265. EXERCISES. 1. Reduce of a quarter of a guinea to the fraction of £1. 2. Reduce 5% of a half-crown to the fraction of 16s. 3. Express 4. Express of £1 as the fraction of £2. 10s. of £2, 4s. 6d. as the fraction of £10. 5. Find the fraction of 1 cwt. which expresses 1 oz. 6. Find the fraction of 1 day which expresses % of a week. 7. What is the fraction of a mile which expresses of 648 yards? 8. Express of of 16 lbs. as the fraction of 1 ton; and 4 of 3% of a square inch as the fraction of a square yard. 9. Reduce of 3960 seconds to the fraction of 1 week. 61 912 10. Find the sum and difference of % of £1, and 3 of 1 guinea. therefore the sum = 16s. 8d.+ 11s. 8d.= £1. 8s. 4d. 11. 12. 3 Therefore the sum=+7=£}}=£1. 8s. 4d. hour+week+ day, to be expressed in positive terms. of £3. 16s.+ of 1⁄2 of £1. 15s. 3ąd.—7 of £1. 6s. 84d. to be expressed in positive terms. 15. Express the result of of 1 ton 4 cwt.-3 of 1 cwt. 3 qrs. 9 lbs. of 1 ton 1 cwt. 1 qr. 1 lb. 16. Express 31 of £5. 18s. 8d.+ of £1. 2s. 6d. - of £6. 14s. 74d. as the fraction of £10. 10s. 10d. 17. Reduce & of 5 yds. 2 ft. 9 in.-2 of 4 of 4 yds. 1 ft. 6 in. to the fraction of 12 yds. 64 266. MISCELLANEOUS EXERCISES IN FRACTIONS. 1. The hands of a watch are together at twelve o'clock. When will that occur again for the first time? Since the minute hand moves round the dial in 1 hour, or 60 minutes, whilst the hour hands moves the 5 minutes' distance, then the minute hand gains 55 min. in 60 minutes' time over the hour hand. But after one hour, or at one o'clock, the hour hand points at one, and the minute hand at twelve; then the minute hand has 5 minutes' distance to gain to overtake the hour hand, Thus they will be again over each other in 1 hr. 5 min. 27 sec. 2. Both hands of a watch are together between six and seven o'clock. The exact time is required. At six o'clock, the minute hand was 30 minutes' distance from the hour hand, then the minute hand had 30 minutes to gain to overtake the hour hand. But 55 min. are gained in 60 min. time, Therefore the time is 32 min. 437 sec. after six o'clock. 3. A watch, which loses 4 minutes in a day, is set right at 12 o'clock on May 10th. What will be the true time on May 17th, when the hands of that watch point to 12 o'clock ? In 7 days the watch loses 7x4 min.=28 min. But the day has 24 × 60 or 1440 minutes, and since it loses 4 minutes, the minute hand moves only 1440-4=1436 minutes; therefore, 1436 min. correspond to 1440 min. 28 min. Thus, on May 17th, when it true time will be 28 min. 434 1448-368. 28 × 360 359 =28 min. 433 sec. is 12 o'clock by that watch, the 4. A reservoir is filled by one pipe in 6 hours, by a second in and second 1 and third .. they fill together in 1 hour +&+&=}}= of reservoir. Now are filled in 1 hour, and, or the whole, is filled in, or 12 hours. 5. Three men, A, B, and C, perform a work in 24 days; A and B can perform it in 32 days, and A and C in 36 days. What part of the work can B and C perform in 18 days? :: A+B+C do in 1 day of the work, 6. A general, detaching of his army to occupy a certain height, and of the remainder to watch the enemy's motions, had only 700 men left. Required, the whole number of his troops. Let the whole army be represented by 1, the unit, and we have of the army on the height, and there remains 1of, or of the army watching the enemy; then soldiers remaining with general=-= of the army. Therefore, of the army amount to 700 men, and T 77-49 121 700 or 25 men; 28 therefore, the whole, or 1,121 × 25, or 3025 men. 28 ΙΣΤ 7. A man, carrying peaches, sold one-half of them to A, one quarter of the remainder to B, and had 6 left. How many had he at first? Let what he had be represented by the unit, or 1; having sold to A, he had left of his fruit; to B he sold of, or of the whole. .. he sold +1, or §. So he had left 1—§, or 3. ..of his fruit=6, and of his fruit=§=2; .. the whole, or 1=8×2=16 peaches. 8. I am the owner of § of a house. I sell of % of my share for £825. 10s. 5d. What is the value of the house? The part sold is of % of 3, or 2 of the house; then 2 of the house are worth £825. 10s. 5d. and of the house is worth £825. 10s. 5d. 25 or £33. Os. 5d., The house, or 2, is worth 72 × £33. Os. 5d., or £2477. 10s. Od. 9. A watch, which goes 3 minutes too fast every day, is set right at 12 o'clock. What will be the exact time when the hands of the watch point at 12 minutes after 7 ? 10. Both hands of a watch point at twelve. It is required to ascertain how many times they will be together from noon to midnight, and at what time will each meeting take place? 11. If of a ship cost £68249. 18s. 5 d., what must of her be sold for to gain £116. 14s. 6d. ? 12. A does a work in 4 days, B in 5 days, C in 33 days, and D in 4 days. How much will each man do in 1 day, and how long will they be performing the work together? 13. I spent first of the contents of my purse, then % of the remainder; now of what I have left amounts to £1. 11s. 4 d. How much had I at first? 14. Find the sum, the difference, the product, and the quotient of of £16. 18s. 10d., and of £2. 17s. 4d. 15. Divide the sum of 4, 5, 64 by the difference of 8 and 53, and multiply the quotient by the sum of 16 and 11§. 16. Add together of a guinea, £, s., 3d., cr., moi., and half-cr. 17. If of a ship be worth £496. 17s. 24d., what part of her is worth £649. 17s. 6d. ? 18. In a plantation, of the trees are oak, chesnut, larch, Scotch fir, and the remaining 497 are beech trees. How many trees are there in the plantation? 19. If of a lottery ticket cost £1. 10s., what is the price of of a ticket? 31 51 20. A horse and saddle are worth £75. The saddle is of the value of the horse. Find the worth of each. 21. If 10 men or 12 women perform a piece of work in 24 days, in what time can 8 men and 8 women do the same? 22. If A accomplishes a piece of work in 12 days, B three times as much in 16 days, and C four times as much in 25 days, in what time can A, B, and C together do five times the same quantity of work? 23. A piece of linen, containing 25 yards, weighs 12 lbs. What is the weight of 1 yard, and how many yards are there in 1 lb. weight? 24. A pedestrian left Worksop for London on the 1st of January, at 8 o'clock in the morning. His rate of travelling, including stoppages, was 53 miles in 24 hours. Another pedestrian left London for Worksop on the 2nd of January, at 1 o'clock in the morning, and his rate of travelling, including stoppages, was 1 miles in hour. The distance between London and Worksop is 144 miles. It is required to answer the following questions :— 1st, what is the rate per hour of the first man? 2nd, in what time does he walk 1 mile ? 3rd, on what day, and at what time, did he reach London ? 4th, how far had he gone when the other man left town? 5th, at what rate per hour did the second man travel? 6th, in what time does he walk 1 mile? |