1. What is the quotient of 751 grapes, divided by 16? 4)751 3 Ans. 4616 DEMONSTRATION OF THE RULE. In 751 grapes there are 187 sets, (say bunches,) with 4 grapes or units in each bunch, and 3 units over. In the 187 bunches there are 46 piles, 4 bunches in a pile, and 3 bunches over. But there are 4 grapes in each bunch; therefore, the number of grapes in the 3 bunches is equal to 4x3=12, to which add 3, the grapes of the first remainder, and we have the entire remainder 15. 2. Divide 4967 by 32. 4)4967 4x8=328)1241 3, 1st remainder 155....1x4+3=7 the true remainder. Ans. 15532. 3. Divide 956789 by 7x8=56. Ans. 4. Divide 4870029 by 8x9=72. Ans. 6763941 5. Divide 674201 by 10xll=110. Ans. 6. Divide 445767 by 12x12=144. Ans. 309587 Q. What is a composite number? (See $ 27, page 50.) How do you divide when the divisor is a composite number? When there is a remainder, how do you find the true remainder. CASE II. § 38. When the divisor is 10, 100, 1000, &c. RULE, I. Cut off from the right hand of the dividend as many figures as there are 0's in the divisor. II. The left hand figures of the dividend will express the quotient, and the figures cut off the remainder. EXAMPLES. Ans. 32100 1. Divide 3256 by 100. In this example there are two O's OPERATION. in the divisor, therefore, there are two 100)32 56 figures cut off from the right hand of the dividend, and the quotient is 32, and 56: 100 DEMONSTRATION OF THE RULE. The quotient ought to be 10, 100, 1000, &c., times less than the dividend. But the same figure is 10, 100, 1000, &c., times greater or less in value, according to its distance from the unit's place. By cutting off figures from the right hand, the unit's place is removed to the left, and consequently the dividend is diminished 10, 100, 1000, &c., times, according as you cut off 1, 2, 3, &c., figures. 2. Divide 49763 by 10. Ans. 49761 3. Divide 7641200 by 100. Ans. 76412. 4. Divide 496321 by 1000. Ans. 496 30200 CASE III. § 39. When there are ciphers on the right of the divisor. RULE. I. Cut off the ciphers by a line, and cut off the same number of figures from the right of the dividend. Il. Divide the remaining figures of the dividend by the significant figures of the divisor, and annex to the remainder, if there be one, the figures cut off from the dividend : this will form the true remainder. EXAMPLES. 1. Divide 67389 by 700. OPERATION. the 89, and then find that 7 is 7100)67389 contained in the remaining 96...1 remains. figures, 96 times, with a 189 true remain. mainder of 1; to this we annex Ans. 96488 89, forming the remainder 189: to the quotient 96 we annex 189 divided by 700 for the entire quotient. re DEMONSTRATION OF THE RULE. The number 700=100x7. Hence it is a composite number of which the factors are 100 and 7. In striking off the two figures 89, from the right of the dividend, we divide it by 100; we then divide the 673 by the other factor 7. We then multiply the remainder 1 by 100 and add 89 to the product, giving 189 for the true remainder, (see § 37.) 2. Divide 8749632 by 37000. 37|000) 8749|632236 74 Ans. 23647633 3. Divide 986327 by 210000. Ans. 4146327 210000 4. Divide 876000 by 6000. Ans. 5. Divide 36599503 by 400700. Ans. 911358 03 400 700 Q. How do you divide by 10, 100, 1000, &c. ? (see 38.) Which part is the quotient? Which part is the remainder? When there are ciphers on the right of the divisor, how do you form the true remainder ? APPLICATIONS IN DIVISION. 1. Divide 80 dollars equally among four men. Here the 80 dollars is to be divided OPERATION. into 4 equal parts, and the quotient 20 4)80 dollars expresses the value of one of the 20 dollars. equal parts. 2. Four persons buy a lottery ticket; it draws a prize of 10000 dollars: what is each one's share ? Ans. dollars. 3. A person dying leaves an estate of 4500 dollars to be divided equally among 5 children: what is each one's share? Ans. 900 dollars. 4. There are 1560 eggs to be packed in 24 baskets : how many eggs will be put in each basket ? Ans. 5. What number must be multiplied by 124 to produce 40796 ? Ans. 329. 6. How many times can 24 be subtracted from 1416 ? Ans. 7. The sum of 19125 dollars is to be distributed among a certain number of men, each is to receive 425 dollars : how many men are to receive the money ? Ans. 8. By the census of 1840 the whole population of the 26 States was 16,890,320 : if each one had contained an equal number of inhabitants, how many would there have been in each state ? Ans. 649,62726 9. If a man walks 12775 miles in a year, or 365 days, how far does he walk each day? Ans. miles. 10. A farmer sells a drove of sheep for 2 dollars a head, and receives 1250 dollars : how many sheep did he sell ? Ans. 625. 11. It is computed that the distance to the sun is 95,000,000 of miles, and that light is 8 minutes travelling from the sun to the earth : how many miles does it travel Ans. 12. By the census of 1840 it appeared that the City of New York contained 312710 inhabitants; allowing 5 to each house, how many houses were there in the city at that time? Ans. 62,542. 13. A merchant has 5100 pounds of tea, and wishes to pack it in 60 chests : how many pounds must he put in each chest? Ans. 14. A person goes to a store and buys a piece of cloth containing 36 yards, for which he pays 288 dollars : how much does he pay per yard ? Ans. dollars. 15. There are 7 days in a week : how many weeks in Ans. 52 weeks and 1 day over. 16. There are 24 hours in a day : how many days in 2040 hours ? Ans. days. 17. Twenty-three persons dined together, their bill was 92 dollars : how much had each one to pay ? Ans. 4 dollars. per minute ? a year of 365 ? GENERAL REMARKS. § 40. Numeration, Addition, Subtraction, Multiplication, and Division, are called the five ground rules of Arithmetic. Q. How many principal rules are their in Arithmetic? What are they? Can Multiplication be performed by Addition ? Can Division be performed by Subtraction ? By how many rules, then, may all the operations in Arithmetic be performed ? § 41. The preceding rules furnish answers to the following questions. Ques. 1. When the cost of each one of several things is given, how do you find their entire cost ? Ans. Add the costs of the several things together, the sum will be the entire cost. What is the entire cost of a bag of coffee at 6 dollars, a chest of tea at 4 dollars, a box of raisins at 2 dollars, and a barrel of sugar at 12 dollars ? Ans. 24 dollars. Q. 2. When you have two unequal numbers, how do you find their difference? A. By subtracting the less from the greater. Q. 3. When the subtrahend and remainder are given, or known, how do you find the minuend? A. By adding the remainder and subtrahend together. Hence the following principles. 1st. If the sum of two numbers be diminished by one of them, the remainder will be the other number. 2d. The less of two numbers added to their difference, will give the greater. The sum of two numbers is 56, one of the numbers is 12: what is the other? Ans. 44. The less of two numbers is 25, and their difference 30: what is the greater ? Ans. The less of two numbers is 35, and their difference 35 : what is the greater ? Ans. 70. Q. 4. When you have the cost of a single thing, how will you find the entire cost of any number of things at the same rate ? A. Multiply the cost of the single thing by the number of things. |