sary to divide sums of money, as dollars, cents, &c. or other articles into a certain number of equal parts. By division this can be effected with great facility. SECTION VII. COMPOUND ADDITION. Compound Addition is the addition of numbers of different denominations. The denominations, however, must be of the same kind. Pounds, shillings, &c. cannot be added to years, months, &c. Shillings, pence, &c. are different denominations from pounds, &c. but of the same kind; viz. that of money. A butcher sold 3 pieces of beef; one piece weighed 12 lbs. 8 oz.; another 15 lbs. 3 oz.; the other 25 lbs. 15 oz. What was the weight of the whole? 12 pounds 8 ounces. 52 26 If we add the pounds as in simple addition, we find 52 of them; and by adding the ounces separately in the same manner, we find 26. But 16 ounces make a pound; so that 26 ounces contain 1 pound and 10 ounces; for 16 taken from 26, leaves 10. If we add this 1 pound with the others, we shall have 53 pounds and 10 ounces. This operation may be more conveniently performed as follows: oz. Lbs. 12 8 15 3 25 15 Whe first add the ounces; 15 and 3 are 18, and 8 are 26, which contains 16 once and 10 besides. We write the 10 ounces under the other ounces, and carry the 1 (pound) to the next column, to be added with the other pounds, which thus increased, make 53, which we write under the column. Thus we find the whole weight to be 53 pounds and 10 ounces. 53 10 Add together 4 years, 6 months, 3 weeks, and 5 days; 8 yrs. 9 mo. 0 w. and 2 d.; and 7 yrs. 11 mo. 1 w. and 3 d. In Compound Addition, we proceed on the same principle as in Simple Addition, viz. of adding those numbers which are of the same denomination together, and when the sum of these is sufficient to make one or more of the next higher denomination, we write the excess (over the exact number which forms one or more of the next higher denomination) under the column added, and reserve the one or more of the higher denomination, to be added with the column which expresses it. EXAMPLE in pounds, shillings, and pence. £. S. d. 435 15 8 125 6 2 215 17 3 8 7 4 We first add the column of pence, which makes 17; this contains 12 (the number of pence in a shilling) once, and 5 excess, which we write under the column. By adding the 1 shilling reserved with the column of shillings, their sum is 46, which contains 20 (the number of shillings in a pound) twice, and 6 excess, which we write under the column. We now add the 2 pounds reserved, with the pounds; their sum is 785; and the whole amount therefore is 785 £. 6 s. and 5 d. 785 6 5 It will be observed, that in this process we added together the units and tens of the shillings as in simple addition; thus, 1 to carry to 7 is 8, and 4 are 15, and 6 are 21, and 5 are 26; 2 to carry to 1 is 3, and 1 are 4, which we conceive to be placed at the left hand of the 6, making 46 shillings. In the same manner also, we added the pounds. From the preceding examples and illustrations, we derive the following RULE. To add compound numbers, write them under each other, placing those of the same denomination in the same column, and the lowest denomination at the right hand, and the higher in regular order towards the left. Add the column containing the lowest denomination. If the sum is sufficient to make one or more of the next higher denomination, write under the column the excess over the exact number which makes one or more of the next, and reserve the one or more to be added with the column of the same name. Proceed in this manner with the next higher denomination, and so on till the work is finished. Numbers are sometimes found where no even number of units makes one of the next greater: as 5 yards make one rod. In these cases the parts are to be written as fractions. METHODS of Proof. The method of proof by casting out the 9's is not applicable to compound numbers. Compound Addition may be proved by cutting off the upper line, and adding the rest. If this sum, added to the upper line, is equal to the sum obtained by the first addition, the work is right. Sums in this rule may be proved, also, by subtracting successively from the sum of the numbers added, all the parts of these numbers; and if the work is right, nothing will remain. £. S. 487 6 625 6 272 9 d. 8 1335 2 10 111 1 0 EXAMPLE. We first add the hundreds in the column of pounds; the sum is 12, which taken