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Lesson 17

RATIO AND PROPORTION 1

Ratio is the language of arithmetic. It is the language with which we relate things of the same kind.

15 people, for example, are more than 5 people. We can express that by saying how much more, that is, 10 more, or by saying how many times more: Three times more. When we say that 15 people are three times more than 5 people, that is called their ratio.

(For a discussion of saying "times more" versus just saying "times," see below.)

Most important, we will see that percents are ratios. 6 people are half of 12 people -- that is the ratio of 6 to 12. Half. In the language of percent, we say that 6 people are 50% of 12 people. Why? Because 50 has that same ratio to 100. Half.

This Lesson depends on Lesson 15: Parts of Natural Numbers

In this Lesson, we will answer the following:

What do we mean by the ratio of two natural numbers?
How do percents express ratios?
Section 2
What is a proportion?
Section 3
How can we express the ratio of any smaller number to a larger number?
When is a ratio expressed with the lowest terms?
How can we know when a ratio is expressed with the lowest terms?
What is the Theorem of the Common Divisor?

1.	What do we mean by the ratio of two natural numbers?

	It is their relationship with respect to relative size that we can express verbally in a sentence. Specifically, one number is a multiple of the other (so many times it), a part of it, or parts of it.

(Euclid, Book VII. Def. 20.)

Example 1. Multiple. What ratio has 15 to 5?

Answer. 15 is three times 5.

That is the ratio -- the relationship -- of 15 to 5. If Jill has $15, and Jack has $5, then Jill has three times more than Jack.

To answer "3 to 1" is not sufficient, because we want to name the ratio of 15 to 5 explicitly. It is true that 15 is to 5 as 3 is to 1 -- but what ratio has 3 to 1?

3 is three times 1.

(The 19th century program to rid mathematics of language and replace it with algebraic relations, successfully put to sleep the subject of ratio and proportion.)

Notice that we answer with a complete sentence beginning with the first number 15, and ending with the last number 5. For, a ratio is a relationship.

The two numbers in a ratio are called the terms; the first and the second.

When the first term is larger, we say it is so many times the smaller. 15 is three times 5.

What ratio has 28 to 7?

28 is four times 7.

Example 2. Part. What ratio has 5 to 15?

Answer. 5 is the third part of 15.

That is called the inverse ratio of 15 to 5. The terms are exchanged.

Notice again that we answer with a complete sentence beginning with the first term and ending with the second. "5 is 15."

Example 3. Parts. What ratio has 10 to 15?

Answer. 10 is two thirds of 15.

"Three times." "The third part." "Two thirds." Those are names of the three types of ratio. One number is a multiple of the other (so many times it), a part of it, or parts of it.

Example 4. What ratio has 12 to 6?

Answer. 12 is two times 6. Or we could say, "12 is twice as much as 6," or "12 is double 6."

These are the various ways of expressing the ratio, the relationship, of 12 to 6.

Inversely, 6 is half of 12.

When trying to express a ratio, if the student will say a sentence, and then consider the truth of that sentence, the fact will speak for itself.

Example 5. What ratio has 80 to 8? Inversely, what ratio has 8 to 80?

Answer. 80 is ten times 8. Therefore, inversely, 8 is that part of 80 with the ordinal form of ten: 8 is the tenth part of 80.

(For the relationship between 8 and 8-with-a-0 after it, i. e. 80, see Lesson 2 and the problems that follow.)

Example 6. What ratio has 800 to 8? Inversely, what ratio has 8 to 800?

Answer. 800 is one hundred times 8. Inversely, 8 is the hundredth part of 800.

Percents are ratios

A percent is a ratio -- a relationship between numbers. What ratio has 6 to 12? 6 is 50% of 12. Why does 50% mean half? Because 50% is half of 100%.

2.	How do percents express ratios?

	Whatever ratio the percent has to 100 percent, that is the ratio we mean.

Example 7. What does 200% mean?

Answer. Since 200% is two times 100%, then 200% means two times. 200% of 8 -- two times 8 -- is 16.

Example 8. How much is 300% of 8?

Answer. 24. 300% of 8 means three times 8, because 300% is three times 100%.

100% is the whole, in this case, 8. 50% means half, because 50% is half of 100%. 50% of 8 is 4.

200% means two times; 300% means three times; 400% means four times; and so on.

Whatever ratio the percent has to 100%, that is the ratio we mean.

We see that any number less than 8 will be less than 100% of 8. While any number more than 8 will be more than 100%.

We have, as it were, two languages: The language of ratio -- "Half," "Three quarters," "Twice as much" -- and the language of percent: 50%, 75%, 200%. The student must become fluent in both languages, and in translating from one to the other.

Example 9. Compare the following:

a) 10 has what ratio to 40? b) 10 is what percent of 40?

c) 7 has what ratio to 21? d) 7 is what percent of 21?

Answers. a) 10 is the fourth part of 40, or a quarter of 40, or a fourth of 40.

b) 10 is 25% of 40.

25% means a quarter, because 25% is a quarter of 100%.

c) 7 is a third of 21. d) 7 is 33

1
3

% of 21.

See Lesson 16, Question 6.

Example 10. How much is 250% of 8?

Answer. 250% means two and a half times. It is 200% + 50%. 250% is a mixed number of times -- expressed as a percent.

250% of 8, therefore, is 16 + 4 = 20.

Example 11. How much is 433

1
3

% of $60?

Answer. 433

1
3

% = 400% + 33

1
3

%: Four and a third times.

433

1
3

% of $60 = $240 + $20 = $260.

Example 12. Calculate mentally: 125% of $7.80.

Answer. 125% means one and a quarter times: 100% + 25%.

Now, to take a quarter of $7.80, we may think of it as

$8.00 − 20 cents.

A quarter of $8.00 is $2.00.

A quarter of 20 cents is 5 cents.

Therefore,

125% of $7.80	=	$7.80 + $2.00 − 5 cents

	=	$9.80 − 5 cents

	=	$9.75.

Practice with these problems continues in Lesson 28.

Saying "times more" versus saying "times"

15 is more than 5. How many times more?

15 is three times more. No? Yet some critics insist that we should not say "15 is three times more than 5" -- because they say that should mean 20. Why? Because three times 5 is 15. So three times more than 5 should mean 5 + 15, which is 20.

There is good reason to reject that argument, however, and it is simply because of the accepted meanings of our spoken language. There is nothing the least ambiguous about saying, "15 is three times more than 5."

As an example of consistency, these critics cite, "How much is 50% more than 10?" To that question it is correct to answer 15. But that is simply another idiomatic use of language when speaking of percent. Though it might seem to follow logically that we should speak that way in all cases, it does not follow linguistically. Mathematicians cannot require us to speak a private language.

As Merriam-Webster's Dictionary of English Usage puts it:

"The question to be asked ... is not whether it makes sense mathematically, but whether it makes sense linguistically � that is, whether people understand what it means. ... The 'ambiguity' of times more is imaginary: in the world of actual speech and writing, the meaning of times more is clear and unequivocal. It is an idiom that has existed in our language for more than four centuries, and there is no real reason to avoid its use."

So, 15 is more than 5. How many times more?

15 is three times more than 5.

At this point, please "turn" the page and do some Problems.

Continue on to the next Section: Proportions

1st Lesson on Parts of Natural Numbers

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