Ratio and proportion, Section 2 -- A complete course in arithmetic

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RATIO AND PROPORTION 1

Lesson 16 Section 2

3.	What is a proportion?

	A statement that two ratios are the same.

Example 1. 5 is to 15 as 8 is to 24.

This is a proportion because 5 is the third part of 15, just as 8 is the third part of 24. (Lesson 14.)

A proportion involves four numbers -- four terms. To explain why they are proportional we start with the first term and state its ratio to the second; then we state the ratio of the third term to the fourth.

Example 2. Why is this a proportion?

16 is to 2 as 80 is to 10.

Answer. This is a proportion because 16 is eight times 2, just as 80 is eight times 10.

Example 3. Why is this a proportion?

10 is to 15 as 2 is to 3.

Answer. This is a proportion because 10 is two thirds of 15, just as 2 is two thirds of 3.

Example 4. Complete this proportion:

8 is to 32 as 9 is to ?

Solution. 8 is the fourth part of 32. And 9 is the fourth part of 36.

The 1st term is to the 2nd as the 3rd is to the 4th.

Example 5. Complete this proportion:

27 is to 3 as ? is to 5

Solution. 27 is nine times 3. And 45 is nine times 5.

If the student will speak and use sentences, the answer will be clear.

Example 6. In each item below, what ratio has a to b?

(a simply means the first term; b means the second.)

a) a is to b as 1 is to 6.

b) a is to b as 10 is to 1.

Answers.

a) Since 1 is the sixth part of 6, then a is the sixth part of b.

b) Since 10 is ten times 1, then a is ten times b.

Symbolic representation

We will let the following signify a proportion:

1
2

4
8

Read this:

"1 is to 2 as 4 is to 8."

Why is this a proportion? Because 1 is half of 2, and 4 is half of 8.

(Can you be innocent and make-believe you never heard of a "fraction"? Fractions, as we will see, are based on ratios, not vice-versa.)

Example 7. Read this proportion, and complete it:

8
2

20
?

Answer. "8 is to 2 as 20 is to what number?"

Now, what ratio has 8 to 2?

8 is four times 2. And 20 is four times 5.

What is taught in most textbooks these days as "ratio and proportion," is not. The innocent student is taught to write the letter x for the unknown term, cross-multiply, and solve an algebraic equation. While that mechanical method gives the answer, it offers no understanding of why. It is not, therefore, educational.

A ratio is relationship expressed in words. It requires understanding.

Example 8. Complete this proportion:

7
21

4
?

Answer. "7 is to 21 as 4 is to what number?"

What ratio has 7 to 21?

7 is the third part of 21. And 4 is the third part of 12.

Example 9. Complete this proportion:

2
3

?
12

Answer. "2 is to 3 as what number is to 12?"

Now, 2 is two thirds of 3. What number is two thirds of 12?