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A R I T H M E T I C

Lesson 18  Section 3

RATIO AND PROPORTIONMixed ratio

Back to Section 1

We have seen that

means

Three times 6 plus half of 6.

Three times 6 is 18;  half of 6 is 3;  therefore, three and a half times 6
is 21.

In other words, the ratio of 21 to 6 is:

21 is three and a half times 6.

That is called a mixed ratio.

 6. What is a mixed ratio? 25 to 10 A ratio in which the larger number is not an exact multiple of the smaller. The larger number will be a multiple of the smaller, plus a part of it.

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 see that we can always express in words the relationship -- the ratio -- of any two natural numbers.

Example 4.   What ratio has 7 to 3?

7 = 6 + 1,

then

"7 is two and a third times 3."

The remainder 1 is a third of 3.

Example 5.   What ratio has 8 to 3?

8 = 6 + 2,

then

"8 is two and two thirds times 3."

The remainder 2 is two thirds of 3.

Example 6.   In a survey there were 5 Yes's for every 2 No's.  There were 406 No's.  How many Yes's were there?

Solution 1.   The ratio of Yes's to No's was 5 to 2.  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 the number of No's -- 406.

Two times 406 is 812.  Half of 406 is 203.  812 + 203 = 1,015.

Solution 2.   Proportionally,

 Yes'sNo's = 52 = ? 406

406 = 203 × 2.  Therefore, the missing term is 203 × 5.

200 × 5  +  3 × 5 = 1000 + 15 = 1,015.

Example 7.   8 out of 50 students got A on an exam.  Assuming that same ratio of the part to the whole:

a)  How many would have got A if 250 students had taken the exam?

 Solution. 8 50 = ?  250

250 is five times 50.  Therefore, five times 8 -- 40 -- students would have got A.

b)  How many would have got A if 75 students had taken the exam?

 Solution. 8 50 = ? 75

What number times 50 is 75?  That is, what ratio has 75 to 50?

 75 = 50 + 25 = 50 + Half of 50.

75 is one and a half times 50.  Therefore, one and a half times 8 -- 12 -- students would have got A.

Example 8.   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 11.)  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. If there were twice as many workers, they could paint twice as many rooms in 5 hours. So how many rooms could 15 workers paint in 5 hours?

15 workers are two and half times 6 workers. (15 = 12 + 3. Compare Example 1.) Therefore in 5 hours they could paint two and a half times as many rooms.

 Two and a half times 4 = Two times 4 + Half of 4 = 8 + 2 = 10 rooms.

15 workers, then, can paint 10 rooms in 5 hours. This implies that they can paint 2 rooms per hour. Therefore to paint 14 rooms will require 7 hours.

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

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