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53. If I pay £10, 4s. 9d. of income-tax, being at the rate of 7d. in the pound, what is my annual income? . . Ans. £351.
54. If a man walk 7 miles in 2 hours 10 min., how many miles will he walk at the same rate in 4 hours ? - Ans. 1211 miles.
55. If 52 cwts. 13 lbs. of beef cost £158, 1s. 84d., how much may be bought for £82, 7s. 2 d. ? . . Ans. 27 cwts. 17 lbs.
56. If 18 men can perform a piece of work in 28 days, how many men could do it in a fourth part of the time ? . Ans. 72 men.
57. If a person can perform a journey in 6 days, riding 9 hours each day, how long will it take him to perform the same journey if he rides 12 hours a day ? . . . . Ans. 41 days.
58. If 4 tons 6 cwts. of railway-bars cost £39, 17s. 5 d., how much must be paid for 1723 tons 1 qr. at the same rate ?
Ans. £15,976, 13s. 8 d. 34 59. If 27 oxen are grazed in a field for 112 days, how many oxen could have been grazed equally well in the same field for 48 days ? . . . . . . . . Ans. 63 oxen.
60. The rents of a parish amount to £1750, and a poor-rate is wanted of £98, 8s. 9d. ; what must be the assessment per pound ? . . . .
. . Ans. ls. 11d. 61. If the quartern-loaf cost 84d. when wheat is at 60s. per quarter, what ought to be its price when wheat is 49s. per quarter ?
. Ans. 62 d. 23 62. A bankrupt's effects are valued at £983, 12s. 4d., and his debts are £1728, 12s. 7 d., how much will his creditors receive per pound ? .
. . Ans. 11s. 4 d. 7024 63. If a clerk has a salary of £72, 18s. a year, commencing on the 1st of January, how much will he have to receive on leaving his situation on the 25th of September ? Ans. £53, 10s. 64d. 187
64. Sold 14 yds. 2 qrs. 1 nail for £10, 15s. 4d. from a piece of cloth which at first contained 28 yds. 1 qr.; what is the value of the remainder at the same rate ? . . Ans. £10, 2s. 4 d. 24
65. If the carriage of 14 cwts. O qr. 23 lbs. for 65 miles comes to a certain sum of money, what weight may I have carried 37 miles for the same sum ? , . Ans. 24 cwts. 3 qrs. 23 lbs.
66. A vessel has provisions for 15 days, but being obliged by certain circumstances to continue at sea for 20 days, to what quantity must the daily ration of 20 lbs. be reduced to make the provisions last during that time ? . . . . Ans. 15 lbs.
67. If the soldiers in a besieged garrison have provisions sufficient for 5 months at the rate of 20 oz. per man a day, how long will they be able to hold out when each man's allowance is reduced to 12 oz. a day? .
. . . Ans. 8 months. 68. If a cistern of 230 gallons has a pipe which discharges 5 gallons in a minute, and another has a pipe which discharges 6 gallons in a minute, and if both cisterns are emptied in the same time, how many gallons does this last cistern contain ?
Ans. 276 gals. EXPLANATION OF THE RULE.-This rule depends on the principle
that, from the nature of the question regarding the three given terms, there is always the same proportion, or ratio (as it is termed in mathematical language) between the two similar terms, as there is between the third term and the unknown quantity.
For example, in the question, If 10 yards of cloth cost £5, what will 20 yards cost?' it is obvious that there is a certain proportion between 10 yards and 20 yards—the one is the double of the other; and from the nature of the case, there must be a similar proportion between £5, the price of 10 yards, and the unknown sum, the price of 20 yards—the one will also be the double of the other.
To express the question distinctly, it is stated thus-as is the proportion of 10 yards to 20 yards, so is the proportion of £5 to the unknown sum. Omitting the words, the figures appear as follows, according to the rule of Simple Proportion:
yds. yds. £
10 : 20 :: 5 : The proportion between two numbers is ascertained by dividing the greater by the less-the proportion being expressed by the quotient: thus-if we divide 20 yards by 10 yards, the quotient is 2; that is, 20 is 2 times 10; or, inversely, 30 is the of 20. In the present example, 20 yards being 2 times 10 yards, it follows that the price of 20 yards will be 2 times the price of 10 yards (£5); and therefore, if we multiply £5 by 2, we will obtain the unknown price required namely, £10.
