HIGHER COURSE OF FRACTIONS,

UNITARY SYSTEM, &c.

GREATEST COMMON MEASURE.

A Measure of a number is any number that will divide it without leaving a remainder. Thus, 2 is a measure of 4, 3 of 9, 5 of 20, &c.

A Common Measure of two or more numbers, is a number which will divide each of them without a remainder. Thus, 2 is a common measure of 4, 6, 10, 12, 18, &c.

The Greatest Common Measure, (G. C. M.,) of two or more numbers is the greatest number that will exactly divide each of them. Thus, 6 is the G. C. M. of 12, 18, 24, 30.

To find the G. C. M. of two numbers.

RULE.-Divide the greater number by the less; then the divisor by the remainder; next the last divisor by the new remainder; and so on until there is no remainder. The last divisor is the G. C. M.

EXAMPLE.What is the G. C. M. of 1825 and 2555?

1825)2555(1
1825

730)1825(2

1460

365)730(2
730

G. C. M. 365 Ans.

Find the G. C. M. of the following numbers:

LEAST COMMON MULTIPLE.

A Multiple of a number is that which contains it a certain number of times exactly, or which is divisible by it, without leaving a remainder Thus, 120 is a multiple of 12, because it contains, or is divisible by, the number 12 exactly 10 times.

A Common Multiple of two or more numbers is a number which is a multiple of each of them. Thus, 6 is a common multiple of 2 and 3, and 12 is a common multiple of 2, 3, 4, and 6.

The Least Common Multiple, (L. C. M.,) of two or more numbers is the least number which is a multiple of each of them. Thus, 36, 24, and 12 are all common multiples of 2, 3, 4, and 6; but 12 is the least number that exactly contains them all, and is therefore their L. C. M.

To find the L. C. M. of two or more numbers.

RULE.-Set down the numbers in a line, and strike out any that are exactly contained in any of the others. Take any one of the uncancelled numbers as a divisor. Set it to the left of the line, and divide the remaining numbers by this divisor, or by any factor of it which will divide them without a remainder. Write down the results in a second line, and again strike out any that are exactly contained in any of the others. Assume any one of the uncancelled numbers in this second line as a new divisor, and proceed with it as before.

Bring down the results into a third line, if necessary, and proceed thus until no two numbers in the last line have any common measure, but unity.

Then, to get the L. C. M., multiply together all the divisors and all the numbers left in the last line.

EXAMPLE. What is the L. C. M. of 16, 24, 28, 30, 32, 36, 40, 44, 45, 48, and 50?

Divide by 40, of which the factors are 4, 8, 10, and 5.

40,16 24 28 30 32 36 40 44 45 48 50

Then L. C. M. = 40 × 6 × 7 x 2 x 11 x 3 x 5 554400 Ans.

=

Here we first strike out 16 and 24, since they are contained exactly in 48. We next assume 40 as divisor, of which 4, one of its factors, reduces 28 to 7, 36 to 9, and 44 to 11. Also 8, another factor, reduces 32 to 4. and 48 to 6. Also 10, another factor, reduces 30 to 3 and 50 to 5. Also 5, another factor, reduces 45 to 9. We then strike out 3 and 9 in the second line, since they are each contained in 9, and assume as a new divisor, of which 3, one of the factors, reduces 9 to 3.

Find the L. C. M. of :

1. 6, 9, and 30. 2. 12, 20, and 24. 3. 8, 10, and 15. 4. 3, 6, 9, 21, and 24. 5. 2, 3, 4, 5, 6, 7, 8. 6. 14, 21, 3, 2, and 63. 7. 24, 16, 18, and 20. 8. 27, 54, 81, 14, and 63 9. 7, 21, 35, 4, and 20. 10. 2, 9, 16, 35, 56, and 63.

11. 27, 36, 45, 42, and 16. 12. 10, 14, 21, 28, and 35. 13. 7, 28, 35, 42, and 63. 14. 4, 5, 8, 24, 40, and 120. 15. 18, 20, 24, 36, and 48. 16. 60, 50, 144, 35, and 18.

17. 8, 9, 11, 22, 72, 32, and 99. 18. 15, 12, 128, 30, 16, 4, 320.96. 19. 3, 6, 9, 12, 48, 21, 24, and 16. 20. 6, 10, 14, 18, 22, 28. and 32. G

VULGAR FRACTIONS.

