Ratios can be written in several different ways. Using a colon. Using the word to. Or as a fraction.

Size: px

Start display at page:

Download "Ratios can be written in several different ways. Using a colon. Using the word to. Or as a fraction."

Bathsheba Atkins
5 years ago
Views:

1 Show Me: Proportional Relationships M7051 Tanyss and her family are redecorating parts of their house. One of their projects is to replace the tiles on the kitchen floor, and they will use this plan of light and dark tiles. In this lesson we will see how Tanyss and her family use ratios and proportions. Mathematics will help them make their home a nicer place to live in. This is the pattern of tiles that Tanyss and her family are using for their kitchen floor. In this square, there are two light tiles and seven dark tiles. We can compare the numbers two and seven by using a ratio. Ratios can be written in several different ways. Using a colon. Using the word to. Or as a fraction. When we express ratios in words, we use the word "to" we say "the ratio of something to something else." We read all three of these as the ratio of two to seven. Here s a question. There are fourteen dark tiles and four light tiles. Which of these is not a ratio of the number of dark tiles to the number of light tiles? Here s what it looks like when Tanyss and her family have laid two sets of tiles. The ratio of dark tiles to light tiles is fourteen to four. When the family lays a third set of tiles, we would not expect the ratio of dark to light tiles to change.

2 Let s check and see if fourteen to four and twenty-one to six are equivalent ratios. To simplify the ratio fourteen to four, we can divide the numerator and denominator by two to get the ratio seven to two. To simplify the ratio twenty-one to six, we can divide the numerator and denominator by their common factor three. Again, the simplified ratio is seven to two. If two ratios can be simplified to the same ratio, then the two ratios are equal. When two ratios are equal, we have a proportion. This proportion can be read fourteen is to four as twenty-one is to six. Try this one. Which of these is a ratio equivalent to three halves? Let s see if we can write a proportion using these two ratios. The ratio five to nine is already in its simplest form because the only common factor of five and nine is one. The ratio ten to twenty has a common factor of ten, so we can simplify by dividing the numerator and denominator by ten. So the simplest form of the ratio ten to twenty is the ratio one to two. The two ratios five to nine and one to two are both in simplest form and do not have the same numerator and denominator.

3 This means that these two ratios, five to nine and ten to twenty, cannot be used to write a proportion. Try this. Which pair of ratios can be used to write a proportion? Let s take a look at this proportion. Four is to five as eight is to ten. In a proportion, we can find what s called the cross product by multiplying the numerator of the first fraction by the denominator of the second fraction and the denominator of the first fraction by the numerator of the second. Let s find the cross products for the first proportion. We multiply the numerator of the first fraction, four, by the denominator of the second, ten. Then multiply the numerator of the second fraction, eight, by the denominator of the first, five. This gives us forty on both sides. So we see that the cross products in this proportion are equal to each other. Here s a second proportion, nine is to three as six is to two. The cross products are nine times two and three times six, which are both eighteen. The cross products of this proportion too are equal. Is there a pattern here? In a proportion, the cross products are always equal. When we write an equation from a proportion, such as four times ten equals five times eight that equation is called the cross product equation. Solve this.

4 Complete the cross product equation for the proportion six is to fifteen as two is to five. Let s take a look at another proportion. We can see that this proportion has a variable that represents an unknown value. The proportion is read X is to four as six is to twenty. One cross product is twenty times X. The other cross product is four times six, or twenty-four. In a proportion the cross products are equal, so we can write twenty X equals twenty-four. Try this one. For the proportion five is to twelve equals ten is to Z, what equation can we write using the cross products? This proportion has a variable that represents an unknown value. Let s see if we can solve this proportion and find the value of Y. The cross products are two times Y and four times five, or twenty. Remember, in a proportion the cross products are equal, so the cross product equation is two Y equals twenty. Dividing both sides of the equation by two, gives us Y equals ten. So if the original proportion is four is to Y as two is to five, then the value of the variable Y is ten. Here s another example.

