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RATIO AND PROPORTION 2 Lesson 17 Section 2
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This illustrates 3 out of 5: the ratio of 3, the part, to the whole 5. "Out of" is often how fractions are introduced -- as ratios but with fractional symbols. A fraction, however, is a number we need for measuring. "Out of" does nothing to explain where the number 3/5 belongs on the number line. See Lesson 19. Example 1. In a class of 40 students, 3 out of 5 got B. How many students got B? Solution 1. Complete the proportion, Part to Whole:
Because that is the same as 3 out of 5 is how many out of 40? Now, 40 -- the term corresponding to 5 -- is 8 × 5. Therefore the missing term will be 8 × 3. 3 out of 5 is 8 × 3 out of 8 × 5: 24 out of 40. 24 students got B. That should be a simple mental calculation. Notice that "3 out of 5" -- a smaller number out of a larger -- makes sense. It would make no sense to say "5 out of 3." "3 out of 5" means, For every 5, there are 3. 
If there are two 5's, 
there will be two 3's. If there are three 5's, there will be three 3's: 
And so on. Therefore, if there are eight 5's, there will be eight 3's. This is the Theorem of the Same Multiple (Section 1). Solution 2. To say that 3 out of 5 got B, is to say that three fifths of the students got B. One fifth of 40 is 8. Therefore, three fifths are 3 × 8 = 24. Example 2. In a recent survey, 7 out of 10 people responded Yes. If 280 people responded Yes, then a) how many people were surveyed? b) how many responded No? Solution. In this Example, 280 is the part that responded Yes. It corresponds to 7.
Now, 280 is what number times 7? 280 is 40 × 7. Therefore the missing term is 40 × 10.
400 is the whole number of people surveyed. b) Since 280 is the number that responded Yes, then the difference, 400 b) will be the number that responded No. 400 Percent: Out of 100 Percent problems are a particularly important kind involving "out of," because percent is how many for each, or out of, 100. Example 3. 8 out of 100 is what percent? Answer. 8 out of 100 is the ratio of 8 to 100, which is 8%:
Note: When we consider the ratio of a smaller number to 100, then we may say "out of." But when we have the ratio of a larger number to 100 -- 200 to 100 -- then 200 "out of" 100 make no sense. In that case, we must say "for each." Example 4. 200 for each 100 is equal to what percent? Answer. 200%. Example 5. 8 out of 25 is equal to what percent? Answer. Now percent is not out of 25, it is out of 100. Therefore, let us complete this proportion:
"8 out of 25 is equal to how many out of 100." Since 100 is 4 × 25, then the third term will be 4 × 8:
But 32 out of 100 is 32%. And since that is equal to 8 out of 25, then 8 out of 25 is also 32% Note that the following questions mean the same: 8 out of 25 is what percent? 8 is what percent of 25? Finding a percent by making the fourth term 100 is called the method of proportions.
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We will go into this more in Lesson 29. Example 6. 18 is what percent of 200? Solution. Proportionally,
"18 out of 200 is how many out of 100?" Now, to go from 200 to 100, we have to divide by 2. Therefore, we have to divide 18 by 2, also:
"18 is the same percent of 200 that 9 is of 100." 18 is 9% of 200. Example 7. a) We know that 6% is 6 out of 100. .6%, then, is 6 out of how many? Answer. Since .6% is .6 out of 100, then on multiplying both terms by 10: .6 out of 100 is 6 out of 1000. b) .06% is 6 out of how many? Answer. On multiplying both terms by 100, .06 out of 100 is 6 out of 10,000. At this point, please "turn" the page and do some Problems. or Introduction | Home | Table of Contents Please make a donation to keep TheMathPage online. Copyright © 2001-2008 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |
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280,