from 13 the beginning of the result in the example, leaves 1, which was produced by what was reserved in the tens in performing the addition. Next add the tens; their sum is 12. Take from this the 3 (tens) increased by the 1 (hundred) that remains from the left hand column, considered as 10 tens, and the remainder is 1 (ten,) which was reserved from the column of units. The sum of the column of units is 14, which, taken from the 5 increased by the 1 (ten,) leaves 1, which was reserved from the column of shillings. The sum of the shillings is 21, which, taken. from 2 increased by the 1 pound 20 shillings, leaves 1, which was reserved from the column of pence. The sum of the pence is 22, which, taken from 10 increased by 1 shilling 12 pence, leaves 0. The work is therefore right. USE OF COMPOUND ADDITION. In the transaction of business, the amount of articles of different denominations, such as pounds, shillings and pence, pounds, ounces, &c. is often required. By this rule the object is attained. SECTION VIII. COMPOUND SUBTRACTION. Compound Subtraction teaches how to find the difference between two sums of different denominations, which is done by taking the one sum from the other. EXAMPLES. £. S. d. 3 9 5 From 823 7 Difference 87 17 10 In this example, as we cannot take 5 pence from 3, we borrow from the minuend in the column of shillings 12 pence (1 shilling,) which we add to the 3 pence, and from the sum take the 5, and the remainder is 10. We therefore carry 1 (shilling) to the 9, which makes 10. Because 10 is greater than 7, we borrow 1 from the column of pounds, equal to 20 shillings, which added to 7, makes 27; from this take 10, and the remainder is 17. We then carry 1 (pound) to the subtrahend in the column of pounds, and subtract it as in simple addition. The reason for borrowing and carrying (tens) was explained in the simple rules. The principle here is the same, though the numbers vary from each other. When the lower figure exceeds the upper, borrow as many as will make one of the next higher denomination. From 4 miles, 2 furlongs, 27 rods, 10 feet, and 4 inches, take 1 mile, 6 furlongs, 38 rods, 17 feet, and 6 inches. M. fur. rods. ft. in. 4 2 27 10 4 1 6 38 14 6 2 3 28 11 10 In this example we borrow 1 foot, which is 12 inches, and add to the 4 inches; from the sum, 16, we take 6, and the remainder is 10. We add 1 to the 14 feet, or, what will amount to the same thing, consider the 10 above it as 9, to which add 161; the sum is 25; from this take 14, and it leaves 11. Add 1 to the 38 rods and take the sum from 67, (40 rods, = 1 furlong, being added to the 27,) and 28 remain. Carry 1 to 6 and subtract it from 10, (8 furlongs being added to the 2,) and the remainder is 3. Carry 1 to the 1, and subtract it from 4, and 2 remain. In the result we have 112 feet and 10 inches. But foot is 6 inches; add these to the 10, and the sum is 16, i. e. 1 Add the foot to the other feet, and the foot and 4 inches. result will read 12 ft. 4 in. instead of 11 ft. 10 in. From the preceding examples and illustrations we deduce the following RULE. Write the sum to be subtracted under the other as in addition, and subtract the lower sum from the upper. If the lower number of any of the denominations exceed the upper, add as many to the upper as make one of the next higher denomination; from the sum take the lower number, and carry one to the lower number of the next higher denomination before subtracting it. In some instances it becomes necessary to borrow twice, or double the number that it takes to make one of the next higher, because it does not always take a simple number of units to make one of the next superior denomination. Having one to carry after we take the feet from the number above, it makes 6 yards. But 5 yards make a rod, and 6 cannot be taken from 54; we therefore take 2 rods from the next column, which is equal to 11 yards. From this number 6 can be taken, and 5 remain. It will be seen by looking at the remainder, though but 1 rod is written in the place of rods, that the 5 yards &c. make more than another rod, and the true remainder is 2 rods, 0 feet, and 3 inches. A few examples occur, where the subtraction cannot be performed, without resolving the given numbers into others, retaining the same value. Example. Rods. yds. ft. in. 0 The difference in the value of the above numbers is 1 inch. |