CIRCULAR MEASURE. (ART. 159.) 95. In 75 degrees, how many seconds? 96. In 8 signs, and 15 degrees, how many minutes? 97. In 12 signs, how many seconds? 98. In 86860 seconds, how many degrees? 99. In 567800 minutes, how many signs? 100. In 25000000 seconds, how many signs? COMPOUND NUMBERS REDUCED TO FRACTIONS. Ex. 1. Change 7s. 6d. to the fraction of a pound. Reducing the 7s. 6d. to pence, (Art. 161, I,) we Numerator 90d. Denominator 240d. have 90d., which Ans. £, or £3. is the numerator of the fraction. Then reducing £1 to the same denomination as the numerator, we have 240d., which is the denominator. Consequently is the fraction required. But may be reduced to lower terms. Thus , or (Art. 120.) Hence, 165. To reduce a compound number to a common fraction. First reduce the given compound number to the lowest denomination mentioned for the numerator; then reduce a UNIT of the denomination of the required fraction to the same denomination as the numerator, and the result will be the denominator. (Art. 161.) OBS. When the given number contains but one denomination, it of course requires no reduction. 2. Reduce 3s. 7d. 2 far. to the fraction of £1. 3. Reduce 9d. 3 far. to the fraction of 1s. 4. What part of a bushel is 3 pecks and 5 qts.? QUEST.-165. How is a compound number reduced to a common fraction? 5. What part of a peck is 5 qts. and 1 pt. ? 8. What part of 1 hogshead is 15 gals. and 3 qts.? 10. What part of 1 hundred weight is 2 qrs. and 7 lbs. ? 13. What part of 1 mile is 10 fur. and 35 rods? 17. What part of £1 is of a penny? numerator. Ans. £ Note. The lowest denomination mentioned in this example, is thirds of a penny. Hence, £1 must be reduced to thirds of a penny for the denominator, and 2, the given number of thirds will be the Ans. £, or £zbu. 18. What part of £1 is 54 shillings? Ans. £23. 19. What part of 1 day is 24 hours? 20. What part of 1 21. What part of 1 day is 4 h. and 8 min. ? hour is 3 min. and 40 sec.? 22. What part of 1 hour is 15 sec.? 23. What part of 1 pound is of an ounce ? FRACTIONAL COMPOUND NUMBERS. REDUCED TO WHOLE NUMBERS OF DIFFERENT DENOMINATIONS. Ex. 1. Reduce of £1 to shillings and pence. 166. To reduce fractional compound numbers to whole numbers. First reduce the given numerator to the next lower denomination; (Art. 161, I;) then divide the product by the denominator, and the quotient will be an integer of the next lower denomination. Proceed in like manner with the remainder, and the several quotients will be the whole numbers required. 2. Reduce of £1 to shillings. 3. How many shillings and pence in £? 4. How many shillings, &c., in £4 ? 7. Change Ans. 12s. 5. In of 1 week, how many days, hours, &c. ? 6. In of 1 day, how many hours, minutes, &c.? of 1 league to miles, &c. 8. Change of 1 mile to furlongs, &c. 9. Reduce of 1 hundred weight to quarters, &c. 10. In of 1 ton, how many hundred weight, &c.? 11. In of 1 bushel, how many pecks, quarts, &c.? 12. In of 1 peck, how many quarts, &c.? 13. Reduce of £1 to shillings. Suggestion. Since the numerator, when reduced to the denomination required, cannot be divided by the denominator, the division must be represented. Ans. s. Note. This, in effect, is reducing of £1 to the fraction of a shilling. 14. Reduce 5 of £1 to pence. Ans. gd. 167. From the last two examples it is manifest, that a fraction of a higher denomination may be changed to a fraction of a lower denomination, by reducing the given numerator to the denomination of the required fraction, and placing the result over the given denominator. QUEST.-166. How are fractional compound numbers reduced to whole ones? 167. How is a fraction of a higher denomination changed to a fraction of a lower denomination? of £1 to the fraction of a shilling. Ans. 178. 15. Reduce 16. Reduce of 1 week to the fraction of a day. 17. Changes of 1 mile to the fraction of a rod. 18. Change of 1 rod to the fraction of a foot. 19. Change 187 of 1 yard to the fraction of a nail. 20. Change of 1 ton to the fraction of a pound. ADDITION OF COMPOUND NUMBERS. 1. What is the sum of £4, 9s. 6d. 2 far.; £3, 12s. 8d. 3 far.; and £8, 6s. 9d. 1 far. ? Operation. £ S. d. far. " " " 4 9 6 2 3" 12 " 8 " " 3 8 6 9 1 " 16" 9 0 Having placed the farthings under farthings, pence under pence, &c., we add the column of farthings together, as in simple addition, and find the sum is 6, which is equal to 1d. and 2 far. over. Set the 2 far. under the column of farthings, and carry the 1d. to the column of pence. The sum of the pence is 24, which is equal to 2s. and nothing over. Place a cipher under the column of pence, and carry the 2s. to the column of shillings. The sum of the shillings is 29, which is equal to £1 and 9s. over. Write the 9s. under the column of shillings, and carry the £1 to the column of pounds. sum of the pounds is 16, the whole of which is set down in the same manner, as the left hand column in simple addition. (Art. 25.) The answer is £16, 9s. Od. 2 far. 168. Hence, we derive the following general RULE FOR ADDING COMPOUND NUMBERS. The 1. Write the numbers so that the same denominations shall stand under each other. QUEST.-168. How do you write compound numbers for addition? Which denomination do you add first? When the sum of any column is found, what is to be done with it? II. Beginning with the lowest denomination, find the sum of each column separately, and divide it by that number which it requires of the column added, to make ONE of the next higher denomination. Set the remainder under the column, and carry the quotient to the next column. III. Proceed in this manner with all the other denominations except the highest, whose entire sum is set down as in simple addition. (Art. 29.) PROOF.-The proof is the same as in Simple Addition. (Art. 28.) OBS. The process of adding numbers of different denominations, is called Compound Addition. It is the same as Simple Addition, except in the method of carrying from one denomination to another. 2. What is the sum of £10, 6s. 7d.; £18, 12s. 10d.; £5, 3s. 4d.? 3. Ans. £34, 2s. 9d. 5. 4. 16 11 0 1 12 9. Add 7 lbs. 9 oz. 16 pwts. 10 grs.; 3 lbs. 10 oz. 8 pwts. 9 grs.; 8 lbs. 3 oz. 1 pwt. 4 grs. 10. An Englishman bought a carriage for £35, 12s. ; a horse for £27, 8s. 10d.; a harness for £7, 16s. 11d.; how much did he give for the whole? QUEST.-What is done with the last column? How is the operation proved? Obs. What is the process of adding compound numbers called? Does it differ from simple addition? |