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: [Illustration] SECOND EXAMPLE. [Illustration] 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 10_s._, what weight ought to be carried 30 miles for £5. 8. 9? _Answer_, 15 cwt.