64. When the divisor is 10, 100, or 1000, &c., we can divide by simply removing the decimal point in the dividend as many places towards the left as there are zeros in the divisor; the number at the right of the point will be the remainder; thus, 368.100 3.68 (or 36%). Hence, if the divisor consists of any number of significant figures with zeros annexed, first cut off the zeros from the divisor and an equal number of figures from the right of the dividend, then divide what remains of the dividend by what remains of the divisor. To the remainder, if any, annex the figures that were cut off in the dividend; thus, 38/00) 968/42 ( 25 76 208 190 1842 Disregarding the tens and units, we find how many times 38 (hund.) is contained in 968 (hund.), which is 25, with a remainder of 18 (hund.); this, with the 4 tens and 2 units left in the dividend, makes the entire remainder 1842, .. 3800 is contained in 96842, 251842 times. 3600 6179 15. Divide 42179 by 1000; by 18000. Ans. 42,1700; 2870% 16. Divide 76532102 by 4800; by 91000. 17. Divide 98000269 by 32600000; by 980000. 18. Bought a farm for $18715, at $95 an acre; how many acres were there in the farm? Ans. 197. 19. Paid $4505 for 27 acres of woodland; what was the price per acre? SOLUTION. If 27 acres cost $4505, one acre will cost one twentyseventh of $4505, which equals, &c. 20. Paid $35328 for 368 acres of land; find the price per acre. 21. The distance from Boston to Albany is 198 miles, from Albany to Buffalo, 298 miles. How long will it take a train to pass over the road at the rate of 28 miles an hour, allowing 2 hours for detentions between Boston and Albany, 1 hour at Albany, and 3 between Albany and Buffalo? Ans. 233g hours. 22. The Ohio Canal descends 1832 feet in 152 locks; what is the average descent in each lock? 23. If 8 presses can coin 19000 pieces of money in an hour, how many pieces can one press coin in a minute, 60 minutes making an hour? 24. In how many days, of 12 hours each, can the president of a bank sign 9000000 bank notes, if he signs 8 in a minute? For Contractions in Division, see Appendix. For Dictation Exercises, see Key. 66. QUESTIONS FOR REVIEW. 1. What is ARITHMETIC? What are numbers? What is an abstract number? a concrete? What is a unit? Define Notation and Numeration. How are numbers represented? Describe the Roman method; the Arabic. Which is more used? Why is this sometimes called the Decimal System? What is the decimal point? By what is the number of units of any order expressed? By what is the order of units expressed? 2. How do you write numbers? How do you read numbers? Name the first seven periods. Name others as far as you can. How are these periods separated? What are the names of the places of each period? 3. What is the least number of figures that will express units? thousands?-billions? trillions? - millions? — quadrillions. 4. In 189654238761, what is the largest number of thousands? of millions? of ten-millions? of nundred-billions ? - of trillions? - of tens? - of hundreds - - of ten-thousands? 5. How will zeros at the right of a number affect it? at the left? 6. What does a figure in the first place at the right of the decimal point represent?in the second place?—in the third? 7. What is ADDITION? What is the sign for Addition? for Equality? How do you arrange numbers to be added? Is this absolutely necessary? Perform and explain an example containing four numbers of at least seven figures each. Give the rule. 8. What is SUBTRACTION? Name and define the terms used. What is the sign for Subtraction? Take 3684 from 7000068, and explain. Give the rule; -the proof. When the minuend and difference are given, how can you find the subtrahend? When the subtrahend and difference are given, how can you find the minuend? 9. What is MULTIPLICATION? Name and define the terms used. What is the sign for Multiplication? Perform and explain an example in which the multiplier has, at least, two figures. Give the rule first method of proof; -second method. How do you multiply by 10, 100, 1000, &c.? How do you proceed if there are zeros at the right of the multiplicand or multiplier ? Tens X units what? Thousands Xtens? Hundreds X tens? Ten-thousands hundreds? Ten-thousands X ten-thousands? is the sign for Division? 10. What is DIVISION? Name and define the terms used. What Perform and explain an example in short Give the proof by multiplication. Perform division. Give the rule. and explain an example in long division. Give the rule. Give the proof by casting out the 9's. 11. How do you divide by 10, 100, 1000, &c.? How do you divide when the divisor contains zeros at the right of significant figures? When the dividend and quotient are given, how can you find the divisor? When the divisor and quotient are given, how can you find the dividend? When the multiplier and product are given, how can you find the multiplicand. When the multiplicand and product are given, how can you find the multiplier? 