SECOND EXAMPLE WORKED OUT. AN INVERSE PROPORTION. NO REDUCTION OR CANCELLING. In what time will 24 men do as much work as 13 men do in 11. days? Example completed. I. BY PROPORTION. Abbreviated Question. If 13. men · ... 24. men Statement and Work. 11. days. Here are two terms workers (13 men and 24 men). The answer will be time. Put time (11 days) in the third term. Will the answer be more or less than the third term? An increased No. of workers will complete the task in a shorter time. For Time and No. of workers are in Inverse Proportion. See page 309 VIII. Therefore, 24′ men will do the work in a shorter time than 13. men. The answer will be less than the third term; therefore The second term must be less than the first. Put 13 (mén) in second place, and 24· (men) in first. 24. d. d. 8. men. 11 X 13 1 4 3 X= == 24 24 3.) 177 = time for Ans. 5 d. 11 ho., taking a working day as 12 hours. II. The same Example worked without reference to ratio. If we proceed in obedience to the general rule given on page 312, the resulting work will be precisely similar to the foregoing. For, in every Inverse Proportion Example, the value of one is brought out by multiplying second and third terms together, which thus completes the first step of the general rule for working without reference to ratio. Without recourse, then, to Proportion, the question would be reasoned out thus: Time for 13 men = 11' days. But, since each of the 13 men works 11 days, one man, to do the whole, must work 13 times 11 days. .. Time for 1· man = 11 Now, 24 men, working together, one man does in 24 days. .. Time for 24∙ men = = = days × 13· = 143⋅ days. will do as much in one day as of time for one man. 522 days 5 d. 11 h. III. = Or, reversing the order of the Multiplication and Division, (PRINCIPLE XVIII.) the question may be solved and explained thus: Time for 13⋅ men = 11. days. And 24 times as many men would do the work in of the time. .. Time for (24 times 13 or) 312. men = 2 of 13·d. 5th. But, 24 men, being of 312′ men, will require 13 times as long. .. Time for 24 men = 51 ho. × 13⋅ = 5⋅d. 111⁄2 ho. .. Answer x 24. - 11∙ day's × 13. ... Dividing both sides by 24 we have (PRINCIPLE XXIV.) Answer, or time for 24 men, = (11·d. × 13·) ÷ 24 THIRD EXAMPLE WORKED OUT. REDUCTION AND CANCELLING IN FIRST TWO TERMS. What will 12 cwt. 15 lb. cost, at the rate of £5 4's. 02d. for 13 cwt. 3. qr. 14. lb.? Statement and Work. cwt. qr. lb. £. 8. d. cwt. qr. lb. : 12 0 15 :: 5 4 0 (c. of 13. 3. 14.): x. 7. 8)36 851 c. of 97 0 14. 411 0c. of 12. 0·152 II. Or thus: reducing also third term, and abstaining from Cancelling. This is the unwieldy method until recently recommended for every example in "The Rule of Three" when any of the terms are compound quantities. If the long Division, which we have suppressed, were written out at full, the work would occupy twenty-two lines under third term, instead of three, which, as is shewn above, are sufficient. The third term should never be "reduced to its lowest name" unless the multiplier (2nd term) be too large for convenient Compound Multiplication. Page 311 6. IMPLICIT CHANGES OF UNIT. The 6216, being employed to divide the farthings, in last Example, are implicitly changed into farthings, because farthings can only be divided by (that is, distributed into lots of) farthings. (PRINCIPLE XIV.) Wishing to ascertain the 21th part of 27167805 far., we proceed to find how many lots of 6216 far. are contained in 27167805 far. Now, 27167805. fars. 4370g lots of 6216 fars. each. = = 6216 lots of 4370g fars. each. •*. 6216 of 27167805· fars. = 4370 fars. See p. 218. In the work under third term of this example there are, also, other six implicit changes of unit, which the pupil will do well to discover and note. III. The same example worked out without reference to ratio. Or thus: by Rule of Three in Vulgar Fractions. Or thus: By Rule of Three in Decimals. cwt. qr. lb. As 13 3 14. cwt. Or, As 8: 7 cwt. : 12. 0.15 :: 5. 4. 02 cwt. £. Or, As 13.875: 12-140625 :: 5·203125(c. of 13·875): x. .. I = £4 11's. Od. §q. Although not in this one, Decimals, in many examples, shorten the work. The quantities being first expressed as Vulgar Fractions, attention to Exercise LXXI. will render it easy to decide when the employment of Decimals will be possible or advantageous. EXERCISE CXVII. EXAMPLES IN THE RULE OF THREE, OR SIMPLE PROPORTION, DIRECT AND INVERSE; IN INTEGERS, IN VULGAR FRACTIONS, AND IN DECIMALS. Work every example in two ways: I. By Simple Proportion, or Rule of Three. II. By Multiplication and Division, without reference to Ratio. Explain every line. Reduce as little as possible. Cancel as much as possible. A. THE RULE OF THREE IN INTEGERS. DIRECT PROPORTIONS. 1. If 3 articles cost 3's. 64d., what will 11 cost, at the same rate? 2. When 9 cost £1. 11-s. 103d. what will 7 cost? 3. What is the value of 10', of which 6 cost £2. 10's. 5d.? 4. Find weight of 5′ when 6. weigh 89-lb. 11 oz. 5'dwt. 6'gr.? 5. The weight of 2. being 18 lb. 7.3 5.3 2. 10 gr. find weight of 9.? 6. If 4 men build 91⋅yd. 2 ft. 2 in. of hedging, what length will 7. build? 7. Supposing 82'yd. 1'qr. 3′na. 1'in. of cloth, to make 5 dresses, how much will make 7.? 8. If 8. men can mow 42'ac. 2′r. 36′po. 28′yd., what area will 11· men mow? |