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MULTIPLICATION OF COMPOUND NUMBERS.
432. Multiplication of Compound Numbers is the process of finding the product when the multiplicand is a compound number.
1. Multiply £15 12 s. 10 d. by 9. SOLUTION.—We write the multiplier under the OPERATION. lowest denomination of the multiplicand, and begin
£ s. d. at the right to multiply. 9 times 10 d. are 90 d.,
15 12 10 which by reduction we find equals 7 s. and 6 d.; we
9 write the 6 d. under the pence, and reserve the 7 s. to add to the next product: 9 times 12 shillings are 108 140 6 shillings, plus the 7 s. equals 115 s., which by reduction we find equals £5 and 15 shillings; we write the 15 s. under the shillings and reserve the £5 to add to the next product: 9 times £15 are £135, plus the £5, equals £140, which we write under the pounds.
Rule.-I. Write the multiplier under the lowest denomination of the multiplicand.
II. Begin with the lowest denomination, and multiply each term in succession as in simple numbers, reducing as in addition of compound numbers.
Proof.—The same as in multiplication of simple numbers. NOTE.-If the multiplier is a large composite number, it will be more convenient to multiply by its factors.
8. A lumberman has 15 piles of wood, each containing 8 cd. 76 cu. ft., how much wood has he?
Ans. 128 cu. 116 cu. ft. 9. A man traveled 25 mi. 224 rd. 5 yd. in one day, 6 times as far the next, 8 times as far the next, and the next as far as the second and third; how much did he lack of traveling 1000 miles ?
Ans. 254 mi. 197 rd. 3. yd.
DIVISION OF COMPOUND NUMBERS. 433. Division of Compound Numbers is the process of finding the quotient when the dividend is a compound number. 434. There are two cases: 1st. To divide a compound number into equal parts. 2d. To divide one compound number by a similar one.
CASE I. 435. To divide a compound number into a number of equal parts.
1. Divide £107 11 s. 6 d. into 6 equal parts. SOLUTION.—We write the divisor at the left of the OPERATION. dividend, and begin at the highest denomination to divide. I of £107 equals £17 and £5 remaining ; 6)107 1. 6 £5 equals 100 s., which added to 11 s. equals 111 s. : of 111 s. equals 18 s. and 3 s. remaining; 3 s. equals
17 18 7 36 d. which added to 6 d. equals 42 d. : 1 of 42 d. is 7d. Hence the following
Rule.-I. Begin with the highest denomination of the dividend and divide each term in succession, as in simple numbers.
II. If there is a remainder, reduce it to the next lower denomination, add it to the term of that denomination, and divide the result as before.
III. Proceed in the same manner until all the terms are divided.
Proof.—The same as in division of simple numbers. NOTE.- When the divisor is large and composite, and the factors not greater than 12, it is perhaps more convenient to divide by the factors. (2)
(4) cwt. lb. oz. lb. oz. pwt. gr. mi. rd. yd. 7)24 98 9 5 )124 6 16 11 8)138 65 2
8. If a car could run 640 mi. 298 rd. 15 ft. in a day, what distance will it average an hour? Ans. 26 mi. 225 rd. 13 ft.
9. The earth revolves around the sun in about 365 da. 5 h. 48 min. 49.7 sec. ; in what time does it move 1 degree?
Ans. 1 da. 20 min. 5849% sec. 10. The moon revolves around the earth in 29 da. 12 h. 44 min. ; in what time does it move 6 degrees ?
Ans. 11 h. 48 min. 44 sec. 11. Venus performs her revolution around the sun in about 224 da. 16 h. 49 min. 10 sec.; in what time does she move 45 degrees ?
Ans. 28 da. 2 b. 6 min. 8 sec.
CASE II. 436. To divide one compound number by a similar one.
1. Divide £78 18 s. 6 d. by £9 11 s. 4 d. SOLUTION.—£78 18s. 6 d. equals
OPERATION. 18942 pence; £9 11 s. 4 d. equals
£78 18 s. 6d 18942 d. 2296 pence; and dividing 18912 d. by 2296 d. we have a quotient of 84.
£9 11 s. 4d. 2296 d. From this solution we have the fol
18942 Rule.-Reduce both dividend and divisor to the lowest denomination mentioned in either, and then divide as in simple numbers.
Proof.—The same as in division of simple numbers. NOTE.—The division may also be made without reducing to the lowest denomination, and this will be shorter when the quotient is integral.
