16. Bought 20 bushels of wheat for 8£. 3s. 11d.; what cost 1 bushel? what cost 200 bushels ?

Ans.

Ans.

17. Paid E. Bradley 81£. 19s. 2d. for 200 bushels of wheat; what cost 20 bushels ?

18. Mr. Day paid 3£. 4s. 2d. for 10yds. of cloth; what should he have paid for 97yds.?

19. If 8 barrels of flour cost 2£. 12s.,

Ans. 31£. 2s. 5d. what cost 29 barrels ? Ans. 9£. 8s. 6d.

20. If 17 bushels of wheat cost 6£. 13s. 2d., what cost 101 bushels? Ans. 39£. 11s. 2d. 21. Gave 10. 4s. 3d. for 19 yards of cloth; what cost 97 yards? Ans. 52£. 2s. 9d.

SECTION XVI.

VULGAR FRACTIONS.

FRACTIONS are parts of an integer, or whole number. An integer is any whole number or quantity, as 1, 7, 11, &c., or a pound, a yard.

VULGAR FRACTIONS are expressed by two numbers, 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 these parts are taken, or expressed by the fraction.

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 or .

3. A single or simple fraction consists of but one numerator and one denominator; as .

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

5. A mixed number is an integer with a fraction; as 7, 59. 6. A complex fraction is a fraction having a fraction or a mixed number for its numerator or denominator, or both; as, 7 1 8 1 81 9' 7'11

, or

7 97

8

7. The terms of a fraction are the numerator and denominator; the numerator being the upper term, and the denominator the lower.

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

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

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

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

12. Numbers are said to be prime to each other, when only a unit measures or divides them both without a remainder; thus, 7 and 11 are prime to each other.

13. Prime factors of numbers are those factors which can be divided by no number but by themselves or a unit; thus the prime factors of 21 are 7 and 3.

14. An even number is that which can be divided into two equal whole numbers.

15. An odd number is that which cannot be divided into two equal whole numbers.

16. A square number is the product of a number multiplied by itself.

17. A cube number is the product of a number multiplied by its

square.

18. A composite number is that produced by multiplying two or more numbers together.

19. The factors of a number are those whose continued product will exactly produce the number.

20. An aliquot part is that which is contained a precise number of times in another.

21. An aliquant part is such a number as is contained in another a certain number of times with some part or parts over. 22. A perfect number is that which is equal to the sum of all its aliquot parts, or is equal to the sum of all the numbers that will divide it without a remainder; thus 6 is a perfect number, because it can be divided by 3, 2, and 1; and the sum of these numbers is 6. But 12 is not a perfect number, because its aliquot parts are more than 12; thus 6+4+3+1 14. 8 is not a perfect number, because its aliquot parts are less than 8; thus 4+2+1=7. But 28, 496, and 8128

=

are perfect numbers. The chief use of a knowledge of these numbers is in the higher branches of mathematics.

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

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

25. Ratio is the relation which two numbers or quantities of the same kind bear to each other, and may be found by dividing one number by the other. For example, the ratio of 12 to 4 is 3, because 12÷4=3; and the ratio of 4 to 8 is, because 4 by 8 = 4.

CASE I.

To find the greatest common measure of two or more numbers, or to find the greatest number that will divide two or more numbers, without a remainder.

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 measure.

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 incommeasurable.

The above rule may be illustrated and demonstrated by the following example.

Let it be required to find the greatest common measure or divisor of 24 and 88.

According to the rule, we first divide 88, the greater number, by 24, the less; for it is evident that no number greater than the less of two numbers can measure or divide those numbers. As therefore 24 will exactly measure or divide itself, if it will also divide 88, it will be the greatest common divisor sought.

OPERATION.

24)88(3 72

16)24(1
16

8) 16(2

16

Now we find that 24 will not exactly measure or divide 88, but there is a remainder, 16. 24, therefore, is not the common divisor of the two numbers. Now as 72, the number which we subtracted from 88, is an exact multiple of 24, we know that any number which will exactly measure or divide 24 will also divide 72; and as 16, the remainder of the division of 88 by 24, is that part of 88

which 24 will not measure or divide, it is a number which must be divided by the common divisor of 24 and 88. Now, since no number can divide 16 greater than 16 itself, and since, if it will divide 24, we know that it will also divide 88, because 88 is a multiple of 24, +16, we proceed according to the rule to try whether 16 will measure or divide 24, and therefore place 24, the last divisor, at the right of 16, the last remainder. We know, also, that if 16 will divide 24 it is the greatest common divisor of 24 and 88; because we have before shown that any number which will divide 24 and 88 must also divide 16.

On dividing 24 by 16 we again find a remainder, 8. Now 8 being the remainder after the division of 24 by 16, we know, according to the reasoning before adopted, that no number greater than 8 can measure or divide 16 and 24, and that if it will measure 16, it will also measure 24, because 24 is a multiple of 16,8, and that for the same reason it will divide 88, for 88 is a multiple of 24, + 16. Making 8, therefore, the divisor, and 16 the dividend, according to the rule, we find that 8 will exactly divide 16, and hence know that 8 is the greatest common divisor of 24 and 88. Q. E. D.

2. What is the greatest common measure of 56 and 168?

Ans. 56.

3. What is the greatest common measure of 96 and 128? Ans. 32.

4. What is the greatest common measure of 57 and 285 ? Ans. 57.

5. What is the greatest common measure of 169 and 175 ? Ans. 1.

6. What is the greatest common measure of 175 and 455 ? Ans. 35.

7. What is the greatest common measure of 169 and 866 ?

12. What is the greatest common measure of 144, 485, and 25? Ans. 1.

13. What is the greatest common measure of 671, 2013, and 4026 ?

14. What is the greatest common measure of

24?

15. What is the greatest common measure of 21,

Ans. 671.
16, 20, and
Ans. 4.
27, and 81 ?
Ans. 3.

CASE II.

To reduce fractions to their lowest terms.

1. Reduce to its lowest terms.

OPERATION.

4)18=4)√1⁄2= Ans.

NOTE. That is equal to may be demonstrated as follows: — 16 is the same multiple of 1, that 48 is of 3, therefore 16 has the same ratio to 48, that 1 has to 3; and as the value of a fraction depends on the ratio which the numerator has to the denominator, it is evident when their ratios are the same that their values are equal; therefore, is equal to 16. Q. E. D.

48.

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 a unit. Or divide the numerator and denominator by their greatest common measure.

To reduce mixed numbers to improper fractions.

1. How many fifths of a gallon in 173 gallons?