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35. A farmer has 29 bushels of rye, which he wishes to put into 8 bags ; how much must each bag contain ?
CASE II. When the divisor is a composite number, proceed as in the following
EXAMPLE. 36. If 42 yards of cloth cost 14£. 3s. 6d., what is the value of 1 yard ?
Ans. 6s. 9d. £. 8. d. In this question, we find the component parts 7)14 3 6 of 42 are 6 and 7; we therefore first divide the 612 0 6 price by 7, and then divide the quotient by 6.
0 6 9 From the above, we deduce the following
RULE. Divide the dividend by one of the component parts, and the quotient thence arising by the other, and the last quotient will be the answer.
Note. – To find the true remainder; multiply the last remainder by the first divisor, and to the product add the first remainder.
and 7T. lcwt. 3qr. 10lb. at 6 cents per pound. Moses Atwood purchased one fourth of the remainder at 6 cents per pound. One half of what then remained I sold to J. Gale at 10 cents per pound. The remaining quantity I sold to J. Smith at 12 cents per pound; but he has become a bankrupt, and I lose half my debt. What have I gained by my purchase ?
QUESTIONS TO BE PERFORMED BY ANALYSIS.
1. If 7 pair of shoes cost $8.75, what will one pair cost ? what will 20 pairs cost ?
Ans. $25.00. 2. If 5 tons of hay cost $ 85, what will 1 ton cost? what will 17 tons cost ?
Ans. $ 289.00. 3. When $0.75 are paid for 3gal. of molasses, what is the value of lgal. ? What cost 37gal.?
Ans. $ 9.25. 4. Gave $ 1.92 for 4lbs. of tea; what cost llb. ? what cost 37lbs. ?
Ans. $ 17.76. 5. For 12lbs. of rice I paid $ 1.08; what was paid for llb.; and what must I give for 25lbs. ?
Ans. $ 2.25. 6. Gave S. Smith $ 63.00 for 9 tubs of butter ; what was the cost of 1 tub? What cost 27 tubs ? Ans. $ 189.00.
7. T. Swan can walk 20 miles in 5 hours; how far can he walk in 1 hour? How long would it take him to walk from Bradford to Boston, the distance being in a straight line 28 miles?
Ans. 7 hours. 8. If a hungry boy would eat 49 crackers in 1 week, how many would he eat in 1 day? how many would be sufficient to last him 19 days ?
Ans. 133 crackers. 9. Gave $ 20 for 5 barrels of flour ; what cost i barrel ? what cost 40 barrels ?
Ans. $ 160.00. 10. For 3lbs. of lard there were paid 36 cents; what was the cost of 37lbs. ?
Ans. $4.44. 11. Paid F. Johnson 72 cents for 9 nutmegs; how many cents were paid for 1 nutmeg; and what should be charged for 37 nutmegs?
Ans. $2.96. 12. Paid 2£. 17s. 5d. for 52lbs. of sugar; what cost llb. ? what cost 76lbs. ?
Ans. 13. Paid 4£. 3s. 11d. for 76 pounds of sugar; what cost 52lbs. ?
Ans. 14. If 52lbs. of sugar cost 2 £. 175. 5d., how many pounds can be purchased for 4£. 3s. 11d. ?
Ans. 15. When 4.£. 3s. 11d. are paid for 76lbs. of sugar, how mWv pounds can be obtained for 2 €. 17s. 5d. ? Ans.
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.
Thus. S Numeratora
Thus, Denominator 11 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 13 org. 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 į of of g of . 5. A mixed number is an integer with a fraction; as 7747, 5%. 6. A complex fraction is a fraction having a fraction or a mixed number for its numerator or denominator, or both; as, 7} } 877 911 7 11 or 91
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 prod.
uct will exactly produce the number. 20. An aliquot part is that which is contained a precise num
ber of times in another. 21. An aliquant part is such a number as is contained in an.
cther 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 l; 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