5-2 Identifying and Writing Proportions

Transcription

1 5-2 Identifying and Writing Proportions Learn to find equivalent ratios and to identify proportions. Course 2

2 equivalent ratios proportion Vocabulary

3 Students in Mr. Howell s math class are measuring the width w and the length l of their heads. The ratio of l to w is 10 inches to 6 inches for Jean and 25 centimeters to 15 centimeters for Pat.

4 These ratios can be written as the fractions and. Since both simplify to 5, they are equivalent. Equivalent 3 ratios are ratios that name the same comparison.

5 An equation stating that two ratios are equivalent is called a proportion. The equation, or proportion, below states that 10 the ratios and 25 are equivalent Reading Math 10 6 = Read the proportion 10 by saying ten is to six 6 = as twenty-five is to fifteen.

6 If two ratios are equivalent, they are said to be proportional to each other, or in proportion.

7 Additional Example 1A: Comparing Ratios in Simplest Forms Determine whether the ratios are proportional. A. 24, = = 9 16 Simplify Simplify Since 8 17 = 9 16, the ratios are not proportional.

8 Additional Example 1B: Comparing Ratios in Simplest Forms Determine whether the ratios are proportional. B. 150, = = 10 7 Simplify Simplify. 63 Since 10 7 = 10 7, the ratios are proportional.

9 Try This: Example 1A Determine whether the ratios are proportional. A. 54, = = 1 2 Simplify Simplify Since 6 7 = 1 2, the ratios are not proportional.

10 Try This: Example 1B Determine whether the ratios are proportional. B. 135, = = 9 5 Simplify Simplify. 45 Since 9 5 = 9 5, the ratios are proportional.

11 Additional Example 2: Comparing Ratios Using a Common Denominator Use the data in the table to determine whether the ratios of rice to water are proportional for both servings of rice. Servings of Rice Cups of Rice Cups of Water Write the ratios of rice to water for 12 servings and for 40 servings. Ratio of rice to water, 12 servings: 3 6 Write the ratio as a fraction. Ratio of rice to water, 40 servings: 10 Write the ratio as a fraction = = = = Write the ratios with a common denominator, such as 114. Since 57 = 60, the two ratios are not proportional

12 Try This: Example 2 Use the data in the table to determine whether the ratios of beans to water are proportional for both servings of beans. Servings of Beans Cups of Beans Cups of Water Write the ratios of beans to water for 8 servings and for 35 servings. Ratio of beans to water, 8 servings: 4 3 Write the ratio as a fraction. Ratio of beans to water, 35 servings: 13 Write the ratio as a fraction = 4 9 = = = Write the ratios with a common denominator, such as 27. Since 36 = 39, the two ratios are not proportional

13 You can find an equivalent ratio by multiplying or dividing the numerator and the denominator of a ratio by the same number.

14 Insert Lesson Title Here Additional Example 3: Finding Equivalent Ratios and Writing Proportions Find a ratio equivalent to each ratio. Then use the ratios to find a proportion. A = = 6 10 B = = 7 4 = 6 Multiply both the numerator and 10 denominator by any number such as 2. = 7 4 Write a proportion. Divide both the numerator and denominator by any number such as 4. Write a proportion.

15 Insert Lesson Title Here Try This: Example 3 Find a ratio equivalent to each ratio. Then use the ratios to find a proportion. A = = 6 9 B = = 4 3 = 6 Multiply both the numerator and 9 denominator by any number such as 3. = 4 3 Write a proportion. Divide both the numerator and denominator by any number such as 4. Write a proportion.

16 Insert Lesson Title Here Lesson Quiz Determine whether the rates are proportional by writing them in simplest form and comparing them , , 3 ; proportional , , 2 ; not proportional 7 3 Determine if the ratios are proportional by finding a common denominator , , 16 ; not proportional , , 9 ; proportional 21 21

17 Insert Lesson Title Here Lesson Quiz 5. In a local pre-school, there are 5 children for every teacher. Write an equivalent ratio to show how many children there would be if there were 4 teachers. 5,

18 5-3 Solving Proportions Learn to solve proportions by using cross products. Course 2

19 Insert Lesson Title Here cross product Vocabulary

20 The tall stack of Jenga blocks is 25.8 cm tall. How tall is the shorter stack of blocks? To find the answer you will need to solve a proportion. For two ratios, the product of the numerator in one ratio and the denominator in the other is a cross product. If the cross products of the ratios are equal, then the ratios form a proportion.

21 2 5 = = = 30 CROSS PRODUCT RULE In the proportion a = c, the cross products, b d a d and b c are equal. You can use the cross product rule to solve proportions with variables.

22 Additional Example 1: Solving Proportions Using Cross Products Use cross products to solve the proportion = m 5 15 m = m = 45 15m 15 = m = 3 The cross products are equal. Multiply. Divide each side by 15 to isolate the variable.

23 Insert Lesson Title Here Try This: Example 1 Use cross products to solve the proportion. 6 7 = m 14 7 m = m = 84 7m 7 = 84 7 m = 12 The cross products are equal. Multiply. Divide each side by 7 to isolate the variable.

24 When setting up a proportion to solve a problem, use a variable to represent the number you want to find. In proportions that include different units of measurement, either the units in the numerators must be the same and the units in the denominators must be the same or the units within each ratio must be the same. 16 mi 4 hr = 8 mi x hr 16 mi 8 mi = 4 hr x hr

25 Additional Example 2: Problem Solving Application If 3 volumes of Jennifer s encyclopedia takes up 4 inches of space on her shelf, how much space will she need for all 26 volumes? 1 Understand the Problem Rewrite the question as a statement. Find the space needed for 26 volumes of the encyclopedia. List the important information: 3 volumes of the encyclopedia take up 4 inches of space.

26 Additional Example 2 Continued 2 Make a Plan Set up a proportion using the given information. 3 volumes 4 inches = 26 volumes x Let x be the unknown space.

27 Additional Example 2 Continued 3 Solve 3 4 = 26 x 3 x = x = 104 3x 3 = x = She needs Write the proportion. The cross products are equal. Multiply. Divide each side by 3 to isolate the variable. inches for all 26 volumes.

28 Additional Example 2 Continued 4 Look Back 3 4 = = = 104 The cross products are equal, so answer is the

29 Try This: Example 2 John filled his new radiator with 6 pints of coolant, which is the 10 inch mark. How many pints of coolant would be needed to fill the radiator to the 25 inch level? 1 Understand the Problem Rewrite the question as a statement. Find the number of pints of coolant required to raise the level to the 25 inch level. List the important information: 6 pints is the 10 inch mark.

30 Try This: Example 2 Continued 2 Make a Plan Set up a proportion using the given information. 6 pints 10 inches = x 25 inches Let x be the unknown amount.

31 Try This: Example 2 Continued 3 Solve 6 10 = x x = x = x 10 = x = 15 Write the proportion. The cross products are equal. Multiply. Divide each side by 10 to isolate the variable. 15 pints of coolant will fill the radiator to the 25 inch level.

32 Try This: Example 2 Continued 4 Look Back 6 10 = = = 150 The cross products are equal, so 15 is the answer.

33 Insert Lesson Title Here Lesson Quiz: Part 1 Use cross products to solve the proportion = 45 t 2. x 9 = = r n 10 = 28 8 t = 36 x = 3 r = 24 n = 35

34 Insert Lesson Title Here Lesson Quiz: Part 2 5. Carmen bought 3 pounds of bananas for $1.08. June paid $ 1.80 for her purchase of bananas. If they paid the same price per pound, how many pounds did June buy? 5 pounds