2. How long will it take a student to walk 376 mi. 220 rd. at the rate of 17 mi. 300 rd. a day? Ans. 21 days.
3. A farmer raises 60 bu. 3 pk. 6qt. 1 pt. of grain on an acre; on how many can he raise 2925 bu. 3 pk.? Ans. 48 A. · 4. In how many hours will a pipe discharge 163 tuns 7 gal. of water, at the rate of 2 tu. 3 hhd. 40 gal. 2 qt. 1 pt. an hour ?
Ans. 56 hours. 6. How long would it take a bird to fly across the Atlantic ocean, 3000 mi., at the rate of 25 mi. 185 rd. 4 yd. an hour ?
Ans. 4 da. 21 h. 16 min. 38+ sec. 6 How long would it take a person to travel around the earth, at the average rate of 15 mi, 62 rd. 3 yd. 2 ft. in 4 h. 20 min. 30 sec.?
Ans. 296 da. 9 h. 36 min.
DIFFERENCE BETWEEN DATES.
437. To find the difference of time between two dates.
1. Sbakespeare was born April 23d, 1564, and died April 25th, 1616; what was his age?
SOLUTION.—Dates are expressed in the number of OPERATION. the year, the month, and the day; hence the date of Yr moda. his birth is 1564 yr. 4 mo, 23 da., and the date of his 1616 4 25 death is 1616 yr. 4 mo. 25 da. ; and the difference of 1564 7 23 these two dates will equal his age, which we find to : be 52 yr. 2 da.
52 0 2 Rule.- Write the number of the year, month, and day of the earlier date under the year, month, and day of the later date, and take the difference of the numbers.
NOTE.—In this method we reckon 30 days to the month ; when greater accuracy is required, we reckon the actual number of days in each month. The exact time between two dates is found by the table, Art. 406.
2. Milton was born Dec. 9th, 1608, and died Nov. 8th, 1675; what was his age ? Ans. 66 yr. 10 mo. 29 da.
3. Andrew Jackson was born Mar. 15th, 1767, and died June 8th, 1845 ; what was his age ?
Ans. 78 yr. 2 mo. 23 da. 4. Thomas was 16 yr. old May 25th, 1865 ; how old will he be Mar. 29th, 1873 ?
Ans. 23 yr. 10 mo. 4 da. 5. How many days from Feb. 12th, 1861, to Sept. 17th of the same year ?
Ans. 217 days. 6. A note was given Aug. 15th, 1860, and paid May 10th, 1865; how long was it on interest ?
Ans. 4 yr. 8 mo. 25 da. 7. A note is dated Jan. 16th, 1860, and is due Nov. 21st, 1860; what is the exact time it has to run ?
Ans. 310 days. 8. A was born Mar. 6tb, 1820; B, July 9th, 1833; both died Sept. 19th, 1865; how much older was A than B?
Ans. 13 yr. 4 mo. 3 da. 9. A was born Jan. 1st, 1741, and B Jan. 1st, 1584; each died exactly 45 years after he was born ; what was the difference of their ages ?
Ans. 1 day.
CASE II. 438. To find the day of the week upon which any given day of the month will fall, the day of the week of some other date being given.
NOTE.-A common year begins one day later than the preceding year. A year following leap year begins two days later.
1. If the 12th of March be on Sunday, on what day of the week will the next 20th of October be?
SOLUTION.-By the table we find the difference of OPERATION. time to be 222 days: dividing by 7, the number of 222 : 7 = 315+5 days in a week, we have 31 weeks and 5 days; the 20th of October must therefore be 5 days after Sunday, or on Friday.
Rule. Find the number of days between the two dates, reduce this number to weeks ; the number of days remaining will be the number of days from the given day of the week to the required day.
2. If the 1st of May is on Tuesday, on what day is the 8th of August of the same year?
Ans. Wednesday. 3. If a leap year begins on Friday, on what day will the 4th of July be ?
Ans. Monday. 4. In 1865 Christmas, the 25th of December, fell on Monday; on wbat day did the year commence? Ans. Sunday.
6. Christmas of 1863 came on Friday ; on what day did 4th of July, 1864, come ?
Ans. Monday. 6. The battle of Bunker Hill was fought on Saturday, June 17, 1775; and Gen. Warren's statue was erected June 17, 1857; on what day was it erected ? Ans. Wednesday.
7. Let the pupils now determine, from the above principles, the day of the week upon which they were born.
LATITUDE AND LONGITUDE. 439. The Latitude of a place is its distance from the equator, north or south. It is reckoned in degrees, minutes, and seconds, and cannot exceed 90°, or a quadrant.
440. The Longitude of a place is its distance, east or west, from a given meridian. It is reckoned in degrees, minutes, and seconds, and cannot exceed 180°, or a semicircumference.