REDUCTION.

Case 1.-To reduce a whole number to a fraction having a given denominator.

RULE.-Express the whole number as a fraction, having 1 for its denominator, and multiply both the numerator and the de nominator by the given denominator.

EXAMPLE.-Reduce 217 to a fraction having 13 for its deno

1. Reduce 7, 9, 27, and 40 to fractions having 11 for denominator.

2. Reduce 7, 23, 101, to fractions having 13 for denominator. 3. Reduce 2, 207, 440, and 9 to fractions having 109 for de nominator.

4. Reduce 4, 37, 126, 73, and 1007 to fractions having 101 for denominator.

5. Reduce 27, 304, 617, and 98 to fractions having 248 for denominator.

6. Reduce 209, 407, 789, and 5 to fractions having 611 for denominator.

Case II. To reduce a mixed number to an improper fraction. RULE.-Multiply the whole number by the denominator of the fraction; add the numerator to this product; and place the sum over the given denominator.

EXAMPLE. Reduce 16 to an improper fraction.

7. Reduce the mixed numbers 73, 18, and 12818 to improper fractions.

8. Reduce the mixed numbers 9, 6, and 14% to improper fractions.

9. Put into fractional forms the mixed quantities 71, 911 41717, 6318, 1861, 673.

10. Express the mixed numbers 713419, 61713, 21407, 356139 as improper fractions.

11. Reduce the mixed numbers 617018, 2345845, 259101 111 to improper fractions.

Case III. To reduce an improper fraction to a mixed number.

RULE.-Divide the numerator by the denominator; the quotient will be the whole number; and, if there be any remainder, place it as the numerator of the fraction.

= 69.

Or, simply dividing 27 by 4, 2

EXAMPLE 2.-Reduce 217 to a mixed number.

5431

1217

5781

112,

25. Represent 127963, 213451, 12845 in the form of mixed quantities.

Case IV. To reduce a fraction to its lowest terms.

RULE 1.-Divide the numerator and denominator by any number that will measure each of them without leaving a remainder; continue the operation as long as it is possible.

This rule will be found the most convenient when the terms of the fraction are small.

EXAMPLE 1.-Reduce 800 to its lowest terms.

NOTE.

Dividing successively by 10, 10, and 3, we get

A fraction can sometimes be reduced to its lowest terms, and the work may almost always be materially lessened, by dividing both numerator and denominator by any number which will go into each of them without a remainder. In order to facilitate this mode of reduction, it is necessary to remember the following facts:

1st. Any number that ends in 5 is divisible by 5.

2d. Any number that ends in 0 is divisible by 10, 5, or 2.

3d. Any number that ends in an even number is divisible by 2.

4th. When the two right-hand figures are divisible by 4 the whole is divisible by 4.

5th. When the three right-hand figures are divisible by 8 the whole number is divisible by 8.

6th. When the sum of the digits of a number is divisible by 9 the sum itself is divisible by 9 or 3.

7th. When the sum of the digits of a number is divisible by 3 the number itself is divisible by 3.

8th. When the sum of the digits standing in the even places is equal to the sum of the digits standing in the odd places the number is divisible by 11. Thus, the number 7416 is divisible by 4 because 16 (the last two digits) is divisible by 4.

7416 is divisible by 8, because 416 (its last three digits) is divisible by 8. is divisible by 9, because the sum of its digits (7 + 4 + 1 + 6 = 18) is divisible by 9.

is divisible by 3, because the sum of its digits (7 + 4 + 1 + 6 = 18) is divisible by 3.

So also the number 4567321 is divisible by 11, since the sum of the digits in the odd places, 1 + 3 + 6 + 4 = 14 = 2 + 7 + 5; the sum of the digits in the even places.

Reduce the following fractions to their lowest terms:

RULE 2.-Find the G. C. M. of the numerator and denominator; then divide each of them by this common measure.

Express the following fractions in their lowest terms :

2726

59. i

NOTE-When the terms of the fraction are large they may be reduced

by Rule 1 before proceeding to find the G. C. M.

EXAMPLE 3.-Reduce 48519 to its lowest terms.

(a) Rule 1. Dividing successively by 5, 3, and 3, 48510 9702 3234 1078

Express the following fractions in their lowest terms: