Sect 5.3 Proportions

Transcription

1 Sect 5. Proportios Objective a: Defiitio of a Proportio. A proportio is a statemet that two ratios or two rates are equal. A proportio takes a form that is similar to a aalogy i Eglish class. For example, a puppy is to a dog as a kitte is to a cat is a aalogy, but we ca write this as a proportio. A puppy is to a dog would be a fractio o oe side of the equal sig ad a kitte is to a cat would be a fractio o the other side: puppy dog kitte cat Write the followig as a proportio: Ex. Fifty feet of fecig is to 5 pouds as thirty feet is to 2 pouds. Fifty feet of fecig is to 5 pouds : thirty feet is to 2 pouds: So, the proportio is 0 ft 2pouds 5pouds 5pouds 0 ft 2pouds. Ex. 2 is to 2. liters as $2.40 is to 0.72 liters. is to 2. liters: 2. liters ; $2.40 is to 0.72 liters: $ liters So, the proportio is 2. liters $ liters. Ex. 4 5 is to 5 7 as is to 0 2. So, the proportio is

2 Objective a2: Itroductio to equatios ad solvig equatios usig the multiplicatio property of equality. 2 I example #, it is fairly easy to see that the proportio is true sice 0 ft reduces to 5pouds 7pouds ad 0 ft 0 ft also reduces to 2pouds 7pouds. I example #2 ad # however, it is much more difficult to check to see if the proportio is true. We eed to fid a more efficiet way to check to see if a proportio is true. To help us do this, we eed itroduce the idea of equatios ad how to solve equatios usig the multiplicatio property of equality. I Algebra, whe there is a umber that we do ot kow its value, we represet the umber usig a letter like x. So, if we wat to write five times a ukow umber, we write 5 x or 5x. Usig this idea, we ca use what are called equatios to help us fid these ukow umbers for a particular example. Let s state a defiitio. A equatio is a statemet that two quatities are equal. A equatio ca be as simple as 4 quarters $, or it could be more complex like x 2.7 ad 2x 2 5x A solutio to a equatio is the value of x that makes the equatio true. For example, x 0. is a solutio to the equatio x 2.7 sice if we replace x by 0. ad do the multiplicatio, we get: (0.) 2.7. Solve the followig: Ex. 4 4x 28 Sice , the x 7 has to be the solutio. We ca also get the solutio by dividig 28 by 4 sice The property the allows us to solve this equatio is called the multiplicatio property of equality. It says we ca multiply or divide both sides by the same o-zero umber. I our example, we are dividig both sides by 4 to solve for x: Multiplicatio Property of Equality: If A B ad C 0, the A C B C ad A C B are equivalet C equatios to A B (i.e., they have the same solutios)

3 Solve the followig: Ex. 5a 2.4x.84 Ex. 5b 4 x 5 We will divide both sides We will divide both sides by the umber i frot of by the umber i frot of x: x: 2.4x x 5 x.84 (2.4) x 5 4 x 4. x Objective b: Usig cross multiplicatio to determie if a proportio is true. With a proportio like a b c, if we multiply both sides by b d, we get: d b d a b b d c d b d a b b d c d d a b c ad bc 4 (write b ad d over ) (reduce) (simplify) This says that a b c if ad oly if ad bc (provided that either b or d d are 0). This gives use a easy way to check to see if a proportio is true by multiplyig the first umerator with the secod deomiator ad seeig if it is equal to the product of the secod umerator with the first deomiator. This techique is called cross multiplicatio: 4 Cross Multiplicatio For b 0 ad d 0, a b c d called the cross products. if ad oly if ad bc. ad ad bc are Now, let s look back at the results from first three examples ad use cross multiplicatio to show that the proportios are true.

4 4 Use cross multiplicatio to determie if the followig proportios are true: Ex. 5pouds 5pouds 0 ft 2pouds 0 ft 2pouds True Ex liters $ liters 2. liters $ liters True Ex ( ) (chage to improper fractios) (simplify) True (reduce)

5 5 Ex False Objective c: Solvig proportios with a ukow umber. Now, we will examie how to solve for a missig umber i a proportio. First, we will cross multiply. Next, we will simplify each cross product. Fially, we will use the multiplicatio property of equality to solve for the ukow umber. Let s try some examples. Solve: Ex. 0 Ex Ex (simplify) (simplify) 5 (divide by ) (divide by 0.8) or or (simplify) (divide by 5.) Sice is a log messy decimal, let s express the aswer as a fractio: (move the decimal ad reduce)

6 Ex ( 4 )(5 4 ) ( ) (chage to improper fractios) or 2 4 (reduce) (simplify) (divide by ) or 2.75 (ivert ad multiply) (reduce) (simplify)