There are now no common measures left in the row, and the least common multiple required is the product of 2, 2, 3, 5, 7, 11, 13, 4, 17, 3, and 19; or 232792560.

SECTION V.
FRACTIONS.

104. Suppose it required to divide 49 yards into five equal parts, or, as it is called, to find the fifth part of 49 yards. If we divide 45 by 5, the quotient is 9, and the remainder is 4; that is (72), 49 is made up of 5 times 9 and 4. Let the line a b represent 49 yards:

A————————————————————B
C———————I—
D———————K—
E———————L—
F———————M—
G———————N—
IKLMN
H||||| |

Take 5 lines, c, d, e, f, and g, each 9 yards in length, and the line h, 4 yards in length. Then, since 49 is 5 nines and 4, c, d, e, f, g, and h, are together equal to a b. Divide h, which is 4 yards, into five equal parts, i, k, l, m, and n, and place one of these parts opposite to each of the lines, c, d, e, f, and g. It follows that the ten lines, c, d, e, f, g, i, k, l, m, n, are together equal to a b, or 49 yards. Now d and k together are of the same length as c and i together, and so are e and l, f and m, and g and n. Therefore, c and i together, repeated 5 times, will be 49 yards; that is, c and i together make up the fifth part of 49 yards.

105. c is a certain number of yards, viz. 9; but i is a new sort of quantity, to which hitherto we have never come. It is not an exact number of yards, for it arises from dividing 4 yards into 5 parts, and taking one of those parts. It is the fifth part of 4 yards, and is called a fraction of a yard. It is written thus, ⁴/₅(23), and is what we must add to 9 yards in order to make up the fifth part of 49 yards.

The same reasoning would apply to dividing 49 bushels of corn, or 49 acres of land, into 5 equal parts. We should find for the fifth part of the first, 9 bushels and the fifth part of 4 bushels; and for the second, 9 acres and the fifth part of 4 acres.

We say, then, once for all, that the fifth part of 49 is 9 and ⁴/₅, or 9 + ⁴/₅; which is usually written (9⁴/₅), or if we use signs, 49/5 = (9⁴/₅).

EXERCISES.