Example 6. If 17 be divided by 6 the quotient is 2, and the remainder 5, So if N be divided by D, and the quotient be Q and the remainder R, we have Thus, if the divisor is x, the quotient y, and the remainder z, the dividend is xy + z. Example 7. A and B are playing for money; A begins with $p and B with q dimes: after B has won $x, how many dimes has each? 3. 4. What is the defect of 2c from 3d? What dividend gives b as the quotient when 5 is the divisor? 5. By how much does 3k exceed k? 6. If 100 be divided into two parts and one part be x what is the other? If a be one factor of b, what is the other? 7. 9. What is the price in cents of a oranges at ten cents a dozen ? 10. What is the price in cents of 100 oranges when x cost six cents ? 11. If the difference of two numbers be 11, and if the smaller be x, what is the greater? 12. If the sum of two numbers be c and one of them is 20, what is the other? 15. 16. If 100 contains x five times, what is the value of x? 17. In x years a man will be 36 years old, what is his present age? 18. 19. take? 20. 21. How old will a man be in a years if his present age is x years? What value of x will make 5x equal to 20? What is the price in dimes of 120 apples, when the cost of two dozen is x cents? 22. hour? How many hours will it take to walk x miles at 4 miles an 23. How far can I walk in x hours at the rate of y miles an hour? 24. In x days a man walks y miles, what is his rate per day? 25. How many minutes will it take to walk x miles at a miles an hour? 26. A train goes x miles an hour, how long does it take to go from New York to Albany, a distance of 144 miles? 27. How many miles is it between two places, if a train travelling p miles an hour takes 5 hours to perform the journey? 28. What is the velocity in feet per second of a train which travels 30 miles in x hours? 29. A man has a dollars and b dimes, how many cents has he? 30. If I spend x half-dollars out of a sum of $20, how many half-dollars have I left? 31. Out of a purse containing $a and b dimes a man spends c cents; express in cents the sum left. 32. By how much does 2x-5 exceed x+1? 33. What number must be taken from a-2b to leave a- -3b? If a bill is shared equally amongst x persons and each pays four dimes, how many cents does the bill amount to? 34. 35. If I give away c dimes out of a purse containing a dollars and b half-dollars, how many dimes have I left? 36. In how many weeks will x horses eat 100 bushels of oats if one horse eats y bushels a week? 37. If I spend x quarters a week, how many dollars do I save out of a yearly income of $y? 38. A bookshelf contains x Latin, y Greek, and z English books: if there are 100 books, how many are there in other languages? 39. I have x dollars in my purse, y dimes in one pocket, and z cents in another: if give away a half-dollar how many cents have I left? 40. In a class of x boys, y work at Classics, z at Mathematics, and the rest are idle: what is the excess of workers over idlers? 76. We subjoin a few harder examples worked out in full. Example 1. What is the present age of a man who x years hence will be m times as old as his son now aged y years? In x years the son's age will be y+x years; hence the father's age will be m(y+x) years; therefore now the father's age is m(y+x)-x years. cent. Example 2. Find the simple interest on $k in n years at ƒ per Example 3. A room is x yards long, y feet broad, and a feet high; find how many square yards of carpet will be required for the floor, and how many square yards of paper for the walls. (1) The area of the floor is 3xy square feet; .. the number of square yards of carpet required is 3.xy = xy 2a (3x+y) 9 Example 4. The digits of a number beginning from the left are a, b, c; what is the number? Here c is the digit in the units' place; b standing in the tens' place represents b tens; similarly a represents a hundreds. The number is therefore equal to a hundreds +b tens+c units =100 a+ 10 b+c. If the digits of the number are inverted, a new number is formed which is symbolically expressed by 100c+10b+a. Example 5. What is (1) the sum, (2) the product of three consecutive numbers of which the least is n? The numbers consecutive to n are n+1, n+2; .. the sum=n+ (n + 1) + (n+2) =3n+3. And the product=n (n+1)(n+2). We may remark here that any even number may be denoted by 2n, where n is any positive whole number; for this expression is exactly divisible by 2. Similarly, any odd number may be denoted by 2n+1; for this expression when divided by 2 leaves remainder 1. Example 6. How many days will a men take to mow b acres if c boys can mow a acres in b days, and each man's work equals that of n boys? 1. Write down four consecutive numbers of which x is the least. 2. Write down three consecutive numbers of which y is the greatest. one. 3. Write down five consecutive numbers of which x is the middle 4. 5. What is the next even number after 2n? What is the odd number next before 2x+1? 6. Find the sum of three consecutive odd numbers of which the middle one is 2n+1. 7. A man makes a journey of x miles. He travels a miles by coach, b by train, and finishes the journey by boat. How far does the boat carry him? 8. A horse eats a bushels and a donkey b bushels of corn in a week; how many bushels will they together consume in n weeks? 9. If a man was x years old 5 years ago, how old will he be y years hence? 10. A boy is x years old, and five years hence his age will be half that of his father. How old is the father now? 11. What is the age of a man who y years ago was m times as old as a child then aged x years? 12. A's age is double B's, B's is three times C's, and C is x years old: find A's age. 13. What is the interest on $1000 in b years at c per cent.? 14. 15. What is the interest on $x in a years at 5 per cent.? What is the interest on $50a in a years at a per cent.? What is the interest on $24xy in x months at y per cent. per annum? 16. 17. A room is x yards in length, and y feet in breadth; how many square feet are there in the area of the floor? 18. A square room measures x feet each way; how many square yards of carpet will be required to cover it? 19. A room is p feet long and x yards in width; how many yards of carpet two feet wide will be required for the floor? 20. What is the cost in dollars of carpeting a room a yards long b feet broad with carpet costing c dimes a square yard? 21. How many yards of carpet x inches wide will be required to cover the floor of a room y feet long and z feet broad? 22. A room is a yards long and b yards broad; in the middle there is a carpet c feet square; how many square yards of oil-cloth will be required to cover the rest of the floor? 23. How many miles can a person walk in 45 minutes if he walks a miles in x hours? 24. How long will it take a person to walk b miles if he walks 20 miles in c hours? 25. If a train travels a miles in b hours, how many feet does it move through in one second? 26. A train is running with a velocity of x feet per second; how many miles will it travel in y hours? 27. How long will x men take to mow y acres of corn, if each man mows z acres a day? 28. How many men will be required to do in x hours what y men do in xz hours? 29. What is the rate per cent. which will produce $y interest from a principal of $1000 in r years? 30. Find in how many years a principal of $a will produce $p interest at r per cent. per annum. |