SECTION II.. Rule of Three.

RULE OF THREE.

238. Suppose it required to find what 156 yards will cost, if 22 yards cost 17s. 4d. This quantity, reduced to pence, is 208d.; and if 22 yards cost 208d., each yard costs ²⁰⁸/₂₂d. But 156 yards cost 156 times the price of one yard, and therefore cost

208 208 × 156 ---- × 156 pence, or --------- pence (117). 22 22

Again, if 25½ French francs be 20 shillings sterling, how many francs are in £20. 15? Since 25½ francs are 20 shillings, twice the number of francs must be twice the number of shillings; that is, 51 francs are 40 shillings, and one shilling is the fortieth part of 51 francs, or ⁵¹/₄₀ francs. But £20 15s. contain 415 shillings (219); and since 1 shilling is ⁵¹/₄₀ francs, 415 shillings is

51 × 415 ⁵¹/₄₀ × 415 francs, or (117) -------- francs. 40

239. Such questions as the last two belong to the most extensive rule in Commercial Arithmetic, which is called the RULE OF THREE, because in it three quantities are given, and a fourth is required to be found. From both the preceding examples the following rule may be deduced, which the same reasoning will shew to apply to all similar cases.

It must be observed, that in these questions there are two quantities which are of the same sort, and a third of another sort, of which last the answer must be. Thus, in the first question there are 22 and 156 yards and 208 pence, and the thing required to be found is a number of pence. In the second question there are 20 and 415 shillings and 25½ francs, and what is to be found is a number of francs. Write the three quantities in a line, putting that one last which is the only one of its kind, and that one first which is connected with the last in the question. Put the third quantity in the middle. In the first question the quantities will be placed thus:

22 yds. 156 yds. 17s. 4d.

In the second question they will be placed thus:

20s. £20 15s. 25½ francs.

This generally comes in the same member of the sentence. In some cases the ingenuity of the student must be employed in detecting it. The reasoning of (238) is the best guide. The following may be very often applied. If it be evident that the answer must be less than the given quantity of its kind, multiply that given quantity by the less of the other two; if greater, by the greater. Thus, in the first question, 156 yards must cost more than 22; multiply, therefore, by 156.

Reduce the first and second quantities, if necessary, to quantities of the same denomination. Thus, in the second question, £20 15s. must be reduced to shillings (219). The third quantity may also be reduced to any other denomination, if convenient; or the first and third may be multiplied by any quantity we please, as was done in the second question; and, on looking at the answer in (238), and at (108), it will be seen that no change is made by that multiplication. Multiply the second and third quantities together, and divide by the first. The result is a quantity of the same sort as the third in the line, and is the answer required. Thus, to the first question the answer is (238)

208 × 156 17s. 4d. × 156 ----------pence, or, which is the same thing, -------------------. 22 22

240. The whole process in the first question is as follows:

yds. yds. s. d. 22 : 156 ∷ 17 . 4 12 --- 208 pence. 156 ---- 1248 1040 208 ----- 22)32448(1474¾d. and ¹⁴/₂₂, or ⁷/₁₁ of a farthing, 22 or (219) £6 . 2 . 10¾-⁷/₁₁. --- 104 88 ---- 164 154 ---- 108 88 -- 20 (228) 4 -- 80 66 -- 14

It is usual to place points, in the manner here shewn, between the quantities. Those who have read Section VIII. will see that the Rule of Three is no more than the process for finding the fourth term of a proportion from the other three.

The question might have been solved without reducing 17s. 4d. to pence, thus:

yds. yds. s. d. 22 : 156 ∷ 17 . 4 156 (227) ---------- 22)£135 . 4 . 0(£6 . 2 . 10¾-⁷/₁₁ (228) 132 --- 3 × 20 + 4 = 64 44 -- 20 × 12 = 240 220 --- 20 × 4 = 80 66 -- 14

The student must learn by practice which is the most convenient method for any particular case, as no rule can be given.

241. It may happen that the three given quantities are all of one denomination; nevertheless it will be found that two of them are of one, and the third of another sort. For example: What must an income of £400 pay towards an income-tax of 4s. 6d. in the pound? Here the three given quantities are, £400, 4s. 6d., and £1, which are all of the same species, viz. money. Nevertheless, the first and third are income; the second is a tax, and the answer is also a tax; and therefore, by (152), the quantities must be placed thus:

£1 : £400 ∷ 4s. 6d.

242. The following exercises either depend directly upon this rule, or can be shewn to do so by a little consideration. There are many questions of the sort, which will require some exercise of ingenuity before the method of applying the rule can be found.

EXERCISES.

If 15 cwt. 2 qrs. cost £198. 15. 4, what does 1 qr. 22 lbs. cost?

Answer, £5 . 14 . 5 ¾ ¹⁸⁵/₂₁₇.

