Secondly, when the quotient is of the same kind. Ex. The wages of a man for 23 yrs. 5 mo. 10 days are £224. 18s. 8d. What is the yearly amount? Here, by the question, we are led to divide £224. 18s. 8d. by 23 yrs. 5 mo. 10 days. But 23 yrs. 5 mo. 10 days=8440 days, or or 440 of 1 year. Therefore, the question 8440 12 × 30 360 360 resolves itself into dividing £224. 18s. 8d. by 8440, which operation is performed as shown in (§ 257). 8440 8440)80976 0 0(£9. 11s. 1014d. 75960 5016 20 100320 8440 15920 8440 7480 12 89760 8440 5360 [Note. The pupil should never neglect to shorten his labour, when possible. It will have been noticed that =211, and the work consists in multiplying the dividend by 9, and dividing the product by 211.] 261. MISCELLANEOUS EXERCISES. 1. In 3 hrs. 25 min. a ropemaker made a rope 76 yds. 2 ft. 9 in. in length. How much of it was made per hour? 2. If 2 cwt. 3 qrs. 1 lb. cost £150. 13s. 10d., how much does 1 lb. cost? 3. Bought 14 yards of velvet for £19. 8s. 8d. What was the cost per yard? 4. Twelve men and a boy have earned £45. 17s. 10d.; the boy is to receive 9 half-crowns. What is each man's share? 5. £300 was paid for 200000 bricks. score? How much is that per 6. If 112 ingots of gold are worth £77878. 5s. 4d., what is the value of 1 ingot ? 7. If 150 reams of paper cost £75, what is that per sheet? 8. Divide £10416. 13s. 4d. among one million of persons. What will each receive? 9. The daily wages of a man are 3s. 6d. In how many days will he earn £3. 19s. 6d. ? 10. The length of the equatorial circumference is 24899 miles. It is divided into 360°, like every circle. 1st, find the length of 1o, and also of l'in yards. 2nd, since the earth rotates in 24 hours, at what speed per hour are objects on the equator carried by that motion ? 11. The wheel of an engine makes 1000000 revolutions in 7 hours. What is its rate, in degrees, &c., per hour? 12. If in 1 minute every part of the circumference of a wheel moves 3° 20', in what time will 100° be described? 13. The circumference of a wheel is 4 yards. What is the length of 1°, and also of 35° 20' ? 14. A schoolmaster receives £8. 7s. 6d. weekly, and each pupil pays 2s. 6d. weekly. How many pupils has he? 15. A mother who was asked the age of her daughter, replied, I am 34 yrs. 7 mo. 8 days 9 hrs. old; my husband's age is 31 yrs. 9 days. Now, if from one-half the sum of our ages 19 yrs. 11 mo. 29 days 13 hrs. be subtracted, the remainder is my daughter's age. Find her age. 16. A merchant sold 13 hhds. 56 gals. of brandy, 9 hhds. 54 gals. of gin, and 10 hhds. 36 gals. of rum in 24 days. How much did he sell each day? 17. A railway carriage went over 346 miles in 14 hrs. 34 min. 45 sec. In what time was each mile performed? 18. At the Great Exhibition, besides other refreshments, there were consumed 870000 plain buns, at 1d. each; 930000 Bath buns, at 2d. each; and 1090000 bottles of ginger beer, &c., at 4d. per bottle. How much money was received for them? 19. If a steam packet sailed 1020 miles in 4 days 18 hrs. 33 min. 36 sec., what was the average rate per hour, and also in what time was each mile performed? 20. Divide £550. 3s. 14d. among 4 men, 6 women, and 8 children, giving to each man double a woman's share, and to each woman triple a child's. 21. In common marching, soldiers take 75 steps a minute; in quick marching, 108. How far would a regiment advance in 8 hours, going 5 hours common, and 3 hours quick marching, reckoning 2 ft. 10 in. for each step? 22. At the Great Exhibition, there were, on a shilling day, 76473 persons admitted; on the next day, when the admission was 2s. 6d., the number amounted to 39582. On which day was the most money received, and how much? 23. A draper bought several pieces of cloth, at the rate of £10. 2s. 9 d. for 5 yards, and sold them again at the rate of £20. 18s. 94d. for 9 yards. On the whole, he gained £7. 12s. 6d. How many yards were there? REDUCTION OF CONCRETE QUANTITIES AS FRACTIONS OF OTHERS. 262. All that has been said upon fractions relates to generalities; they have been considered more as parts of abstract units. We shall now treat more particularly of the application of fractions to concrete quantities. Our first case shall be to express a given quantity in terms of or as the fraction of another given quantity. Ex. Express 5s. 6d. as the fraction of £1. As there are 240 pence in £1, therefore 1d. is But 5s. 6d. 66d., hence 66d. are 66 × of £1, or 1 of £1. : of £1. of £1., or 240 From which we infer this law reducing both proposed quantities to the same denomination, the result of the first quantity is the numerator, and the result of the other quantity the denominator of the fraction required. Ex. What fraction of a ton is 2 cwt. 2 qrs. 14 lbs. ? 29 2240 therefore 2 cwt. 2 qrs. 14 lbs., or 294 lbs. are 2 of 1 ton, or 2 of 1 ton. 16 Or thus since 14 lbs. are To of 1 ton, therefore 2 cwt. 2 qrs. 14 lbs., or 21 × 14 lbs. are of 1 ton. Ex. Reduce £3. 14s. 6d. to the fraction of £5. 14s. 9d. and £5. 14s. 9d.=2754 halfpence. therefore 1789 halfpence are 1782 of £5. 14s. 9d. Ex. Express 1 cwt. 6 lbs. as the fraction of 12 lbs. therefore 1 cwt. 6 lbs., or 118 lbs., are 13, or 59 of 12 lbs. 263. EXERCISES. 1. Reduce 11s. 6d. to the fraction of £1. 2. Reduce £1. 10s. 6d. to the fraction of 6s. 91d. 3. Express 16 hrs. 45 min. 45 sec. as the fraction of 1 day. 4. Express 11 oz. 6 drs. as the fraction of 1 lb. 5. What fraction of £1. 16s. 8ğd. are 11 crowns 2s. 2d.? 6. What fraction of 6 ac. 2 ro. are 3 ac. 0 ro. 28 pl.? 7. Express 6 ft. 611 in. in terms of 3 yds. 9 in. 8. Express 16 lbs. in terms of 3 qrs. 18 lbs. 9. Reduce 3 wks. 5 days 14 hrs. 24 min. to the fraction of a lunar month. 10. Express 3 hhds. 38 gals. 1 pt. in terms of 1 ton. 11. Express 8 quires 18 sheets in terms of 1 ream. 264. Secondly, we shall determine a method of expressing a fraction of one given quantity as the fraction of another Ex. Express s. in terms of £1. then s.10, and s.1, or 24: From which it follows that we must multiply the given fraction by that fraction which shows what part the lower denomination is of the higher. Ex. Express of 1 guinea as the fraction of 10s. |