This Lesson depends on Parts of Natural Numbers 1
In this Lesson, we will answer the following:
Example 1. How much is two and a half times 8?
Answer. "Two and a half times 8" means
Two times 8 plus half of 8.
Two times 8 is 16. Half of 8 is 4. 16 plus 4 is 20.
Example 2. A cheese sells for $6 a pound, and you buy three and a half pounds. How much do you pay?
That is, "Three and a half times 6" means
Three times 6 plus half of 6.
18 + 3 = 21.
That is a mixed number of times: A whole number of times plus a part.
Example 3. How much is five and a quarter times 8?
Example 4. How much is a third of 120?
Answer. Ignore the 0. Then
A third of 12 is 4.
A third of 120 is 40.
A third of 1,200 is 400.
A third of 120,000 is 40,000.
It is a 4 followed by four 0's.
Answer. If we ignore all the 0's, then we cannot take an eighth of 4. But if we ignore only two 0's, then
An eighth of 40 is 5.
An eighth of 4000 is 500.
Finding a part by dividing
We have seen in Lesson 10 that to divide a number into equal parts, we divide. We can now restate that in the language of parts. And everything we know about division will follow.
To find half of a number, divide by 2; to find third, divide by 3; to find a fourth, divide by 4; and so on. (This is a theorem whose proof is indicated in Lesson 10, Example 5.)
Example 6. How much is half of 112?
Solution. We must divide 112 by 2. But to do that, we can easily decompose 112 into two numbers that are obviously divisible by 2 -- that is, two numbers whose half we know (Lesson 10):
112 = 100 + 12.
Half of 100 is 50. Half of 12 is 6. Therefore, half of 112 is 56.
In other words, a part can be distributed.
(Euclid, VII. 5.)
Half of 100 + Half of 12 = Half of 112.
This is also true for adding parts:
Three fourths of 100 + Three fourths of 100 = Three fourths of 200.
Example 7. How much is a third of 252?
Solution. Upon decomposing 252 into 240 + 12:
Example 8. Three people go to lunch and the bill is $32.40. How much does each one pay?
Solution. We must find a third of $32.40. Now, a third of $30 is $10. A third of the remainder, $2.40, is $.80. Each one pays $10.80.
Example 9. How much is a fifth of $37.50
Solution. What number closest to 37 has an exact fifth part? 35. Therefore, decompose $37.50 as
$35 + $2.50.
A fifth of $35 is $7.
A fifth of $2.50 is $.50.
Therefore, a fifth of $37.50 is $7.50.
We will see below another way to find a fifth.
Example 10. How much is a tenth of $62? How much is a hundredth?
Answer. To find a tenth of a number, divide by 10. To divide a whole number by 10, separate one decimal place. (Lesson 3, Question 5.) Therefore,
A tenth of $62 is $6.20.
To find a hundredth of a whole number, separate two decimal places:
A hundredth of $62 is $.62.
Now, a tenth of a number is 10% of it, and a hundredth of a number is 1%. (Lesson 3, Questions 8 and 9.) Therefore we have found 10% and 1% of $.62.
Example 11. How much is a fifth or 20% of $240?
Solution. A tenth of $240 is $24. Therefore a fifth is 2 × $24 = $48.
To see that a fifth is twice as much as a tenth, divide the green line into tenths, that is, into ten equal pieces. If we divide that same line into fifths, that is, into five equal pieces, then we can see that each fifth is twice as much as a tenth.
Also, in Lesson 3, Question 9, we saw that 20% is twice as much as 10%, which is a tenth.
Example 12. How much is 20% of $345?
Solution. 10% or a tenth of $345 is $34.50. Therefore a fifth is
Example 13. The percent that means a third.
a) In a recent exam, a third of the class got A. What percent got A?
Answer. Since the whole class is 100%, then a third of the class will be a third of 100%. We must divide 100 by 3. It will not be a whole number. (Lesson 10.)
Again, percents are parts of 100%. (Lesson 14.) Just as 50% means half because 50 is half of 100 and 25% means a quarter, because 25
b) What percent means two thirds?
In Section 2 we will see a simple way to find a quarter or 25% of a number.
Example 14. Calculator problem. How much is five eighths of $650.16?
Solution. To find five eighths, we must first find one eighth. Press
650.16 ÷ 8
Five eighths will be
5 × 81.27 = 406.35
On a simple calculator, the problem can be done in sequence by pressing
650.16 ÷ 8 × 5 =
At this point, please "turn" the page and do some Problems.
Continue on to the Section 2.