4.4Solving Applied Problems Involving Proportions (284)
Objectives:
Solve applied problems using proportions
4.4.1 Solving Applied Problems Using Proportions
Again we can use the Book’s Mathematics Blueprint or the Given, Find, How, Solve, Solution,
Check methods.
We still cross multiply and simplify.
PP1 Yesterday an automobile assembly line produced 243 engines, of which 27 were defective. If the same rate is true each day, how many of the 4131 engines produced a month are defective? (285)
Given:27 engines defective out of 243 engines examined
Total of 4131 engines produced
Find:Number of the 4131 expected to be defective
How:Use proportion of number found/ number examined = number expected/number produced.
Solve:27 / 243 = x / 4131 (hint: 243 = 27 × 9)
27 × 4131 = 243 × x
459 = x
Solution:Of the 4131 produced, 459 engines are expected to be defective
Check:30 / 200 = x / 4000; 200x = 12000; x = 600 which isn’t that close
Note: I could have used the inverse of both sides of the proportion
Normally, we have to compute things like miles per gallon.
PP2 Cindy’s car travels 234 miles on 9 gallons of gas. How many gallons of gas will Cindy need to take a 312 mile trip? (286)
Given:234 miles use 9 gallons
Trip of 312 miles
Find:Gallons required for the whole trip
How:Use proportions miles for test/gal for test = miles for trip/gal for trip
Solve:234 / 9 = 312 / x
234 × x = 9 × 312
x = 12
Solution:It should take 12 gallons of gas for a trip of 312 miles
Check:200/10 = 300/x; 200x = 3000; x = 15, sort of close
PP3 Alicia must pedal at 80 revolutions per minute to ride her bicycle at 16 mph. If she pedals at 90 revolutions per minute, how fast will she be riding? (287)
Given:80 rev/min is 16 mph
She upgrades to 90 rev/min
Find:What speed will she be moving at this increased revs
How:Use proportions revs normal/speed normal = rev upgrade/speed upgrade
Solve:80 / 16 = 90 / x
80 × x = 16 × 90
x = 18
Solution:At 90 revs per minute, she will be travelling at 18 mph
Check:80 / 20 = 90 / x; 80x = 180; x = 25, sort of close
Gear ratio is used frequently in real life. Big Ben would be an example. [Draw a picture]
PP4 For every 4050 people who walk into Tom’s Souvenir Shop, 729 make a purchase. Assuming the same conditions, if 5500 people walk into Tom’s, how many people may be expected to make a purchase.
Given:729 make a purchase if 4050 come in
5500 expected to come in
Find:How many expected to make a purchase with the increased traffic
How:Use proportions # purchase normal/ normal come in = expected purchase / increased traffic
Solve:729 / 4050 = x / 5500
729 × 5500 = 4050 × x
990 = x
Solution:If 5500 people come in Tom’s, 990 are expected to purchase something
Check:700 / 4000 = x / 6000; 4000x = 4300000; x = 1075, sort of close
PP5 A park ranger in Alaska captures and tags 50 bears. He then reduces them to range through the forest. Sometime later he captures 50 bears. Of the 50, 4 have tags from the previous capture. Estimate the number of bears in the forest.
Given:4 tagged bears out of 50 bears tagged
50 bears originally tagged.
Find:Total number of bears in the forest
How:Use proportions # tagged caught/ total caught = total caught/ total number
Solve:4 / 50 = 50 / x
4 × x = 50 × 50
x = 625
Solution:There should be 625 bears in the forest
Check:4 / 50 = 50 / x (hey, that’s what we just did)
Developing Your Study Skills” Getting the right type of help is essential in math. Go to class regularly, taken useful notes, read the text, and do your homework. If you are bashful to ask questions in class, see your teacher before or after class. There is also the math help center. There might be tutors available. If you are having problems, I suggest again you buy the solution manual.