5 Solve this proportion: seven is to four as Z is to two. Think about how to solve this, and then click Solution to see the answer. Solution We know that in a proportion, the cross products are equal, so seven times two, or fourteen, equals four times Z. We then solve the equation by dividing by four. When we simplify, we get a true proportion with Z equals seven over two. It s time for a question. Which set of steps shows a possible way to solve the proportion six is to three as eight is to W? Tanyss s family needs to order carpet for the den, which is forty percent of the area of the living room. The living room has one hundred forty square feet of carpeting. Tanyss has to find out what forty percent of one hundred forty is. Remember, forty percent means forty out of one hundred, which we can write as the ratio forty to one hundred. We can solve percent problems by using a proportion. This diagram shows how to use information from the problem to set up the proportion. Notice that we have one side for percents and the other for numbers. Think about the problem in terms of percents. The

6 den is being compared to the living room. So, the whole living room is one hundred percent and the den is forty percent. The whole is one hundred percent. The part is forty percent. Now try this. Place the numbers forty and one hundred into the proportion. The living room, or the whole, is one hundred forty square feet. We don t know how many square feet the den is. We can let X be the amount of square feet in the den. Now we can set up the proportion, forty is to one hundred as X is to one hundred forty. This proportion can be used to solve the problem: What is forty percent of one hundred forty? First we find the cross products, forty times one hundred forty equals fifty-six hundred, and one hundred times X, and make them equal to one another. Then we divide both sides by one hundred. This leaves fifty-six equals X. That means the floor of the den is fifty-six square feet. Here s another example. What proportion would we use to solve this problem: What is thirty-five percent of three hundred? Try to figure out the answer, then click Solution to check.

7 Remember, thirty-five percent means thirty-five out of a hundred. We can write this as a ratio, of thirty-five over one hundred. Then we know that three hundred is the whole, so that goes in the denominator, and the unknown, X, goes in the numerator. So the correct proportion is thirty-five is to one hundred as X is to three hundred. Time for another question. The area of the kitchen pantry is eighteen percent of the area of the kitchen. The area of the kitchen is one hundred thirty-five square feet. What is the area of the pantry? Let s look at another example. What percent of two hundred twenty-four is twenty-eight? We can use a proportion to solve the problem. Let s think about the problem in terms of percents first. The whole is one hundred percent. The part is an unknown percent. We will let X be the unknown value. This will give us the ratio X to one hundred. Our second ratio will be twenty-eight, the part, over two hundred twenty-four, the whole. Now we have a proportion. Finding the cross product equation and solving, we get X equals twelve point five.

8 Tanyss and her family are replacing all the outlet covers in their house. There are sixty covers in all. They have already replaced forty-eight of the covers. What percent of the covers have they finished replacing? Think about the answer, and then click Solution to check. Solution For the percents, the whole is one hundred percent and the part is the unknown value. Let P be the unknown value. For the numbers, the whole is sixty outlet covers. The part is forty-eight outlet covers. With these numbers, we make the proportion. In a proportion, the cross products are equal, so we can make the cross product equation, then simplify. Our answer, P equals eighty, means that forty-eight is eighty percent of sixty. Tanyss and her family have replaced eighty percent of the outlet covers in their home. Tanyss's house has twenty-five windows. Her brother has washed twenty-two of the windows. What percent of the windows has her brother washed? Outside Tanyss's garage, sixty-two percent of the driveway is covered with asphalt. This area measures ninety-six point one square feet. We can use proportions to help us figure out the total area of Tanyss s driveway. In other words we need to find sixty-two percent of what number is ninety-six point one?

9 We know the whole is one hundred percent of the driveway and the asphalt part is sixtytwo percent. We can write this as the ratio sixty-two to one hundred. We also know that part of the driveway measures ninety-six point one square feet. We don t know the whole area of the driveway. We can let X be this unknown value. Our second ratio is ninety-six point one over X. The proportion sixty-two is to one hundred as ninety-six point one is to X can be used to solve this problem. The cross product equation is sixty-two X equals one hundred times ninety-six point one. Divide both sides of the equation by sixty-two and simplify each side to get the solution, one hundred fifty-five. So the area of the driveway is one hundred fifty-five square feet. Here s a question. Which proportion would help us solve the problem sixty percent of what number is fortyfive? Here s a summary of points covered in this lesson. A proportion is an equation in which each side of the equation is a ratio. In a proportion, the cross products are equal. Find the cross products by multiplying each numerator by the other denominator. Sometimes there is a variable in a proportion. Solve a proportion by forming the cross products and then solving the cross product equation.

10 A proportion can be used to solve different kinds of percent problems. Here are some of them. If you d like to review this activity again, click Review. If you re ready to exit, click Done.

17. [Exploring Numbers]

17. [Exploring Numbers] . [Exploring Numbers] Skill. Comparing whole numbers. Compare the size of the digits in the same place, one at a time. Work from left to right across each number. Q. Which number is the A ) 06 B ) 60 C