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RATIO AND PROPORTION  2

Lesson 17  Section 3

Mixed ratio

In Lesson 15, we saw that

"Three and a half times 6"

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.

This is called a mixed ratio.
 6.   What is a mixed ratio?
25 to 10
  A ratio in which the larger number is a multiple of the smaller number, plus a part of the smaller number.

Example 1.   What ratio has 25 to 10?

Answer.  We can decompose 25 into a multiple of 10 plus a remainder:

25 = 20 + 5.

25 is made up of two 10's, plus a remainder of 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,

Yes's
 No's 		 =  5
2 		 =    ? 
406

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

200 × 5  +  3 × 5 = 1,015.

For another kind of problem involving mixed ratio, see Lesson 25, Example 15.


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

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