In multiplying by 663, explain why we may find of the multiplicand,

and annex two 0's to the result.

2. Divide 8250 by 371.

8250378250 ÷ 3 of 100 = 8250 × 380 = 220.

Fractional parts of a hundred involving fifths and tenths are not included in these lists for the reason that multiplying or dividing by 20, 30, etc., cannot be made simpler than by the direct method.

Solve the following, using short methods:

27. At 75 a bushel, what will 320 bushels of rye cost? 28. At 831 a yard, what will 156 yards of cloth cost?

29. At 871 a pair, what will 48 pairs of rubbers cost? 30. If cloth is worth 75¢ a yard, how many yards can be bought for $18?

31. At the rate of 33 miles a day, how many days will it take a man to walk 200 miles?

32. James missed 12% of his 16 examination questions. How many did he miss? If 100 represented the highest grade possible, what was his mark?

33. A man who owned 25% of an estate sold 20% of his share to his brother. What per cent of the estate did he sell?

34. Mr. Perry owned a fifth interest in a storehouse, and sold 25% of his share to each of his four associates. What part of it did each buy?

35. In an orchard

of the trees bear apples, 40% peaches, and the remaining 50 cherries. How many trees are in the orchard?

36. At 12 a pound, what will a 500-pound bale of cotton bring?

37. A man paid $5075 for 871 acres of land. How much was that an acre?

38. If 6 pounds of tea cost $5, how many pounds of tea can be bought for $9.60?

39. If a dealer pays 371 for 16 boxes of toothpicks, how many boxes can he buy for $2.25?

40. How much would be gained by buying 320 lb. of sugar at 5 a pound and selling it at 61 a pound ?

41. A man bought 16 bu. of corn at paid for it with lard at 121 a pound. of lard were required?

621¢ a bushel and How many pounds

42. At 621 a rod, what is the cost of fencing a rectangular field that is 195 rd. long and 65 rd. wide?

43. A man receives a salary of $1860 a year, and 81% of his salary equals 163 % of his savings. What sum does he save per annum?

44. James's school report showed that he had 5 studies, and that his marks in them were 75, 85, 90, 95, 93. Find his average grade, using a short method based on 5 = 10 ÷ 2.

THE EQUATION

341. Hitherto in all computations you have used figures to represent numbers. It is sometimes more convenient to represent numbers by letters. When we speak of 3 dollars or of 5 gallons, we mean a definite number of dollars or gallons; but when we speak of n dollars or of x gallons, the number of dollars or gallons is indefinite. The reasoning, however, is the same whether numbers are represented by letters or by figures. Thus, if x stands for the number of dollars in your purse, 2x stands for twice that number, 3x for three times that number, etc. (Note that x is used for 1x.)

1. If n represents the number of marbles you have, what does 5 n represent? 7n? 10n?

2. If one bucket holds a quarts and another 3 a quarts, how many quarts do both hold? a+3a=

3. If Elizabeth has 5n yards of ribbon and gives 2n yards to Ann, how many yards has she left? 5n-2n

=

4. At x dollars per head, what will 10 sheep cost? 10 x x = What will they cost at 2x dollars per head? 10 × 2x=

5. What is of 6 dollars? Of 6x dollars? Of 12x? 6. If 5 sheep cost 10x dollars, what will 1 sheep cost? of 10x=

7. The expression 2(4+7) means that the sum of 4 and 7 is to be multiplied by 2. What does 6(5—3) mean? Indicate that the sum of 2x and 3x is to be multiplied by 5,

342. The symbol of equality, , is read equals and indicates that the numbers or expressions before and after it are equal in value. Thus, 4+5=9 means that the sum of 4 and 5 equals 9; and 9-45 means that the difference between 9 and 4 equals 5.

Such statements of equality are called Equations.

For example, 3+ 5 = 8, and 4+2=6 are equations. Is 5+7=7+9 an equation? Why not?

343. 1. When does a scale beam balance?

2. When balanced, if a weight is added to one scale pan, what must be added to the other to preserve the balance?

3. When balanced, if a weight is taken from one pan, what

must be done to the other to preserve the balance?

4. How can the balance be maintained, if the weights

in one pan are doubled? If halved?

344. You may consider the equal numbers in an equation as measuring the equal or balancing weights in the two scale pans.

=

1. If we add 5 to the first side of the equation, 7+ 3 155, how much must be added to the second side to make the sides equal again? 7+3+5=15-5+5;

2. If we take 5 from the first side, what must be done to the second to preserve the equality? 5-5; that is, 5 5.

=

3. How much is 5 times (7+3)?

7+ 3 - 5 = 15 –

How much is 5

times (15-5)? 5 (7+3)=5 (155); that is, 50 =50,