miles a day. How far apart are they at the end of 6 days? 13. Four children inherited $2250 each; but one dying, the remaining three inherited the whole. What was the share of each? Ans. $3000. 14. Two men travel in the same direction, one at the rate of 35 miles a day, and the other at the rate of 40 miles a day. How far apart are they at the end of 6 days? 15. A man was 45 years old, and he had been married 19 years. How old was he when married? Ans. 26 years old 16. Upon how many acres of ground can the entire population of the globe stand, supposing that 25000 persons can stand upon one acre, and that the population is 1000000000? Ans. 40000 acres. 17. Add 384, 1562, 25, and 946; subtract 2723 from the sum; divide the remainder by 97; and multiply the quotient by 142. What is the result? Ans. 284. 18. How many steps of three feet each would a man take in walking a mile, or 5280 feet? Ans. 1760 steps. 19. A man purchased a house for $2375, and expended $340 in repairs; he then sold it for railroad stock worth $867, and 235 acres of western land valued at $8 an acre. What did he gain by the trade? Ans. $32. 20. The salary of a clergyman is $800 a year, and his yearly expenses are $ 450. If he is worth $1350 now, in how many years will he be worth $4500? Ans. 9 years. 21. How many bushels of oats, at 40 cents a bushel, must be given for 1600 bushels of wheat at 75 cents a bushel? Ans. 3000 bushels. 22. If you deposit $2.25 each week in a savings bank, and take out 75 cents a week, how much money will you have left at the end of the year? •? Ans. $78. 23. I bought 325 loads of wheat, each load containing 50 bushels, at $2 a bushel. What did the wheat cost? 24. The product of two numbers is 31383450, and one of the numbers is 4050. What is the other number? 25. A certain railroad is 700 miles long, and cost $31647000. What did it cost per mile? Ans. $45210. 26. What number is that, which being divided by 7, the quotient multiplied by 3, the product divided by 5, and this quotient increased by 40, the sum will be 100? Ans. 700. 27. How many cows, at $27 apiece, must be given for 54 tons of hay at $17 a ton? 28. A mechanic receives $56 for spends $2 a day for the whole time. has he left? 26 days' work, and How many dollars Ans. $4. 29. If 7 men can build a house in 98 days, how long would it take one man to build it? 30. If the cash value of 11137 $5301212, what is the average value? Ans. 686 days. school-houses is Ans. $476. 31. A cistern whose capacity is 840 gallons has two pipes; through one pipe 60 gallons run into it in an hour. and through the other 39 gallons run out in the same time. In how many hours will the cistern be filled? Ans. 40 hours. 32. The average beat of the pulse of a man at middle age is about 4500 times in an hour. How many times does it beat in 24 hours? 33. How many years is it from America, in 1492, to the year 1900? Ans. 108000 times. the discovery of What year is the 400th centennial of the discovery of America? 34. A man bought 40 acres of ground at $15 an acre, and 80 acres at $25 an acre. He sold 90 acres for $4500, and the remainder at $60 an acre. For how much did he sell the whole land? How much did he gain? a 35. What is the remainder after dividing 62530000 by 87900 ? Ans. 33100. 36. A pound of cotton has been spun into a thread 8 miles in length; allowing 235 pounds for waste, how many pounds will it take to spin a thread to reach round the earth, supposing the distance to be 25000 miles ? Ans. 3360 pounds. 37. John has $8546, which is $ 342 less than 4 times as much as Charles has. How many dollars has Charles ? Ans. $ 2222. 38. The quotient of one number divided by another is 37, the divisor 245, and the remainder 230. What is the dividend ? Ans. 9295. 39. What number multiplied by 72084 will produce 5190048 ? Ans. 72. 40. There are two numbers, the greater of which is 73 times 109, and their difference is 17 times 28. What is the less number? Ans. 7481. 41. The sum of two numbers is 360, and the less is 114. What is the product of the two numbers ? Ans. 28044. 42. What number added to 2473248 makes 2568754 ? 43. A farmer sold 35 bushels of wheat at $2 a bushel, and 18 cords of wood at $3 a cord; he received 9 yards of cloth at $4 a yard, and the balance in money. How many dollars did he receive ? Ans. $ 88. 44. A farmer receives $ 684 a year for produce from his farm, and his expenses are $375 a year. What will the save in 5 years ? 45. If a salt manufacturer pays 58 cents for wood to boil one barrel of salt, 10 cents for boiling, 5 cents for the brine, 28 cents for the barrel, and 3 cents for packing and weighing, and receives $1.25 from the purchaser, what does he make on a barrel ? Ans. 21 cents а PROOFS. CASTING OUT NINES. 87. The remainder arising from dividing any number by 9 is always the same as the remainder that arises from dividing the sum of all its digits by 9. This arises from the fact that every 10, 100, or 1000 of a number, when divided by 9, has a remainder of 1. Thus, 9 into 10, once and 1 over; 9 into 100, 11 times and 1 over, etc.; 9 into 20, twice and 2 over; into 200, 2000, etc., 2 over; into 30, 300, 3000, etc., 3 over. Hence, if we divide by 9 separately the parts of a number, the remainders will be expressed by the digits of the number. Thus, if we divide 2345 by 9, first dividing the 2000, remainder 2 ; dividing 300, remainder 3; dividing 40, remainder 4; dividing 5, remainder 5. Total remainder, 2+3+4+5=14; excess of 9's in 14 = 5, which is the same as the excess of 9's in 2345. Methods of proving addition, subtraction, multiplication, and division have already been given. The casting out of 9's furnishes another proof, briefer but not so sure. EXAMPLES 1. Prove that 3523 + 6414 + 1894 + 2129 = 13960. Solution. — Adding the 3523, excess of 9's 4 digits in the first number, 6414, excess of 9's 6 the excess of 9's is 4; in the second, 6; in the third, 1894, excess of 9's 4 4; and in the fourth, 5. 2129, excess of 9's 5 The sum of the excesses is 13960 19 = two 9's and 1. Divid. Exc. of 9's = 1. Exc. of 9's ing 13960 by 9, we find the remainder to be 1. Since the excess in both cases is the same, the answer is probably correct OPERATION. = = 19 : 1. the excess of 9's in 6532; hence the answer is probably correct. 3. Prove that 62 × 42 = 2646. in which the excess of 9's is 3. The excess of 9's in 2604 is 3; hence the answer is probably correct. 4. Prove that 207 ÷ 23 = 9. OPERATION. 23) 207 (9 Exc. 5. Exc. 0. Exc. = 0. SOLUTION. Casting out the 9's from divisor and quotient, the excess is 5 and 0. Their product is 0. Casting out the 9's from the dividend, the excess is 0. Hence the answer is probably correct. RULE. -I. In Addition, find the excess of 9's in the sum of the digits in each addend; add these excesses, and find the excess of 9's in their sum. If it agrees with the excess of 9's in the answer, the answer is probably correct. II. In Subtraction, subtract the excess of 9's in the minuend from that in the subtrahend, casting out the 9's from the remainder. Compare the excess with excess in answer. NOTE. If the excess of 9's in the subtrahend is greater than that in the minuend, add one 9 to the minuend and then subtract. III. In Multiplication, from the product of the excess of 9's in the multiplicand and multiplier, cast out the 9's. Compare the excess with excess in answer. IV. In Division, find the excess of 9's in the quotient and divisor, and take their product. Cast out the 9's from Compare this excess with excess in dividend. this product. |