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A R I T H M E T I C

Lesson 18  Section 2

# RATIO AND PROPORTION

## "Out of"

 "Out of":  The ratio Part   Whole

 4. What does "out of" signify? 3 out of 5 The ratio of a smaller number to a larger, of a part to the whole.

This illustrates 3 out of 5:  the ratio of 3, the part, to the whole 5.
3 is three fifths of 5;  Lesson 17.  (Here, "part" refers to whatever is less than the whole.)

"Out of" is often how fractions are introduced -- as ratios, but with fractional symbols. A fraction, however, is a number we need for measuring. "Out of" does nothing to explain where the number 3/5 belongs on the number line. See Lesson 20.

Example 1.   In a class of 40 students, 3 out of 5 got B.  How many students got B?

Solution 1.  Complete the proportion,

 35 = ? 40

"3 out of 5  is  how many  out of 40?"

Now, 40 -- the term that corresponds to 5 -- is 8 × 5.  Therefore the missing term will be 8 × 3.

3 out of 5  is equal to  8 × 3  out of  8 × 5:

24 out of 40.

24 students got B.

That should be a simple mental calculation.

Notice that "3 out of 5" -- a smaller number out of a larger -- makes sense.  It would make no sense to say "5 out of 3."

"3 out of 5" means, For every 5, there are 3.

If there are two 5's,

there will be two 3's.  If there are three 5's, there will be three 3's:

And so on.  Therefore, if there are eight 5's, there will be eight 3's.  This is the theorem of the same multiple (Section 1).

Solution 2.  To say that 3 out of 5 got B, is to say that three fifths of the students got B.  One fifth of 40 is 8.  Therefore, three fifths are 3 × 8 = 24.

Example 2.   In a recent survey, 7 out of 10 people responded Yes.  If 280 people responded Yes, then

a)  how many people were surveyed?      b)  how many responded No?

Solution.  In this Example, 280 is the part that responded Yes.  It corresponds to 7.

 Part  Whole = 7  10 = 280  ?

Now, 280 is what number times 7?  280 is 40 × 7.  Therefore the missing term is 40 × 10.

 7 10 = 280 40 × 7 400 40 × 10

400 is the whole number of people surveyed.

b)   Since 280 is the number that responded Yes, then the difference,

400 − 280,

b)    will be the number that responded No.

400 − 280 = 120.

Percent: Out of 100

Percent problems often involve the expression "out of," because percent is how many for each, or out of, 100.  Therefore any number out of 100  is that percent.

Example 3.   8 out of 100 is what percent?

 8  100 = 8%.

Note:  When we consider the ratio of a smaller number to 100, then we may say "out of."  But when we have the ratio of a larger number to 100 -- 200 to 100 -- then 200 "out of" 100 make no sense.  In that case, we must say "for each."

Example 4.   200 for each 100 is equal to what percent?

Example 5.   8 out of 25 is equal to what percent?

Answer.   Now percent is not out of 25, it is out of 100.  Therefore, let us complete this proportion:

 8 25 = ?  100

"8 out of 25 is equal to how many out of 100."

Since 100 is 4 × 25, then the third term will be 4 × 8:

 8 25 = 32 4 × 8 100 4 × 25

But 32 out of 100 is 32%.  And since that is equal to 8 out of 25, then 8 out of 25 is also 32%

Note that the following questions mean the same:

8 out of 25 is what percent?

8 is what percent of 25?

Finding a percent by making the fourth term 100 is called the method of proportions.

 5. How do we find a Percent by the method of proportions? Make the 4th term of a proportion 100. The 3rd term is then the Percent that the 1st is of the 2nd.

We will go into this more in Lesson 30.

Example 6.   18 is what percent of 200?

Solution.  Proportionally,

 18200 = ? 100

"18 out of 200  is  how many  out of 100?"

Now, to go from 200 to 100, we have to divide by 2.  Therefore, we have to divide 18 by 2, also:

 18200 = 9     18 ÷ 2 100 200 ÷ 2

18 is 9% of 200.

Example 7.

a)  We know that 6% is 6 out of 100.  .6% is 6 out of how many?

.6 out of 100.

Therefore on multiplying both terms by 10:

.6 out of 100   is  6 out of 1000.

b)  .06% is 6 out of how many?

Answer.   On multiplying both terms by 100,

.06 out of 100   is  6 out of 10,000.

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