In this proportion the first fraction, 7, and its denominator, 11, are the extreme terms, the other fraction, 7, and its denominator, 13, the means; therefore two fractions which have the same numerator are reciprocally proportional to their denominators, or are to each other in the inverse ratio of their denominators. 343. The relation between the unknown quantity or last term of a proportion, and the given term of the same kind, may depend on many conditions which it may be necessary to combine in order to determine the unknown quantity. If it be supposed, for instance, that 20 workmen in 18 days execute 500 yards of a certain work, and from this it be required to determine the number of days in which 76 workmen can execute 1265 yards of the same work, the number of days must evidently depend upon the number of workmen employed and the number of yards to be executed. The data in this question (Ex. 5) are, The number of workmen in each of two parties, 20 and 76: The time in which the first party of 20 workmen can execute 500 yards And the problem is, to find the time taken by 76 men, working at the same rate, to execute 1265 yards of the same work. The answer to the question being time the term 18 days is the third in the proportion. If the time taken by 76 men to execute 500 yards of the work were known, the time required for the execution of 1265 yards at the same rate could be found; let the first be represented by x, the second by x'. The first step, therefore, is to find x, the time taken by 76 men to execute that amount of work (500 yards) which is done by 20 men in 18 days. Now, if 20 men execute a work in 18 days, 76 men execute it in less than 18 days. 20 Therefore 76 men: 20 men :: 18 days: x days=18x days. 76 The second step is to find x', the time taken by 76 men to execute 1265 yards of work at the rate of 500 yards in x days (=18×g days). 500 yards, the term related to the time x days, being less than 1265 yards, the term related to a days, the third term is less than the fourth, which is therefore given by the proportion. 500 yds. 1265 yds.::r days: a' days=xx Restoring the value of x from the first proportion, 1265 500* This result is composed of the continual product of three factors; of these the first is that quantity which is of the same kind as the answer to the question. The second is a fraction having for denominator the antecedent and for numerator the consequent of the first ratio of a proportion of which the first pair of quantities of the same kind and that quantity which is of the same kind as the answer form three terms. The third is a fraction having for denominator the antecedent and for numerator the consequent of the first ratio of a second proportion of which the second pair of quantities of the same kind and the fourth term of the first proportion form three terms. The numerical value of the result is obtained by multiplying the third term of the first proportion by the product of the numerators of these fractional factors, and dividing the result by the product of their denominators (Art. 194), that is, by multiplying the number which is of the same kind as the answer by the product of the consequents of the first ratios of all the proportions, and dividing the result by the product of the antecedents of the first ratios of the same proportions. 344. The antecedent and consequent of each pair of terms of the same kind in these proportions can be determined without forming the fourth terms. Let a be the antecedent, c the consequent of the first pair of terms of the same kind. second Then act: tx=4th term of 1st proportion. с a a Whence the arrangement of the terms a', c', determined by comparison with t, the third term of the 1st proportion, must agree with the arrangement of a', c', determined by comparison with tx portion. Again, d" : c' :: t× a": the 4th term of the 1st pro a' Whence the arrangement of the terms a", c", determined by comparison with t, must also agree with the arrangement of a", c", determined by comparison with t×, the 4th term of the second proportion. . . . . с a The calculation of the fourth terms of these proportions is, therefore, unnecessary. All that is requisite is, to determine the antecedent and consequent of each pair of terms of the same kind by comparison with the number which is of the same kind as the answer to the question, to multiply this number by the product of all the consequents, and to divide the result by the product of all the antecedents. 6th Example. If 15 men working 10 hours a day take 18 days to execute 450 yds. of a certain work, how many men of equal strength, who work 12 hours a day, are required to complete 480 yds. of the same work in 8 days? The answer to the question being the number (x) of a party of men able to execute a given work under the conditions of the question, 15 (men) is the third term of the proportion. The pairs of numbers of the same kind, from the comparison of which with 15 the antecedent and consequent of each pair are to be determined, are, 10 hours term related to 15 men; 12 hours term related to x men. Let f', f", "", respectively denote the fourth terms of the three proportions which must be employed to determine which of these quantities are antecedents and which consequents; then, 1st. In 10 hours a certain work is executed by 15 men; in 12 hours the same work is done by fewer than 15 men ; ... 12 hrs 10 hrs:: 15 men: f' men, or 6: 5::15:ƒ' (Art. 337). 2d. To execute this work in 18 days 15 men are required; to execute it in 8 days more than 15 men are required; .. 8 days: 18 days::15 men :ƒ" men, or 4 : 9::15 : ƒ“. 3d. To execute 450 yds., under given conditions, 15 men are required; to execute 480 yds., under the same conditions, more than 15 are required; .. 450 yds.: 480 yds.::15 men : ƒ"" men, or 15 : 16::15 : ƒ"". Collecting and arranging the first ratios of these proportions, multiplying the general third term by the product of all the consequents of these ratios, and dividing the result by the product of all the antecedents, as under: 12 hrs. : 10 hrs. = 6: 5 8 days 18 days = 4: 9 ::15 men x men: 450 yds. 480 yds.