12. In 7T. 14cwt. 1gr. 20lbs., Avoirdupois, how many pounds? 13. In 15445lb., Avoidupois, how many tons, cwts., qrs., and lbs. 14. How many grains of silver in 4lb., 6oz., 12dwt. and 7grs.? 15. How many pounds, ounces, pennyweights, and grains of gold, in 704121 grains? 16. In 5, 13, 15, 19, 2gr., Apothecaries' weight, how many grains? 17. In 174947 grains, how many pounds, ounces, drachms, scruples and grains? 18. In 6 yards 2 feet 9 inches, how many inches? 19. In 5 miles, how many rods, yards, feet and inches? 20. In 2730 inches, how many yards feet and inches? 21. In 56 square feet, how many square yards? 22. In 355 perches, or square rods, how many acres, roods and perches? 23. In 456 square chains, how many acres ? 24. In 3A., 2R., 8P., how many perches? 25. In 14 tons of round timber, how many cubic inches? 26. In 31 cords of wood, how many cubic feet? 27. In 56320 cubic feet, how many cords? 28. In 157 yards of cloth, how many nails? 29. In 192 Ells Flem., how many yards? 30. 97yd., 3qr., how many Ells English? 31. In 4hhd. Wine measure, how many quarts? 32. In 7560 pints, Wine measure, how many hogsheads? 33. In 7 hogsheads of ale, how many pints? 34. In 74304 half pints of ale, how many barrels ? 35. In 31 bushels, Dry measure, how many pints? 36. In 2110 pints, Dry measure, how many bushels? 37. In 2 years of 365d. 5h. 48m. 48sec., each, how many seconds? 38. How many months, weeks and days in 254 days, reckoning the month at 30 days? ADDITION. 42. THE SUM of two or more numbers is a number containing as many units as all the numbers taken together. ADDITION is the operation of finding the sum of two or more numbers. 1. What is the sum of 769 and 437 ANALYSIS.-Write the numbers thus: draw a line beneath them, sum of the units, sum of the tens, OPERATION. 769 487 16 14 11 12696 The example may be done in another way, thus: Set down the number as before then say, 7 and 9 are 16 set down 6 in the units place, and the 1 ten under the 8 in the column of tens. Then say, 1 to 8 are 9, and 6 are 15. Set down the 5 in the column of tens, and the 1 hundred in the column of hundreds. We then add the hundreds, and find their sum to be 12: hence, the entire sum of 1256. OPERATION. 769 487 116 1256 NOTE 1.-Observe, that units of the same value are always written in the same column, since every collection must contain units of the same kind. 2. When the sum in any column, exceeds 9, it produces a unit of a higher order, which belongs to the next column at the left. In that case, write down the excess over tens, and add the tens to the next column. This is called carrying to the next column. The number to be carried, should not, in practice, be written under the column at the left, but added mentally. Beginners, however, had better set down the numbers to be carried, What is Addition? 42. What is the sum of two or more numbers? How are numbers written down for Addition? What do you do in simple numbers when the sum of any column exceeds 9 ? What is the general rule for the addition of numbers ? What is this called? each under its proper column, as in the examples below. 5. What is the sum of 35 dollars 4 dimes 6 cents 5 mills, 4 dollars 7 mills, and 97 cents 3 mills? NOTE. Write the units of the same value in the same column, separating the dollars from the cents and mills by a comma (Art. 40): then add the columns as in simple numbers. 6. Let it be required to find the sum of £14 7s. 8d. 3far., and £6 18s. 9d. 2far. £ OPERATION. $35,465 4,007 ,973 $40,445 111 OPERATION. s. d. far. 78 3 14 ANALYSIS.-Write the numbers, as before, so that units of the same value shall fall in the same column. Beginning with the lowest denomination, we find the sum to be 5 farthings. But since 4 farthings make a penny, we set down 1 farthing, and carry one penny to the column of pence. The sum of the pence then becomes 18, which is 1 shilling and 6 pence over. Set down the pence, and carry the 1 shilling to the column of shillings, the sum of which becomes 26; that is, 1 pound and 6 shillings. Setting down the 6 shillings and carrying 1 to the column of pounds, we find the entire sum to be £21 6s. 6d. 1far. Hence, for the addition of all numbers, 21 6 6 1 I. Write the numbers so that units of the same value shall fall in the same column. II. Add the units of the lowest denomination, and divide their sum by so many as make one unit of the denomination next higher. Set down the remainder and carry the quotient to the next higher denomination; proceed in the same manner through all the denominations and set down the entire sum of the last column. PROOF. 43. The proof of an operation, in Addition, consists in showing that the answer contains as many units as there are in all the numbers added. There are three methods of proof: I. Begin at the top of the units column and add, in succession, all the columns downwards. If the two results agree, the work is supposed to be right; for, it is not likely that the same mistake will have been made in both additions. II. Divide the given numbers into parts, and add the parts separately then add together the partial sums; if the results agree, the work is supposed to be right; for, a whole is equal to the sum of all its parts. III. Find the excess of 9's in each number, and place it at the right (Art. 20). Add these numbers and note the excess of 9's This excess should be equal to the excess of 9's in in their sum. the sum of the numbers. 44. The pupil should be early taught to omit the intermediate 43. What is the proof of an operation in Addition? methods of proof are there? Explain each separately? 44. What is the reading process, in Addition? How many words in the addition of columns of figures. Thus, in the above example, instead of saying 7 and 0 are 7; 7 and 4 are eleven; 11 and 6 are seventeen; he should simply say, seven, eleven, seventeen. Then, in the column of tens he should say, five, eleven, eighteen, twenty-seven; and similarly, for the other columns at the left. This is called reading the columns. Let the pupils be often practised in the readings, both separately and in concert in the class. 5. What is the sum of 1376, 38940, 8471, 23607, 891? 6. What is the sum of 3480902, 3271, 567321, 91243, 6001, 169 ? 7. What is the sum of 42300, 6000, 347001, 525, 47? |