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11. What is the smallest sum of money for which I can purchase either sheep at $4 per head, or cows at $21, or oxen at $49, or horses at $72 ?
Ans. $3528 12. A can dig 4 rods of ditch in a day, B can dig 8 rods, and C can dig 6 rods; what must be the length of the shortest ditch, that will furnish exact days' labor either for each working alone or for all working together?
Ans. 72 rods. 13. The forward wheel of a carriage was 11 feet in circumference, and the hind wheel 15 feet; a rivet in the tire of each was up when the carriage started, and when it stopped the same rivets were up together for the 575th time; how many miles had the carriage traveled, allowing 5280 feet to the mile?
Ans. 17 miles 5115 feet.
CANCELLATION. 157. Cancellation is the process of rejecting equal factors from numbers sustaining to each other the relation of dividend and divisor.
158. It is evident that factors common to the dividend and divisor may be rejected without changing the quotient, (117, III). 1. Divide 1365 by 105.
ANALYSIS. We first in
dicate the division by wri. 1365 8 x $ x x 13
13 ting the dividend above a 105 8x 6 x 7
horizontal line and the di
visor below. Then factoring each term, we find that 3, 5, and 7 are common factors; and crossing, or canceling these factors, we have 13, the remaining factor of the dividend, for a quotient.
159. If the product of several numbers is to be divided by the product of several other numbers, the common factors should be canceled before the multiplications are performed, for two
1st. The operations in multiplication and division will thus be abridged.
2d. The factors of small numbers are generally more readily detected than those of large numbers. 2. Divide 20 times 56 by 7 times 15.
Analysis. IIaving first indi. 4 8
cated all the operations required gø x $8 32
by the question, we cancel 7 =103
from 7 and 56, and 5 from 15 to X A6 3
and 20, leaving the factors 3 in 3
the divisor, and 8 and 4 in the dividend. Then 8 x 4= 32, which divided by 3, gives 103, the quotient required.
Hence the following RULE I. Write the numbers composing the dividend above a horizontal line, and
the numbers composing the divisor below it. II. Cancel all the factors common to both dividend and divisor,
III. Divide the product of the remaining factors of the dividend by the product of the remaining factors of the divisor, and the result will be the quotient.
Notes, - 1. When a factor is canceled, the unit, 1, is supposed to take its place.
2. By many it is thought more convenient to write the factors of the dividend on the right of a vertical line, and the factors of the divisor on the left.
EXAMPLES FOR PRACTICE. 1. What is the quotient of 18 X 6 X 4 X 42 divided by 4 x 9 x 3 x 7 x 6?
4, Ans. 2. Divide the product of 21 x 8 x 60 x 8 x 6 by 7 x 12 x 3 x 8 x 3.
Ans. 80. 3. The product of the numbers 16, 5, 14, 40, 16, 60, and 50, is to be divided by the product of the numbers 40, 24, 50, 20, 7, and 10; what is the quotient?
4. Divide the continued product of 12, 5, 183, 18, and 70 by the continued product of 3, 14, 9, 5, 20, and 6.
5. If 213 x 84 x 190 x 264 be divided by 30 x 56 x 36, what will be the quotient ?
6. Multiply 64 by 7 times 31 and divide the product by 8 times 56, multiply this quotient by 15 times 88 and divide the product by 55, multiply this quotient by 13 and divide the product by 4 times 6.
Ans. 403. 7. How many cords of wood at $4 a cord, must be given for 3 tons of hay at $12 a ton ?
8. How many firkins of butter, each containing 56 pounds, at 15 cents a pound, must be given for 8 barrels of sugar,
each containing 195 pounds, at 7 cents a pound?
Ans. 13. 9 A grocer sold 16 boxes of soap, each containing 66 pounds at 9 cents a pound, and received as pay 99 barrels of potatoes, each containing 3 bushels; how much were the potatoes worth a bushel ?
10. A farmer exchanged 480 bushels of corn worth 70 cents a bushel, for an equal number of bushels of barley worth 84 cents a bushel, and oats worth 56 cents a bushel; how many bushels of each did he receive ?
Ans. 240. 11. A merchant sold to a farmer two kinds of cloth, one kind at 75 cents a yard, and the other at 90 cents, selling him twice as many yards of the first kind as of the second. He received as pay 132 pounds of butter at 20 cents a pound; how many yards of each kind of cloth did he sell?
Ans. 22 yards of the first, and 11 yards of the second. 12. A man took six loads of potatoes to market, each load containing 20 bags, and each bag 2 bushels. He sold them at 44 cents a bushel, and received in payment 8 chests of tea, each containing 22 pounds; how much was the tea worth a pound?
Ans. 60 cents.
DEFINITIONS, NOTATION, AND NUMERATION.
160. When it is necessary to express a quantity less than a unit, we may regard the unit as divided into some number of equal parts, and use one of these parts as a new unit of less value than the unit divided. Thus, if a yard, considered as an integral unit, be divided into 4 equal parts, then one, two, or three of these parts will constitute a number less than a unit. The parts of a unit thus used are called fractional units; and the numbers formed from them, fractional numbers.. Hence
161. A Fractional unit is one of the equal parts of an integral unit.
162. A Fraction is a fractional unit, or a collection of fractional units.
163. Fractional units take their name, and their value, from the number of parts into which the integral unit is divided. Thus,
If a unit be divided into 2 equal parts, one of the parts is called one half. If a unit be divided into 3 equal parts, one of the parts is called one third. If a unit be divided into 4 equal parts, one of the parts is called one fourth.
And it is evident that one third is less in value than one half, one fourth less than one third, and so on.
164. To express a fraction by figures, two integers are re. quired; one to denote the number of parts into which the integral unit is divided, the other to denote the number of parts taken, or the number of fractional units in the collection. The former is written below a horizontal line, the latter above. Thus, One half is written
One fifth is written ร !
1 Two fifths
1 Three eighths
165. The Denominator of a fraction is the number below the line.
It denominates or names the fractional unit, and it shows how many fractional units are equal to an integral unit.
166. The Numerator is the number above the line.
It numerates or numbers the fractional units; and it shows how many are taken.
167. The Terms of a fraction are the numerator and denominator, taken together.
168. Since the denominator of a fraction shows how many fractional units in the numerator are equal to 1 integral unit, it follows,
I. That the value of a fraction in integral units, is the quotient of the numerator divided by the denominator.
II. That fractions indicate division, the numerator being a dividend and the denominator a divisor.
169. To analyze a fraction is to designate and describe its numerator and denominator. Thus is analyzed as follows:
7 is the denominator, and shows that the units expressed by the numerator are sevenths.
5 is the numerator, and shows that 5 sevenths are taken.
5 and 7 are the terms of the fraction considered as an expression of division, 5 being the dividend and 7 the divisor.
EXAMPLES FOR PRACTICE.
Express the following fractions by figures :
1. Four ninths.
6. One thousand eight hundred fifty-seven nine thousand five hundred twenty-firsts.
7. Twenty-five thousand eighty-sevenths.