=15: 16 r, the number of men required, is therefore 30. = =30; The reduction in the ratios of the quantities of the same kind is made by Article 337; and the omission of factors common to the numerator and denominator of the value of x by Article 192. When three terms of a proportion are given to find the fourth the proportion is simple. But when five terms are given to find the sixth, seven to find the eighth, &c., the proportion is said to be compound. Examples 5 and 6 are instances of compound proportion. 7th Example. Find a third proportional to the numbers 4, 6; and a mean proportional between the numbers 3, 27. Let r be a third proportional to the numbers 4, 6; And a mean proportional between the numbers 3, 27; 6×6 36 and 3: x':: x′ : 27.*.x22=3×27=81 and x = √/81=9. Hence, a third proportional to two given numbers is obtained by dividing the square of the mean term by the given extreme, and a mean proportional between two given extremes is obtained by extracting the square root of the product of the given extremes. 345. General rule for proportion. When three terms of a proportion are given to find the fourth, write as the third term that number which is of the same kind as the answer to the question; then, if it appears from the meaning of the question that the answer ought to be less than this third term, make the greater of the two remaining numbers the antecedent and the less the consequent of the first ratio; but if it appears that the answer ought to be greater than this third term, make the less of the two remaining numbers the antecedent and the greater the consequent of the first ratio. When five, seven, nine, .... terms of a proportion are given to find a sixth, an eighth, a tenth,... term, distinguish the number which is of the same kind as the answer, and separate the other given numbers into two, three, four, pairs of quantities, each pair consisting of like numbers. ... Compare the first pair of numbers with the number like the answer, and determine the antecedent and consequent of a proportion of which the two numbers of the first pair are the first and second terms and the number like the answer the third term. Determine, in the same manner, the antecedent and consequent of each pair of terms of the same kind. Arrange these ratios under each other in two vertical columns, the first containing all the antecedents, the second all the consequents, and write the number, which is like the answer, as their general third term. Reduce the single antecedent and consequent in simple proportion, and the antecedent and consequent of each pair of terms of the same kind in compound proportion, to the same denomination; reduce also the third term to the lowest denomination contained in it. Then multiply the third term by the consequent or product of the consequents, and divide the result by the antecedent or product of the antecedents; the number thus obtained is the fourth term of the proportion or unknown quantity. a. To find a third proportional to two given numbers, divide the square of the second number by the first number, the quotient is the third proportional required. And to find a mean proportional between two given numbers, multiply one given number by the other, and extract the square root of the product; this square root is the mean proportional required. 346. Exercises in the rules of proportion. 1st. Find a third proportional to the numbers 24 and 60 ? 6th. Find a fourth proportional to the numbers 12, 28, and 42? 10th. Find a mean proportional between 9 and 225?.......Ans. 45. 13th. What is the price of 96 yds. of cloth at the rate of 8 yds. 14th. A person's annual income being 146 £., how much is that per day? 15th. If the rent of 9 acres of land is 5£. 12 sh., what is the rent of 72 acres at the same rate ?...... Ans. 44 £. 16 sh. 16th. If 7 cwt. 1 qr. of sugar cost 26 £. 10 sh. 6 d., what is the price of 43 cwt. 2 qrs. ?................. .Ans. 159 £. 2 sh. 17th. How many workmen are required to execute in 15 days as much work as is done by 5 men of equal strength in 24 days?......... .......Ans. 8 men. 18th. If 3 paces make 2 yds, how many yards do 160 paces make ?.......... .......Ans. 1063 yards. 19th. What is the price of 6 bushels of oats at the rate of 1 £. 14 sh. 6 d. per chaldron of 36 bushels ? Ans. 5 sh. 9 d. 20th. What is the value of a bar of pure silver weighing 73 lb. 5 oz. 15 dwt., at the rate of 5 sh. 9 d. per ounce ? Ans. 253 £. 10 sh. 03 d. 21st. A garrison of 536 men have provisions for a year (365 days); how long will these provisions last, at the same rate for each man, if the garrison is increased to 1124 men?............ .......Ans. 174.16 days. 281 22d. What is the amount of tax upon 763 £. 15 sh. at the rate of 3 sh. 6 d. per pound sterling? Ans. 133 £. 13 sh. 13 d. 23d. What quantity of corn can be bought for 42 £. at the rate of 6 sh. per bushel ?........................................................ Ans. 39 qrs. 3 bush. 24th. A party of labourers working 4 hours daily complete a certain work in 12 days, what time ought the same party to take if they work 6 hours a day ?..........Ans. 8 days. 25th. What cost 20 pieces of lead, each weighing 1 cwt. 12 lb., at the rate of 16 sh. 4 d. per cwt.?......Ans. 18 £. 1 sh. 8 d. 26th. A plain of a certain extent supplied 3000 horses with forage for 20 days; for how many days ought the same plain to have supplied 2000 horses ?.........Ans. 27 days. 27th. The debts of a bankrupt amount to 977 £., and his property to 420 £. 6 sh. 31 d.; how much per pound can he pay to his creditors?.......... ..Ans. 8 sh. 74 d. 28th. Suppose that a gentleman's income is 525 £. a year, and that his average daily expenditure is 19 sh. 7 d.; how much does he save yearly?.........Ans. 167 £. 12 sh. 1 d. 29th. The governor of a besieged fortress has bread for 54 days, at the rate of 1 lbs. per day for each soldier; at what rate must he fix the daily allowance if he intends to hold out 81 days?.... .... Ans. 1 lb. 30th. If the penny loaf weighs 9 ounces when the bushel of wheat costs 6 sh. 3 d., what ought it to weigh when the price of wheat is 8 sh. 21 d. per bushel? Ans. 6 oz. 13137 dr. 31st. At 1£. 7 sh. 8 d. the acre; what is the rent of 173 ac. 2 ro. 14 poles of land?.............Ans. 240 £. 2 sh. 7 d. 32d. If 5 yds. of cloth cost 14 sh. 2 d., what must be given for 9 pieces, each measuring 21 yds. 1 qr.? Ans. 27 £. 1sh. 101d. 33d. If a gentleman's estate is worth 2107 £. 12 sh. a year, what may he spend per day in order to save 500 £. of his yearly income?...... ...Ans. 4 £. 8 sh. 1365 19 d. |