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136. TO FIND WHAT PART ONE NUMBER IS OF

ANOTHER.

NOTE.Younger pupils may omit this article.

Ans. 1 is of 2,

What part of 2 is 1, or 1 is what part of 2? because it is 1 of the 2 equal parts into which 2 may be divided.

1 is what part of 3? of 5? of 7? of 9? of 8? why? 1 is what part of 19? of 11? of 6? of 15? of 33?

ILLUSTRATIVE EXAMPLES.

1. 3 is what part of 10? 1 is of 10, .. 3 must be of 10. Ans. 2. is what part of 7?

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OPERATION.is of }, ..or 1 whole one is of .

4. What part of is 2?

OPERATION.

2 X 9

7

Ans.

is of, ... 3, or 1 is of 3, and 2 must Ans. 18.

be 2 X, or 18 of 7.

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is

5. What part of 1

OPERATION.

12 X 3 =

11 X 5

, Ans. must be

of 11.14, or 1 is 11 of 11, and of 1 of 11, or § of 1}. Ans. I§.

From the above we derive the following

RULE. To ascertain what part one number is of another: Divide the number expressing the part, by that of which it is a part.

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18. If by a pipe a cistern can be filled in 3 hours, what part of the cistern will be filled in 1 hour? in 2 hours?

19. If a piece of work can be performed in 9 days, what part of the work can be performed in 7 days?

20. A can perform a journey on foot in 7 days; what part of it can he perform in 21 days?

21. Mr. Bailey has $54, and pays $18 for a coat; what part of his money does he spend?

22. Charles picks 25 quarts of blackberries, and Eben 5% quarts. If Eben's blackberries are worth one dollar, what part of a dollar are Charles's worth?

23. A and B hired a pasture together. A pastured 12 cows, and B 13 cows in it; what part of the price should each pay?

24. Four men were hired to work on a farm; A mowed 7 acres ; B mowed 5 acres ; C, 4 acres, and D, 2 acres. They received $27. What was each one's share?

25. Mr. Snow, dying, left $75000 to his wife and three sons. To his wife, $30,000; to his oldest son just as large a part of the remainder as his wife's portion was of the entire property; to his 2d son of what his eldest received, and to his youngest the rest. What was each son's share ?

For Dictation Exercises, see Key.

MULTIPLES OF NUMBERS.

137. A Multiple of a number is any number that will contain it without a remainder; thus, 8, 12, 16, and 20, are multiples of 4.

138. A Common Multiple of two or more numbers is a number that will contain each of them without a remainder; thus, 20 is a common multiple of 5 and 2.

139. The Least Common Multiple of two or more numbers is the least number that will contain each of them without a remainder; thus, 10 is the L. C. M.* of 2 and 5.

EXERCISES.

Name any 6 multiples of 5. the multiples of 11 up to 140. and 6. Of 3, 6, and 5.

Name 3 multiples of 12. Name all
Name any common multiple of 10

140. TO FIND THE LEAST COMMON MULTIPLE OF TWO OR MORE NUMBERS.

The common multiple of two or more numbers must contair

* Least Common Multiple.

all the factors of those numbers, and the least number that con tains all their factors must be the least common multiple.

ILL. Ex. Find the L. C. M. of 4, 6, 10 and 15.

OPERATION.

4 = 2 X 2

6 = 2 × 3

10 = 2 X 5

15 3 X 5

We find the factors of 4 to be 2 and 2, of 6 to be 2 and 3, of 10 to be 2

and 5, of 15 to be 3 and 5. To contain 4, the L. C.

M. must contain the fac

L. C. M2 X 2 X 3 X 560, Ans. tors 2 and 2, which we note. To contain 6, it must contain 2 and 3; we have already noted 2, so we need introduce only the 3. To contain 10, it must contain 2 and 5; as we have noted 2, we introduce only the 5. To contain 15, it must contain 3 and 5; we have already noted these factors, ..2 × 2 × 3 × 5 60, must be the L. C. M. Hence the

RULE. To find the L. C. M. of two or more numbers: Separate the numbers into their prime factors. Find the product of all the different prime factors, taking each factor the greatest number of times it occurs as a factor in any one number.

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3. 3, 5, 8, 12, 20, 36, and 45.

NOTE.

Ans. 2520.

Ans. 336.

Ans. 360.

- When one of the given numbers is a factor of another, it may be disregarded in the operation; thus, in the preceding example, 3, 5, and

12 may be rejected. Why?

Find the L. C. M. of

4. 18, 36, 40, 60, and 72.

5. 12, 16, 42, 56, and 70.

7. 9, 18, 32, 48, and 52.
8. 8, 16, 28, 35, and 63.
9. Of the nine digits.

6. 13, 28, 35, 39, and 49. When several numbers are prime to each other, what must their L. C. M. equal?

141. The above is the better method for finding the L. C. M. when the numbers are easily separated into their prime factors. For larger and more difficult numbers observe the following method:

ILL. Ex. Find the L. C. M. of 36, 112, 76, and 60.

OPERATION.

2) 36, 112, 76, 60

2)18, 56, 38, 30

Here, by repeated divisions, we take out all the factors that

3) 9, 28, 19, 15

3, 28, 19, 5

L. C. M.

are common, 2,

2×2× 3 × 3 × 28 × 19 × 5=95760. 2, and 3; the

least

common

multiple must contain these factors and those which are not common; .. 2 × 2 × 3 × 3 X 28 X 19 X 5-95760, must be the L. C. M. sought. Hence the

RULE. To find the L. C. M. of two or more numbers: Divide by any prime factor which is contained in two or more of the numbers without a remainder, writing the quotient and undivided numbers in a line beneath, and thus proceed till no two numbers can be divided by the same prime. The product of all the divisors and the numbers remaining is the L. C. M.

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18. What is the width of the narrowest street, across which stepping stones either 4, 5, or 8 feet long will exactly reach?

19. What is the narrowest box that will exactly pack ribbons either 3, 4, or 5 inches wide?

20. What is the smallest bill that may be paid by using either dimes, three-cent pieces, or quarter dollars?

21. What is the smallest-sized cistern the contents of which may be exactly measured by using either 15, 28, or 36 gallon casks?

For Dictation Exercises, see Key.

142, REDUCTION OF FRACTIONS TO EQUIVALENT FRAC TIONS HAVING A COMMON DENOMINATOR.

When the denominators of fractions are alike, they are said to have a Common Denominator.

ILL. EX. Reduce, 2, and to fractions having a common denominator.

We can change these fractions to fractions of any given denominator; but the most convenient denominator for most purposes is that which is the least common multiple of the denominators of the given fractions; and, in the following examples, such denominators are always required. In the preceding example, we must first find the L. C. M. of 3, 4, and 6, which is 12; and then reduce 3, 4, and § to twelfths. 1 · 13, ·· }=} of 1 or, and =244=&. By the same process we find that = 12, and = 18. Ans. 11⁄2

Hence the

.*.

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18.

RULE. To reduce fractions to equivalent fractions having a common denominator : Reduce the fractions to their simplest forms; find the least common multiple of the denominators for the common denominator; multiply the numerator of each fraction by the number by which you would multiply its denominator to produce the common denominator.* The respective products will be the numerators of the required fractions.

ILL. EX. Reduce §,, and to fractions having the L. C. D.

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Here 72 is the L. C. M.; and as 8= 2 × 2 × 2, it must be multi

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* If that number is not readily seen, it may be found by dividing the common denominator by the denominator of the original fraction.

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