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25. A laborer receives $1.40 per day, and spends $.75 for his support. How much does he save in a week? 26. What is the cost of 23487 feet of hemlock boards, at $4.50 per 1000 feet? Ans. $105.6915. 27. A man has an income of $1200 a year. How much must he spend per day to use it all?
28. I bought 28 firkins of butter, each containing 56 pounds, at $.17 per pound. What was the whole cost?
29. What do 4868 bricks cost, at $4.75 per M?
30. A farmer sold 27 bushels of potatoes, at $.331 per bushel; 28 bushels of oats, at $.25 per bushel; and 19 bushels of corn, at $.50 per bushel. What did he receive for the whole? Ans. $25.50.
31. A man bought an overcoat for $364, and a coat for $281, and gave in payment one fifty, and two ten dollar bills. How much change should he receive? 32. If 4 barrels of flour cost $32.3, what will 71⁄2 barrels cost? Ans. $51. 33. If .875 of a ton of coal costs $5.635, what will 94 tons cost? Ans. $59.57.
34. For the first three years of business, a trader gained $1200.25 a year; for the next three, he gained $1800.62 a year, and for the next two he lost $950.87 a year. Supposing his capital at the beginning of trade to have been $5000, what was he worth at the end of the eighth year? Ans. $12100.87. 35. What will be the cost of 18640 feet of timber, at $4.50 per 100? Ans. $838.80. 36. A gardener sold, from his garden, 120 bunches of onions at $.12 a bunch, 18 bushels of potatoes at $.621 per bushel, 47 heads of cabbage at $.07 a head, 6 dozen cucumbers at $.18 a dozen; he expended $1.50 in spading, $1.27 for fertilizers, $1.87 for seeds, $2.30 in plant ing and hoeing. What were the profits of his garden?
37. If a man has a capital of $ 3495 and loses .25 of it, how much has he left? Ans. $2621.25.
38. I paid a man $90 for the use of $1500 for 1 year. How much did I pay him for the use of each dollar?
39. A man bought a carriage for $150. He paid $15 for repairs and then sold it at a gain of .15 of the total cost. For how much did he sell it? Ans. $189.75.
40. If at a closing out sale I can get $8 worth of goods for every $4, how much can I get for $75?
41. I loaned a man a sum of money. He repaid me $24, which was .75 of the sum loaned him. How much does he still owe me? Ans. $8. 42. A merchant sells goods amounting to $ 5346. If he reserves as commission .02 of the amount sold, what is his commission? Ans. $106.92.
43. A merchant sold goods for $5346 and received $106.92 commission. How much commission did he receive for each dollar's worth of goods he sold?
44. A man has a capital of $50000; he invests .6 of it in business, .01 of it in stocks, and the remainder in bonds and mortgages. How much does he invest in bonds and mortgages? Ans. $19500.
45. I paid a man $6 for the use of $100 for 1 year. At the same rate, how much would I pay him for the use of the same sum for 5 years? Ans. $30. 46. At the same rate, what would I pay him for the use of $500 for 1 year? For 5 years?
Ans. $30. $150. 47. If I import $10000 worth of goods on which the duty is .12 of the value of the goods, how much duty must I pay ?
48. A and B engaged in trade, each with gained .12 of his capital and B .27 of his. more did B gain than A?
$5000. A How much Ans. $725
175. A scale is a series of units, increasing or de creasing by fixed multipliers or divisors.
In the ordinary decimal notation this multiplier is 10. The decimal notation will be better understood when pupils have learned something of scales in gen. eral and of the possibility of writing numbers in any scale.
Scales may be uniform or varying.
In the uniform scale the multiplier or divisor is the same. In the varying scale it varies.
The Arabic system of notation and all the tables of the metric system are based on a uniform scale, the decimal. Most of the other tables in Denominate Numbers are based on varying scales.
The number of units which make one of the next higher order is the root or radix of the scale.
Thus, in the decimal scale 10 units make 1 ten; 10 tens, 1 hundred, etc.; therefore 10 is the radix.
Other uniform scales besides the decimal are: the binary (radix 2), ternary (3), quaternary (4), quinary (5), senary (6), septenary (7), octary (8), nonary (9), undenary (11), duodecimal (12), which derive their names from their radix.
A scale contains as many digits as there are units in its radix; and every scale must have the digit, 0.
10 digits, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.
9 digits, 1, 2, 3, 4, 5,
The octary scale has eight digits, 1 to 7 and 0; the septenary seven, 1 to 6 and 0; the senary six, 1 to 5 and 0; the quinary five, 1 to 4 and 0; the quaternary four, 1 to 3 and 0; the ternary three, 1, 2, 0; the binary two, 1, 0; the undenary eleven, 1 to 9, and one more (which may be expressed by any character, as a), and and the duodecimal twelve, 1 to 9, a, b, 0.
By combining the digits in any scale, we obtain the notation of that scale.
In the decimal scale when we pass 9, we must use two digits for the next number, which is 10. In like manner in the quaternary scale, when we pass 3, we must use two figures; therefore 4 in the decimal scale is written 10 in the quaternary scale. In the quinary scale, when we pass 4, we must use two figures; therefore 5 in the decimal scale is written 10 in the quinary. In the decimal scale, when we pass 99, we have to use three digits for the next number, which is 100; but the highest number we can write in the quinary scale in two figures is 44, which corresponds to 24 in the decimal scale. Therefore 25 in the decimal scale would be written 100 in the quinary; and for the same reason 125 would be written 1000; 625, 10000, etc.
In the ternary scale, 3 would be written 10, 9 would be written 100, 27 would be written 1000, 81, 10000, etc. In the senary scale, 6 would be written 10, 36 would be 100, 216 would be 1000, 1296 would be 10000, etc.
The following table shows the notation in the various scales with the corresponding value of the numbers, in the decimal scale:
DECIMAL. 1 2 3 4 5 6 7 8 9 10 11 12 13
1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101
1. Write in the duodecimal scale the numbers corresponding to those from 14 to 36 in the decimal scale.
2. Write in the undenary, septenary, and nonary scales the numbers from 13 to 44 in the decimal scale. 3. Write the numbers from 1 to 100 in the following scales:
4. Express in the decimal scale the number 10000, as used in the ternary and quatenary scales.
5. Express in the decimal scale the number 1000000, as used in the senary scale.
6. Express in the decimal scale the numbers 100, 1000, 10000, 100000, as used in the quinary scale.
175a. Since varying scales do not increase by one constant multiplier, in working examples in such scales we must bear in mind how many units of each order make one of the next higher.
Thus in the scale, 24 grains (gr.) make 1 pennyweight (pwt.); 20 pwt., 1 ounce (oz.); 12 oz., 1 pound (lb.); 24 units of the first order make 1 of the second; 20 of the second, 1 of the third, and 12 of the third, 1 of the fourth. Hence, to reduce grains to pounds, we must multiply successively by 24, 20, and 12; to reduce pounds to grains, we divide successively by 12, 20, and 24.
Numbers written in a uniform scale, as the decimal,
are called simple numbers.
Those written in the
varying scales are compound numbers.
Numbers may be added, subtracted, multiplied, and divided by the same rules as apply to decimal notation, modified to adapt them to the particular scale in which the numbers are written.