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4-1 Ratios & Proportions

4-1 Ratios & Proportions. Notes. A ratio is a comparison of two quantities. . Ratios can be written in several ways . 7 to 5, 7:5, and name the same ratio. 15 ÷ 3 9 ÷ 3. bikes skateboards. Example 1: Writing Ratios in Simplest Form.

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4-1 Ratios & Proportions

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  1. 4-1 Ratios & Proportions

  2. Notes A ratio is a comparison of two quantities. Ratios can be written in several ways. 7 to 5, 7:5, and name the same ratio.

  3. 15 ÷ 3 9 ÷ 3 bikes skateboards Example 1: Writing Ratios in Simplest Form Write the ratio 15 bikes to 9 skateboards in simplest form. 15 9 Write the ratio as a fraction. = 5 3 Simplify. = = 5 3 The ratio of bikes to skateboards is , 5:3, or 5 to 3.

  4. shirts jeans 24 ÷ 3 9 ÷ 3 Check It Out! Example 2 Write the ratio 24 shirts to 9 jeans in simplest form. Write the ratio as a fraction. 24 9 = 8 3 Simplify. = = 8 3 The ratio of shirts to jeans is , 8:3, or 8 to 3.

  5. Practice • 15 cows to 25 sheep • 24 cars to 18 trucks • 30 Knives to 27 spoons

  6. When simplifying ratios based on measurements, write the quantities with the same units, if possible.

  7. 3 yards 12 feet 9 ÷ 3 12 ÷ 3 = = = 3 4 3 4 The ratio is , 3:4, or 3 to 4. Example 3: Writing Ratios Based on Measurement Write the ratio 3 yards to 12 feet in simplest form. First convert yards to feet. 3 yards = 3 ● 3 feet There are 3 feet in each yard. Multiply. = 9 feet Now write the ratio. 9 feet 12 feet Simplify.

  8. 36 inches 4 feet 36 ÷ 12 48 ÷ 12 = = = 3 4 3 4 The ratio is , 3:4, or 3 to 4. Check It Out! Example 3 Write the ratio 36 inches to 4 feet in simplest form. First convert feet to inches. 4 feet = 4 ● 12 inches There are 12 inches in each foot. = 48 inches Multiply. Now write the ratio. 36 inches 48 inches Simplify.

  9. Practice • 4 feet to 24 inches • 3 yards to 12 feet • 2 yards to 20 inches

  10. Notes Ratios that make the same comparison are equivalent ratios. To check whether two ratios are equivalent, you can write both in simplest form.

  11. 1 9 1 9 12 15 3 27 27 36 2 18 Since , the ratios are equivalent. B. A. = and and 2 18 3 27 2 ÷ 2 18 ÷ 2 3 ÷ 3 27 ÷ 3 = = = = 4 5 3 4 Since , the ratios are not equivalent.  12 15 27 36 12 ÷ 3 15 ÷ 3 27 ÷ 9 36 ÷ 9 = = = = Example 4: Determining Whether Two Ratios Are Equivalent Simplify to tell whether the ratios are equivalent. 1 9 1 9 4 5 3 4

  12. Practice

  13. 8 30 30 7 1 2 12 45 Possible answer: , Possible answer: , 4 15 7 21 3. 4. 1 3 14 42 Lesson Quiz: Part I Write each ratio in simplest form. 1. 22 tigers to 44 lions 2. 5 feet to 14 inches Find a ratio that is equivalent to each given ratio.

  14. 36 24 16 10 5. 6. 8 64 16 128 and ; yes, both equal 28 18 32 20 8 5 3 2 14 9 8 5 1 8 =  ; yes ; no Lesson Quiz: Part II Simplify to tell whether the ratios are equivalent. and and 7. Kate poured 8 oz of juice from a 64 oz bottle. Brian poured 16 oz of juice from a 128 oz bottle. Are the ratios of poured juice to starting amount of juice equivalent?

  15. Vocabulary • A proportion is an equation stating that two ratios are equal. To prove that two ratios form a proportion, you must prove that they are equivalent. To do this, you must demonstrate that the relationship between numerators is the same as the relationship between denominators.

  16. Examples: Do the ratios form a proportion? x 3 Yes, these two ratios DO form a proportion, because the same relationship exists in both the numerators and denominators. 7 21 , 10 30 x 3 ÷ 4 8 2 No, these ratios do NOT form a proportion, because the ratios are not equal. , 9 3 ÷ 3

  17. Example ÷ 5 3 7 = 8 40 ÷ 5

  18. Cross Products • When you have a proportion (two equal ratios), then you have equivalent cross products. • Find the cross product by multiplying the denominator of each ratio by the numerator of the other ratio.

  19. Example: Do the ratios form a proportion? Check using cross products. 4 3 , 12 9 These two ratios DO form a proportion because their cross products are the same. 12 x 3 = 36 9 x 4 = 36

  20. Example 2 5 2 , 8 3 No, these two ratios DO NOT form a proportion, because their cross products are different. 8 x 2 = 16 3 x 5 = 15

  21. Solving a Proportion Using Cross Products • Use the cross products to create an equation. • Solve the equation for the variable using the inverse operation.

  22. Example 1: Solve the Proportion Start with the variable. 20 k = 17 68 Simplify. Now we have an equation. To get the k by itself, divide both sides by 68. 68k 17(20) = 68k = 340 68 68 k 5 =

  23. Example 2: Solve the Proportion Start with the variable. Simplify. Now we have an equation. Solve for x. 2x(30) 5(3) = 60x = 15 60 60 x ¼ =

  24. Example 3: Solve the Proportion = Start with the variable. Simplify. Now we have an equation. Solve for x. (2x +1)3 5(4) = 6x + 3 = 20 x =

  25. Example 4: Solve the Proportion = Cross Multiply. Simplify. Now we have an equation with variables on both sides. Solve for x. 3x 4(x+2) = 3x = 4x + 8 x -8 =

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