31. 21 is 32. 30 is 33. 14 is 34. 12 is 35. 18 is

ΤΟ

8

of what number? Is 3? Is? Is fr ?
of what number? Is ?
of what number? Is

of what number? Is

Is g? Is ?
Is T ? Is ?

? Is? Is ?
? Is 20

of what number? Is ? Is 13

1

?

36. Samuel Page sold a pair of oxen for $48, which was of their cost. What did he lose?

37. Bought a horse for $72, which was of his real value; what did I gain?

38. 72 is & of what number?

39. Sold a quantity of depreciated money for $81, which was of its nominal value; what was the sum sold? 40. Having improved a chaise 15 years, it was sold for $25, which was only of what it cost. What was the original price?

41. A Loafer shot at a flock of pigeons on a tree, and killed 24, which was of the number. How many pigeons will remain on the tree?

Section 20.

VULGAR FRACTIONS.

FRACTIONS are parts of an integer.

VULGAR FRACTIONS are expressed by two terms, called the Numerator and Denominator; the former above, and the latter below a line.

The Denominator shows into how many parts the integer, or whole number, is divided.

The Numerator shows how many of those parts are taken.

1. A proper fraction is one whose numerator is less than the denominator, as .

2. An improper fraction is one whose numerator exceeds, or is equal to, the denominator; as 12 or §.

3. A simple fraction has a numerator and denominator only; as,.

4. A compound fraction is a fraction of a fraction, connected by the word of; as 7 of & of § of 2.

5. A mixed number is an integer with a fraction; as 76, 58.

6. A compound mixed fraction is one whose numerator or denominator, or both, is a mixed number; as 71

or

44

7

7. The greatest common measure of two or more numbers is the largest number, that will divide them without a remainder.

8. The least common multiple of two or more numbers is the least number, that may be divided by them without a remainder.

9. A fraction is in its lowest terms, when no number but a unit will measure both its terms.

10. A prime number is that which can be measured only by itself or a unit ; as 7, 11, and 19.

11. A perfect number is equal to the sum of all its aliquot parts; as 6, 28, 496, &c.

12. A fraction is equal to the number of times the numerator will contain the denominator.

13. The value of a fraction depends on the proportion, which the numerator bears to the denominator.

I. To find the greatest common measure of two or more numbers; that is, to find the greatest number that will divide two or more numbers.

1. What is the common measure of 84 and 132; that is, what is the largest number, that will divide both of these numbers without a remainder? Ans. 12.

vide 48 and 132 without a remainder. It is, therefore, its greatest common measure. Hence the following

RULE.

Divide the greater number by the less, and if there be a remainder, divide the last divisor by it, and so continue dividing the last divisor by the last remainder, until nothing remains, and the last divisor is the greatest common meas

ure.

If there be more than two numbers, find the greatest common measure of two of them, and then of that common measure and the other numbers. If it should happen, that 1 is the common measure, the numbers are prime to each other, and are incommensurable.

2. What is the greatest common measure of 85 and 95 ? Ans. 5. 3. What is the greatest common measure of 72 and 168 ? Ans. 24. 4. What is the greatest common measure of 119 and 121? Ans. 1. 5. What is the largest number that will divide 324 and 586 ?

6. What is the largest number that will 684 ?

7. What is the greatest common measure

Ans. 2. divide 582 and Ans. 6.

of 32 and 172? Ans. 4.

8. What is the largest number that will divide 84 and 1728 ?

Ans. 12.

9. What is the greatest common measure of 16, 20, and 26? Ans. 2.

10. What is the greatest common measure of 12, 18, 24, and 30?

Ans. 6.

II. To reduce fractions to their lowest terms. NOTE. A fraction is said to be in its lowest terms, when no number but a unit will divide its numerator and denominator.

1. Reduce to its lowest terms.

OPERATION.

5)

Ans.

We find by the last Rule, that 5 is the largest number, that will divide both the numerator and denominator

of the fraction; and having divided them both by it, we

find the result to be, and that is equal to is evident from the fact, that the ratio of 5 to 15 is equal to the ratio of 1 to 3. And, as the value of a fraction depends on the ratio, which the numerator bears to the denominator, if their ratios are equal, the fractions are also equal. Q. e. d. Hence the following

RULE.

Divide the numerator and denominator by any number that will divide them both without a remainder; and so continue until no number will divide them but unity. Or, divide the numerator and denominator by the greatest common

8. Reduce to its lowest terms.

9. Reduce it to its lowest terms.

Ans.

Ans..

Ans..

Ans..

Ans..

Ans. 18.

Ans. +.

Ans.

10. What is the lowest expression of ? Ans. 173.

III. To reduce mixed numbers to improper fractions.

MENTAL OPERATIONS.

1. In 3 dollars how many halves? How many thirds? 2. In 7 apples how many tenths? How many twelfths? 3. In 83 dollars how many sevenths?

oranges how many fourths?

gallons how many elevenths? quarts how many fifths of quarts?

4. In 3

5. In 9

6. In 7

OPERATION.

7

5

35

3

38

5

We analyze this question by saying, as there are 5 fifths in one quart, there will be 5 times as many fifths as quarts; therefore, in seven quarts and three fifths, there will be 38 fifths, which should be expressed thus, 38. And this fraction, by definition 2d, on page 76, is an improper fraction. Hence the following

RULE.

Multiply the whole number by the denominator of the fraction, and to the product add the numerator, and place their sum over the denominator of the fraction.

7. Reduce 8 to an improper fraction.

8. Reduce 15

9. In 187 how 10. In 161

to an improper fraction.

many ninths?

Ans. Ans. 187.

Ans. 162.

Ans. 1884.

how many one hundred and seventeenths?

117

11. Change 4311 to an improper fraction Ans. 142. 12. What improper fraction will express 27?

13. Change 111 to an improper fraction?

Ans. 36.

Ans. 12322.

IV. To change improper fractions to integers or whole numbers.

MENTAL OPERATIONS.

1. How many dollars in 4 halves? In 5 halves? In 6 halves? In 7 halves? In 12 halves? In 19 halves? 2. How many dollars in 5 quarters? In 9 quarters ? 3. How many dollars in 10 eighths? In 20 eighths?

FOR THE SLATE.

4. How many dollars in 7 dollars?

OPERATION.

16) 37 (21%

32

Ans. 2.

This question may be analyzed by saying, as 16 sixteenths make one dollar, there will be as many dollars in 37 sixteenths as 37 contains 16, which is 21% times, $2. This answer is called a mixed number by definition 5th, page 77. Hence we see the propriety of the following

5

RULE.

Divide the numerator by the denominator, and if there be a remainder, place it over the denominator at the right hand of the integer.

5. Change 17 to a mixed number.

Ans. 10.