1. Add nine billion, six hundred ninety-two million, eighty-one thousand sixty-four; eighty-nine trillion, six hundred thirty-two million, ninety-one thousand eighteen; eighty-seven thousand thirty-four; and two hundred sixty-eight quadrillion, nine hundred eighty-four trillion, ninety-eight million, one thousand ninety-four. 986234681) take (7682+ 3. Multiply (3684291 +3642) by (8643264-8321628.) 4. Divide (368729186) by (3684 +232.) 5. If 892 is one factor, and 28544 the product, what is the other factor? 6. 365 times what number = 298570? 7. If the dividend is 38493, and the divisor 4277, what is the quotient? 8. If the dividend divisor? 42777, and the quotient 9, what is the 9. There were 52 schools in Antigua in 1858, with 4467 scholars; required the average number in each. * In examples 2, 3, and 4, first perform the operations indicated within the parentheses. 10. David was born 1085 years B. C., and Washington 1732 A. D.; what time elapsed between these events? 11. What do I save a year, my income being $1600 a year, and my expenses $24 a week, 52 weeks making the year? 12. Illinois produced in 1860, 1515594 pounds of maple sugar; what was its value at 8 cents per pound? 13. Mississippi produced 1195699 bales of cotton; what was its value at 13 cents per pound, 400 pounds to the bale? 14. Missouri produced 4164 tons of lead, worth $356660, what was the value per ton? 15. The population of Chicago in 1850 was 29963; in 1860, 109260; what was the average increase a year ? 16. If 8 men can do a piece of work in 24 days, how long will it take one man to do it? 17. If 768 be one factor, and 861-237 the other factor, what is the product? 18. Smith & Co. consume 74 tons of coal in a year; how much more must they pay for their coal in 1864, when coal is $14 a ton, than in 1860, when it was $8 a ton? 19. From the invention of parchment to the invention of paper was 782 years; to the use of quills in writing 741 years more; to the invention of printing, 804 years more; to the invention of stereotyping, 345 years more; how many years from the invention of parchment to that of stereotyping? 20. Parchment was invented 887 years B. C.; when was paper invented? Ans. 105 B. C. 21. When were quills first used in writing? Ans. A. D. 636. 22. When was printing invented? 23. When was stereotyping invented? 24. 76854 divided by what number, gives a quotient of 56 and a remainder of 22? 25. What number divided by 87, gives a quotient of 3842 and a remainder of 76? 26. In 1853, Wheeler & Wilson made 799 sewing machines; in 1854, 956; in 1855, 1171; in 1856, 2210; in 1857, 4591; in 1858, 7978; in 1859, 21306; in 1860, 10265; in 1861, 19725. Required the amount. 27. If a sewing machine can set 640 stitches in a minute, how many can it set in an hour? — in a day of 12 hours? —in 6 working days, or a week?-in 52 weeks, or a year? 28* There was sert to the U. S. mint, from 1823 to 1836, $4377984 worth of gold; what was the average value sent a year? If gold was worth 16 dollars an ounce, how many pounds were sent, allowing 12 ounces to a pound? 29.* In the Pacific Mills, 200000000 gallons of water are used in a day. How many weeks would it take a man to pump it if he could pump a gallon in six strokes of the pump, 20 strokes a minute for 16 hours a day, allowing 6 working days per week? 30* If the earth is 95000000 of miles from the sun, and the moon at its full is 224000 miles farther on, and light travels at the rate of 191500 miles a second, how many seconds is it in passing from the sun to the moon and back to the earth? Ans. 498810 seconds. 31. If 3871 be divided by 79, and the quotient be multiplied by 133, to this product 6523 be added, the amount divided by 40, and that quotient multiplied by 970, what will be the product? 32. (172)3?t Ans. 316,220. 33. (73) X 2 =?t 34. (1863+7982) × 3† 35. 7 193609 + 2 + 43† 368 36 (2+1 x 7 + 4) ÷ 5 + (86) x 2 = ?t (2+1×7+ 37* (819)÷10+67 +(2+3×7+7)÷6t=? For Dictation Exercises, see Key. + A vinculum, ? or parenthesis (), signifies that the same operation is to be performed upon all the quantities thus connected. In solving examples, it is generally better first to reduce all quantities connected by a vinculum, or parenthesis, to their simplest forms. Thus, ir Ex. 32. (17 — 2) ÷ 3 =15—3—5. Ex. 33. (7+3) × 2 — 10 × 2—20. Ex. 36. (2+1 x7+4)+5+ (8+6) x 2 = (3 × 7+4)+5+/ 14 X 2 = NOTE. - Examples with stars are "optional examples." They may be omitted by younger pupils until a review, or altogether, if the teacher prefers. |