If a horse go 14 m. 3 fur. 27 yds. in 3ʰ 26ᵐ 12ˢ, how long will he be in going 23 miles?

Answer, 5ʰ 29ᵐ 34ˢ(²⁴⁶²/₂₅₃₂₇).

Two persons, A and B, are bankrupts, and owe exactly the same sum; A can pay 15s. 4½d. in the pound, and B only 7s. (6¾)d. At the same time A has in his possession £1304. 17 more than B; what do the debts of each amount to?

Answer, £3340 . 8 . 3 ¾ ⁹/₂₅.

For every (12½) acres which one country contains, a second contains (56¼). The second country contains 17,300 square miles. How much does the first contain? Again, for every 3 people in the first, there are 5 in the second; and there are in the first 27 people on every 20 acres. How many are there in each country?--Answer, The number of square miles in the first is 3844⁴/₉, and its population 3,321,600; and the population of the second is 5,536,000.

If (42½) yds. of cloth, 18 in. wide, cost £59. 14. 2, how much will (118¼) yds. cost, if the width be 1 yd.?

Answer, £332. 5. (2⁴/₁₇).

If £9. 3. 6 last six weeks, how long will £100 last?

Answer, (65¹⁴⁵/₃₆₇) weeks.

How much sugar, worth (9¾d). a pound, must be given for 2 cwt. of tea, worth 10d. an ounce?

Answer, 32 cwt. 3 qrs. 7 lbs. ³⁵/₃₉.

243. Suppose the following question asked: How long will it take 15 men to do that which 45 men can finish in 10 days? It is evident that one man would take 45 × 10, or 450 days, to do the same thing, and that 15 men would do it in one-fifteenth part of the time which it employs one man, that is, in (450 ÷ 15) or 30 days. By this and similar reasoning the following questions can be solved.

EXERCISES.

If 15 oxen eat an acre of grass in 12 days, how long will it take 26 oxen to eat 14 acres? Answer, (96¹²/₁₃) days.

If 22 masons build a wall 5 feet high in 6 days, how long will it take 43 masons to build 10 feet? Answer, (6⁶/₄₃) days.

244. The questions in the preceding article form part of a more general class of questions, whose solution is called the DOUBLE RULE OF THREE, but which might, with more correctness, be called the Rule of Five, since five quantities are given, and a sixth is to be found. The following is an example: If 5 men can make 30 yards of cloth in 3 days, how long will it take 4 men to make 68 yards? The first thing to be done is to find out, from the first part of the question, the time it will take one man to make one yard. Now, since one man, in 3 days, will do the fifth part of what 5 men can do, he will in 3 days make ³⁰/₅, or 6 yards. He will, therefore, make one yard in ³/₆6 or (3 × 5)/30 of a day. From this we are to find how long it will take 4 men to make 68 yards. Since one man makes a yard in

3 × 5 3 × 5 ----- of a day, he will make 68 yards in ----- × 68 days, 30 30

3 × 5 × 68 or (116) in ---------- days; and 4 men will do this in one-fourth 30

3 × 5 × 68 of the time, that is (123), in ---------- days, or in 8½ days. 30 × 4

Again, suppose the question to be: If 5 men can make 30 yards in 3 days, how much can 6 men do in 12 days? Here we must first find the quantity one man can do in one day, which appears, on reasoning similar to that in the last example, to be 30/(3 × 5) yards. Hence, 6 men, in one day, will make

6 × 30 12 × 6 × 30 ------ yards, and in 12 days will make ----------- or 144 yards. 5 × 3 5 × 3

From these examples the following rule may be drawn. Write the given quantities in two lines, keeping quantities of the same sort under one another, and those which are connected with each other, in the same line. In the two examples above given, the quantities must be written thus:

SECOND EXAMPLE.

Draw a curve through the middle of each line, and the extremities of the other. There will be three quantities on one curve and two on the other. Divide the product of the three by the product of the two, and the quotient is the answer to the question.

If necessary, the quantities in each line must be reduced to more simple denominations (219), as was done in the common Rule of Three (238).

EXERCISES.

If 6 horses can, in 2 days, plough 17 acres, how many acres will 93 horses plough in 4½ days?

Answer, 592⅞.

If 20 men, in 3¼ days, can dig 7 rectangular fields, the sides of each of which are 40 and 50 yards, how long will 37 men be in digging 53 fields, the sides of each of which are 90 and 125½ yards?

2451 Answer, 75----- days. 20720

If the carriage of 60 cwt. through 20 miles cost £14 10s., what weight ought to be carried 30 miles for £5. 8. 9?

Answer, 15 cwt.

If £100 gain £5 in a year, how much will £850 gain in 3 years and 8 months?

Answer, £155. 16. 8.