Example 1. What ratio has 25 to 10?
Answer. 25 is composed of two 10's, plus a remainder of 5.
25 = 20 + 5.
The remainder 5 is a part of 10, namely half. Therefore we say,
"25 is two and a half times 10."
Two times 10 is 20; half of 10 is 5; 20 plus 5 is 25.
We always say that a larger number is so many times a smaller number. 25 is two and a half times 10.
Example 2. What ratio has 14 to 4 -- that is, 14 is how many times 4?
Answer. Again, we can decompose 14 into a multiple of 4 plus a remainder:
14 = 12 + 2.
14 is made up of three 4's with remainder 2, which is half of 4. Therefore we say,
"14 is three and a half times 4."
Again, we say that a larger number is so many times a smaller. And when the first term is larger, the word "times" will immediately precede the second term. "14 is . . . times 4"
Example 3. What ratio has 50 to 40?
Answer. 50 is one and a quarter times 40.
For, 50 = 40 + 10.
50 contains 40 one time with remainder 10, which is a quarter of 40.
What is most important is that we now see that we can always express in words the relationship -- the ratio -- of any two natural numbers.
Example 4. What ratio has 13 to 3?
Answer. Since
13 = 12 + 1,
then
"13 is four and a third times 3."
For, the remainder 1 is a third of 3.
Example 5. What ratio has 14 to 3?
Answer. Since
13 = 12 + 2,
then
"13 is four and two thirds times 3."
The remainder 2 is two thirds of 3.
Example 6. In a survey, the ratio of Yes's to No's was 5 to 2. There were 406 No's. How many Yes's were there?
Solution 1. What ratio has 5 to 2?
"5 is two and a half times 2."
5 = 4 + 1.
The number of Yes's, then, is two and half times 406. Two times 406 is 812. Half of 406 is 203. 812 + 203 = 1,015.
Solution 2. Proportionally,
Example 7. If 6 workers can paint 4 rooms in 5 hours, how long will it take 15 workers to paint 14 rooms?
Solution. We must find out how many rooms 15 workers could paint in ONE hour. Why? Because that will tell us the number of rooms per hour. (Lesson 3.) It will then be a simple matter to know how many hours will be needed to paint 14 rooms.
(This is the standard procedure in what is called a "jobs" problem.)
Now, 6 workers can paint 4 rooms in 5 hours. How many rooms could 15 workers paint in 5 hours?
15 workers are two and half times 6 workers. (15 = 12 + 6. Compare Example 1.) Therefore in 5 hours they could paint two and a half times as many rooms.
15 workers can paint 10 rooms, then, in 5 hours. This implies that they can paint 2 rooms per hour. Therefore to paint 7 times 2 rooms will require 7 hours.