To express the process briefly : the principle of the rule consists in dividing the second term by the first, to find the proportion between them, and then multiplying the third term by the quotient or ratio, in order to find the unknown quantity.
In practice, instead of first dividing and then multiplying, as in No. 1, below, it is usually more convenient to multiply first, and then divide, as stated in the rule (see No. 2, below). (1.)
(2.) yds. yds. £
yds. yds. £ 10 : ) 20 :: 5 It will be observed that the
10 : 20 :: 5 result is the same in both cases.
10)100 £10 Ans.
£10 Ans. Direct Proportion is where one number increases in proportion as another increases, or diminishes as another diminishes, as in example 1, page 57.
Inverse Proportion is where one number increases as another diminishes, or diminishes as another increases, as in example 2, page 57.
RATIO.-This is a mathematical term, as has been already stated, expressing the comparative magnitude of two numbers of the same kind; or how many times the one number is greater or smaller than the other : the ratio is ascertained by dividing the greater number by the less.
The ratio of a larger number to a smaller is expressed by the quotient on dividing the one by the other: thus-the ratio of 24 to 8 is 3; or as 3 to l; meaning that 24 is 3 times 8.
The ratio of a smaller number to a larger, is expressed by the quotient inversely: thus-the ratio of 8 to 24 is }; or as 1 to 3: meaning that 8 is onethird (3) of 24.
The two numbers of a ratio must always be of the same kind-as 10 pounds and 20 pounds; 50 miles and 100 miles. There cannot be a ratio between numbers of different kinds, such as 10 pounds and 100 miles.
The Rule of Proportion is founded on the similarity of the ratio of two given quantities, to the ratio of two other quantities.
COMPOUND PROPORTION. COMPOUND PROPORTION is the rule employed instead of Simple Proportion, when more than three terms (usually five) are given to find the unknown quantity; as, for example, if 12 persons earn £30 in 25 days, how much will 18 persons earn, at the same rate, in 56 days ?
The principle of the rule is the same as in Simple Proportion.
RULES. I. RULE FOR STATING.-1. Write down for the third term that number which is of the same kind as the answer required.
2. For the first and second terms, take two numbers of the same kind, and state them as in Simple Proportion, placing the greater and the less, first and second or the reverse, as the case requires."
3. Take other two terms of the same kind with each other, and state them in like manner, placing them directly under the preceding two; and so on with any others, till all the numbers are stated.*
II. RULE FOR WORKING.-1. Reduce the numbers of the first and second terms to the same denomination, and if the third term is compound, reduce it to a simple number, as in Simple Proportion.
2. Multiply the numbers of the first term together: then those of the second term; thus reducing the different numbers of these two terms into a single quantity for each.
3. Proceed with the three terms thus found, as in Simple Proportion.
NOTE.—The terms may be cancelled, when practicable, on the same principle as in Simple Proportion.
Example.- If I give 16 men £45 for 28 days' work, what must I give, at the same rate, to 20 men for 35 days' work ?
16 men : 20 mon :: £45 Here money being the answer re28 days 35 days
quired, £45 is written for the third term : then 16 men and 20 men
being terms of the same kind, 16 and 32 45 £ s. d. 20 are written as the first and second, 448 ) 31500( 70 6 3 Ans. according to the rule in Simple Pro
portion : 28 days and 35 days being
also of the same kind, 28 and 35 are placed respectively below 16 and 20. The 16 is then multiplied by 28, making 448 for the first term : the 20 is next multiplied by 35, making 700 for the second : the three terms, 448, 700, and 45, are then proceeded with as in Simple Proportion.
* THE OTHER METHOD of stating Simple Proportion, page 59, may also be used in Compound Proportion.
Exercises. 1. If 12 persons consume 18 lbs. of beef in 4 days, how many pounds will 42 persons consume in 6 days ? . . Ans. 941 lbs.
2. If 6 cows produce 28 gallons of milk in 2 days, how many gallons will 13 cows produce in seven days ? Ans. 2127 gallons.
3. If 17 boys wear among them 28 pair of shoes in a year, how many pair should 22 boys wear in a year and a half ?
Ans. 541 pair. 4. A ship at sea had 112 persons on board, with provisions which gave an allowance of 1 lb. 6 oz. per day to each individual for 28 days; a starving ship's company, consisting of 49 persons, were picked up, increasing the number to 161; what quantity of provisions must all now be reduced to daily, in order to give each man a fair share during 26 days ? . . Ans. 1644 oz.
