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A R I T H M E T I C

RATIO AND PROPORTION  3

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 a larger number is a multiple of a smaller number, plus a part of the smaller number.

Example 1.   What ratio has 25 to 10?

Answer.  How many 10's are there in 25?

25 is made up of two 10's and a remainder of 5.  The remainder 5 is half of 10.  Therefore we say,

"25 is two and a half times 10."

Two times 10 is 20; half of 10 is 5;  20 + 5 = 25.

We always say that a larger number is so many times a smaller number.  25 is two and a half times 10.

We could discovered that by dividing 25 by 10:

 2510 ("25 divided by 10,"  Lesson 10)  =  2 R 5.

That reveals the relationship between ratio and division, and why we can use the division bar to signify ratio.  For, the quotient of two numbers indicates their ratio.  The ratio of 15 to 5, for example, is indicated by 15 ÷ 5 = 3.  This implies the ratio of 15 to 5:

15 = 3 × 5.

"15 is three times 5."

The traditional way to symbolize a ratio is 15:3. This is the division sign ÷ but without the bar.

Example 2.   What ratio has 14 to 4 -- that is, 14 is how many times 4?

Answer.   14 is made up of three 4's with remainder 2.

14 ÷ 4 = 3 R 2.

The remainder 2 is a part of 4, namely half.  Therefore we say,

"14 is three and a half times 4."

Again, we say that a larger number is so many times a smaller.

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.  That is,

50 ÷ 40 = 1 R 10.

50 contains 40 one time with remainder 10, which is a quarter of 40.  Therefore,

50 is one and a quarter times 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 11 to 2?

Answer.  If we express the quotient as a mixed number,

 11 2 = 5½

then this tells us,

"11 is five and a half times 2."

This is an easy way, of course, to get the answer.  But the student should not be content just to get the answer.  The student should see the fact.  11 is five and a half times 2.

Example 5.   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 ÷ 2 = 2 R 1; and 1 is half of 2.

"5 is two and a half times 2."

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 = 52 = ? 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.

 7. What does it mean to express a relationship between numbers in the language of ratio? It means to say only the ratio of the numbers, without naming the numbers themselves.

Example 1.   Joan earns \$1600 a month, and pays \$400 in rent.  Express that fact in the language of ratio.

Answer.  "A quarter of Joan's salary goes for rent."

That sentence, or one like it, expresses the ratio of \$400 to \$1600, of the part that goes for rent to the whole.  We are not concerned with the numbers themselves, but only their ratio.

Example 2.   In Jim's class there are 30 pupils, while in Jane's there are only 10.  Express that fact in the language of ratio.

Answer.  "In Jim's class there are three times as many as in Jane's."

This expresses the ratio of 30 pupils to 10.

Example 3.   In a class of 24 students, there were 16 B's.  Express that fact in the language of ratio.

Answer.  "Two thirds of the class got B."

This expresses the ratio of the part that got B to the whole number of students:  16 out of 24.  Their common divisor is 8.  8 goes into 16 two times and into 24 three times.  16 is two thirds of 24.

Example 4.   This month's bill is \$75, while last month's was only \$30.  Express that fact in the language of ratio.

Answer.  "This month's bill is two and a half times last month's."

75 is equal to two times 30 (60) with a remainder of 15, which is half of 30.

75 = 60 + 15

75 is two and a half times 30.

The student will hear this language, will read it, and should be able to understand it and speak it.

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

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Copyright © 2001-2007 Lawrence Spector

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