saying 5 inches every 10 seconds. Therefore, and this is very important, we are going to set up our proportion by saying inches is to seconds. On one side we have 5 describing inches to seconds. On the other side we have to 10 again use the same comparison, inches to seconds. We don t know the inches, so we ll call it n. Where will the 50 go in the ratio, top or bottom? Bottom, because it describes seconds good deal. So now we have = n 50 Now, we can find n by equivalent fractions or we could use property 3 and cross multiply = n 50 10n = 5x50 10n = 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 we had a ratio on one side comparing inches to seconds, then we must write inches to seconds on the other side. If we compared the number of boys to girls on one side, we would have to write the same comparison on the other side, boys to girls. We could also write it as girls to boys on one

6 side as long as we wrote girls to boys on the other side. The first Property of Proportion, tipping the ratios upside down, permits this to happen. In the above examples, I could have simplified the fractions before cross multiplying. By simplifying first, that keeps the numbers smaller. You get the same answers.

7 Measurement Conversions Converting measures is as easy as multiplying by one. Why, because you are multiplying by one to convert measurements. To do conversions, you do need to know conversion factors such as 3 feet = 1 yard, 60 minutes = 1 hour, 16 ounces = 1 pound, and 4 quarts = 1 gallon. Let s start off with a simple problem you would normally do in your head. Let s convert 5 yards to feet. To do this conversion I want to multiply the original measurement by one and at the same time introduce the new measurement (feet) into the problem. In our case, we start off with 5 yards, I will multiply that by one in the form of a fraction 3feet 3feet 5 yards x 3feet 3feet 1. I also know that 3 feet is 1 yard. Making that substitution, I have, Notice, I am multiplying 5 yards by 1, since 3 feet divided by 3 feet is 5 yards x 3 feet 1yard. I wrote yards in the denominator purposely so the labeling for yards divide out, so I am left with 5 yards x 3 feet = 15 feet. 1 yard Convert 5 gallons to cups. To do this, I will multiply 5 gallons by one using conversion factors I know that will cancel out the labeling until I have cups. 5 gallons x 4quarts 1gallon x 2pint s 1quart 3 days x 24hours 1day 4 km x 1000m 1km x 2cups 1pint Convert 3 days to seconds. x 60minutes 1hour Convert 4 km to cm. x 100cm 1m = 40,0000 cm x 60seconds 1min ute = 80 cups* = 259,200 seconds *N.B. When multiplying by 1 in the examples, I am deliberately choosing the labeling in the denominators to cancel out the labeling in the numerators until I am left with the desired measurement.

8 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 Find the value of x given these two rectangles are similar. 4 in. x 10 in. 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 s to the length = x or 10x = x = 20 x = 2