5. If 5 furnaces consume 30 tons of coal in 6 days, how many tons at the same rate of consumption should 3 furnaces consume in 11 days ? .
. . . . . . Ans. 33 tons. 6. If the carriage of 2 tons 15 cwts. for 40 miles be £1, 8s., how much should be the carriage of 5 tons 11 cwts. for 72 miles ? . . .
. Ans. £5, 1s. 8£d. 196 7. If a person travel 120 miles in 4 days, by walking 9 hours a day, what time will be required for him to travel 386 miles, by walking 7 hours a day? . . . . . Ans. 1645 days.
8. If 75 men cut down 45 acres of corn in 4 days, how many acres will 108 men, working equally well, cut down in 25 days?
. . Ans. 405 acres, 9. If 16 persons can be maintained for 60 days on £84, how much money would be required to support in similar circumstances 96 men for 365 days? . . . . . Ans. £3066.
10. If 18 compositors can set up 24 sheets in 8 days, how many sheets could 45 compositors set up in 14 days ? Ans. 105 sheets.
11. If 25 horses consume 8 bushels of oats in 3 days, how many bushels would 42 horses consume in 15 days, at the same rate of living? . .
. Ans. 67} bushels. 12. If 3000 copies of a book of 11 sheets require 66' reams of paper, how much paper will be required for 5000 copies of a book of 12 sheets ? . . .
. Ans. 125 reams. 13. If a man can travel 360 miles in 12 days of 8 hours each, how many miles, at the same rate of walking, will he travel in 60 days of 6 hours each ? . .
. .' Ans. 1350 miles. 14. If 18 men eat 15s. worth of bread in 3 days, when wheat is selling at 54s. per quarter, what value of bread will 54 men eat in 27 days, when wheat is selling at 50s. per quarter ? Ans. £18, 15$.
15. A road-contractor engaged to finish 24 miles of road in 84 days; but after employing 60 men for 54 days, he found that they had only finished 880 yards; how many additional men must he engage, so that the work may be finished within the prescribed
. . . . Ans. 372 men.
THE GREATEST COMMON MEASURE, OR DIVISOR.
THE GREATEST COMMON DIVISOR of two numbers, is the greatest number that will divide each of them, without a remainder: thus 5 is the greatest common divisor of 10 and 15. TO FIND THE GREATEST COMMON DIVISOR OF TWO NUMBERS.
RULE.—Divide the greater by the less; then the divisor by the remainder if any; then the last divisor by the last remainder; and so on in the same way, till there is no remainder: the last divisor is the number sought.
Tliis rule is useful in the reduction of Fractions, see page 68. Example. Find the greatest common divisor of 24 and 42. 24 ) 42(1
Find the greatest common divisor of the following numbers :1. 252 and 348, . . Ans. 121 3. 620 and 2108, . Ans. 124 2. 493 899, .. 29 4. 4081 1 5141, . . 53
THE LEAST COMMON MULTIPLE. THE LEAST COMMON MULTIPLE of several numbers, is the least number that contains each of them a certain number of times exactly : thus-24 is the least common multiple of 4, 6, 8, and 12. TO FIND THE LEAST COMMON MULTIPLE OF SEVERAL NUMBERS.
RULE.—Write the given numbers in a line, one after the other; cancel such of them as divide any of the others exactly; then divide as many of the rest as practicable, by some number* hat divides them without a remainder, placing the quotients and any undivided numbers in the line below; again cancel, divide, &c., as before, carrying on the process till no numbers remain that have a common divisor; then multiply together the numbers used as divisors, and any undivided numbers in the last line; and the product is the least common multiple required.
This rule, like the preceding, is useful in the reduction of Fractions. Example. Find the least common multiple of 12, 4, 9, 30, 8. 2)12, 4, 9, 30, 8
Here, after cancelling and dividing accord3)6, 9, 15, 4
ing to the rule; the divisors employed and
the numbers in tho lowest line (which have 2 3, 5, 4
no common divisor) are multiplied together 2x3x3x5X4=360 Ans. for the answer.
Find the least common multiple of the following numbers :1. 2 4 5 8,. Ans. 401 3. 12 16 15 24, Ans. 240 2. 3 6 9 16,
144 | 4. 16 24 32 78, 1248
• Take whatever number will divide more of them than any other that could be taken.