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=

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.

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 ?

7. What is the greatest common measure of 169

8. What is the greatest common measure of 47

Ans. 35. and 866 ? Ans. 1. and 478 ? Ans. 1.

9. What is the greatest common measure of 84 and 1068 ?

Ans. 12.

10. What is the greatest common measure of 75 and 165 ?

Ans. 15.

11. What is the greatest common measure of 78, 234, and Ans. 78.

468 ?

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 ? Ans. 671. 14. What is the greatest common measure of 16, 20, and 24? Ans. 4. 15. What is the greatest common measure of 21, 27, and 81 ? Ans. 3.

CASE II.

To reduce fractions to their lowest terms.

1. Reduce to its lowest terms.

OPERATION.

4)=4)= Ans.

NOTE.-That is equal to 4 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 18. Q. E. D.

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?

OPERATION.

170 5

We analyze this question by saying, that, as there are 5 fifths in 1 gallon, there will be 5 times as many fifths as gallons. Therefore in 17 gallons and 3 fifths of a gallon there will be Ans. 88 fifths, which should be expressed thus, 3. And this fraction by definition 2d, page 89, is an improper fraction.

88

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.

17

Ans. 7. Ans. 191. Ans. 4878. Ans. 20329. Ans. 2577.

137

16

59

Ans. 10146. Ans. 9554. Ans. 107. Ans. 69691.

61

2. Reduce 16 to an improper fraction. 3. Reduce 14 to an improper fraction. 4. Reduce 1261 to an improper fraction. 5. Reduce 149119 to an improper fraction. 6. Reduce 161 to an improper fraction. 7. Reduce 1713 to an improper fraction. 8. Reduce 984 to an improper fraction. 9. Reduce 1168 to an improper fraction. 10. Reduce 718 to an improper fraction. 11. Reduce 100188 to an improper fraction. Ans. 2008. 12. Reduce 478 to an improper fraction. Ans. 55483. 13. Reduce 871 to an improper fraction. Ans. 101039. 14. Reduce 167194 to an improper fraction. Ans. 19632. 15. Reduce 613304 to an improper fraction. Ans. 188292. 16. Reduce 159109 to an improper fraction. Ans. 32052. 17. Reduce 99999 to an improper fraction. Ans. 999999. 18. Reduce 7 to an improper fraction.

20

199•

00

Ans. 1. Ans. Y.

Ans. +.

Ans. 100.

NOTE. To reduce a whole number to an equivalent fraction having a given denominator, we multiply the whole number by the given denominator, and we then place the product over the given denominator.

22. Reduce 11 to a fraction whose denominator shall be 7.

OPERATION.

11 x7 = 77; 77 Ans.

23. Reduce 5 to a fraction whose denominator shall be 17.

24. Reduce 19 to a fraction whose denominator shall be 13.

Ans. 247.

CASE IV.

To reduce improper fractions to integers or mixed numbers.

1. How many yards in of a yard?

OPERATION.

114

This question may be analyzed by say

19)117(6,3 Ans. ing, as 19 nineteenths make one yard, there will be as many yards as 117 contains 19, which is 6 times and 3 nineteenths times, which is written thus, 6; and this ex

3

pression, by definition 5th, page 89, is a mixed number.

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.

2. Reduce 167 to a mixed number. 3. Reduce 16 to a mixed number. 4. Reduce to a mixed number. 5. Reduce 7 to a mixed number. 6. Change 1000 to a mixed number. 7. Change 4123 to a mixed number. 8. Change 125 to a whole number. 9. Change 37 to a whole number.

CASE V.

Ans. 11.

Ans. 14,16

Ans. 744.

Ans. 314.

Ans. 111.
Ans. 912.
Ans. 125.

To reduce complex fractions to simple ones.

1. Reduce to a simple fraction.

OPERATION.

Ans. 1.

In performing this question, we divide the numerator by the denominator; be

== Ans. cause all fractions are equal to the number of times the numerator contains the denominator.

RULE. If the numerator or denominator be a whole or a mixed number, let it be reduced to an improper fraction. Then multiply the denominator of the lower fraction into the numerator of the upper fraction for a new numerator, and the denominator of the upper fraction into the numerator of the lower fraction for a new denominator; or, we may invert the denominator of the complex fraction, when reduced, and place it in a line with the numerator, then multiply the two numerators together for a new numerator, and the denominators together for a new denominator.

NOTE. - Every fraction denotes a division of the numerator by the denominator, and its value is equal to the quotient obtained by such division. Hence the necessity of inverting the denominator of the complex fraction.