2.4 Ratios and Proportions

Transcription

1 . Ratios and Proportions RATIOS A ratio between two numbers compares their relative size, using division, and this relative size will not change, no matter how large or small the numbers may get. The ratio of two numbers, A and B, is written as either A : B or A B. In words, this is read, A to B. Consider this: Two brothers, Kenny and Jimi decided to go into business together selling lemonade. Kenny, the older brother, contributed $ to get the business started and Jimi contributed $. When the profits were counted at the end of the summer, Kenny got $7 and Jimi got $. Is this a fair sharing of the profits To answer this question, look at the relative size, based on the ratio Kenny : Jimi, of (1) the investments each made: $ : $ or and () the profits each received: $7 : $ or If these two ratios are equivalent, then the share of the profits is fair. One way we can see if the ratios are equivalent is to simplify each and see if they simplify to the same ratio or fraction: simplifies by a factor of to 3 7 simplifies by a factor of 1 to 3 7 Since each fraction simplifies to 3, it is safe to conclude that the sharing of profits was fair. A second way to check if the ratios are equivalent is to set them net to each other and cross multiply (the question mark over the equal sign means we re trying to find out if the fractions are equivalent or not): Set the two fractions net to each other. Cross multiply. (Multiply the denominators across the equal sign; the numerators remain where they are.) Ratios and Proportions page. - 1

2 If, after cross multiplying, the products are not the same, then we say that the ratios are not equivalent. Eample 1: Determine whether or not the ratios are equivalent by cross multiplying. 1 b) 1 0 Procedure: Multiply the denominators, across the equal sign, to the numerators on the other side. Answer: 1 b) So, and 1 are equivalent. So, 0 and 1 are not equivalent. Eercise #1: Determine whether or not the ratios are equivalent by cross multiplying. Use Eample 1 as a guide. 1 b) 9 1 c) PROPORTIONS When we set two equivalent ratios equal to each other we form an equation, called a proportion. Eercise #: Based on the definition of proportion, above, which of these are proportions (Refer to Eercise 1.) 1 b) 9 1 c) Ratios and Proportions page. -

3 When a proportion has a variable within it maybe in one or both of the numerators or one or both of the denominators we can solve for that variable by cross multiplying. Eamples of proportion equations are w 7 1, and Of course, each proportion equation has fractions, so it s common for us to solve by clearing the fractions, and cross multiplying allows us to do that: Caution: If the equation involves more than just two equal fractions, then it is not a proportion and we cannot use cross multiplication to clear the fractions. Here are some eamples of equations with fractions and some eamples of proportions. Equations with fractions Proportions We cannot use cross multiplication to solve these. Ratios and Proportions page. - 3

4 Eample : Solve for in each proportion. 3 b) Procedure: Use cross multiplication to get the equation into a more familiar form. Then solve the resulting equation. Answer: 3 b) Eercise #3: Solve for in each proportion. Use Eample as a guide. 1 1 b) 1 c) 1 7 d) If one ratio has no variable (is only numerical), then it might be able to be simplified before cross multiplying. Simplifying first is helpful because the numbers are easier to work with. Ratios and Proportions page. -

5 Eample 3: Solve for in each proportion. 1 1 b) 30 Procedure: Simplify one of the ratios, if possible. Use cross multiplication to get the equation into a more familiar form. Answer: 1 1 b) 30 First: Simplify 1 1 by a factor of 3 Simplify 30 by a factor of Now cross multiply and solve: Eercise # Solve for in each proportion by first simplifying the numerical ratio. Use Eample 3 as a guide. 0 b) 1 1 c) 3 0 d) Ratios and Proportions page. -

6 Sometimes proportions have unknown information in more than one place. For eample, the proportion + equation can still be solved by cross multiplying. First, though, we should recognize that because the division bar is a grouping symbol the left side numerator is a quantity: ( + ) Now cross multiply. ( + ) As we cross multiply, the parentheses remain until we distribute: + 1 Now we solve as we do any linear equation by isolating the variable. This process remains true even if both variable terms are in the denominator. Eample : Procedure: Solve for in each proportion. Check the answer. Answer: + b) 1 + Use parentheses to group all quantities; then use cross multiplication to get the equation into a linear form. Isolate the variable by solving the linear equation. + b) 1 + Show parentheses: ( + ) ( 1) ( + ) Cross multiply: ( + ) ( + ) ( 1) Isolate the variable: Check: + + Check: true! true! Ratios and Proportions page. -

7 Eercise : Solve each proportion. Check your answer. + 3 b) w w + 1 c) 3 9 d) y 3y e) p + 7 p 3 f) c + c To this point, we ve worked with only whole number values. However, many times a ratio will include fractional values in the numerator or denominator. Ratios and Proportions page. - 7

8 For eample, if it takes 3 cups of milk for pancakes, then we can double the recipe to make pancakes, but it will take 3 cups of milk. This can be seen in the proportion with ratios milk : pancakes. 3 3 It might also be that 3 is epressed as a mied number, 11 3 : Eample : Solve for in each proportion. Write any improper fraction answer as a mied number. Procedure: 3 b) 1 3 c) 3 Answer: Use cross multiplication to get the equation into a linear form. 3 c) 3 3( ) Write 1 3 as an improper fraction Cross multiply: b) Write this answer as a mied number. 1 Ratios and Proportions page. -

9 Eercise : Solve each proportion. Write any improper fraction answer as a mied number. 1 b) 1 w c) d) 3 y 3 y + e) 3 p 3 f) 3 3 m Ratios and Proportions page. - 9

10 Answers to each Eercise Section. Eercise #1: 10 b) 7 7 c) not equivalent equivalent equivalent Eercise #: not a proportion b) is a proportion c) is a proportion Eercise #3: b) c) d) 1 Eercise #: b) 1 c) 1 d) Eercise #: b) w c) d) y e) p 7 f) c 9 Eercise #: 3 b) w c) 3 d) y e) p f) m Ratios and Proportions page. -

11 Section. Focus Eercises 1. Use cross multiplication to determine whether the pair of fractions is equivalent or not b) 1 c) 1 d) e) f) 1 9 g) h) 1. Solve for the variable in each proportion. 3 b) 3 c c) 1 1 d) 1 p 30 e) f) 1 9 g) h) Ratios and Proportions page. - 11

12 i) y 1 j) Solve each proportion. Write any improper fraction answer as a mied number b) 1 p c) 1 m 11 d) 3 1 e) f) Ratios and Proportions page. - 1