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76. Compound Subtraction teaches to find the difference of two given compound numbers of the same kind, by taking the less from the greater.
RULE. Place the less compound number below the greater, so that like denominations may stand under each other as in Compound Addition.
Beginning at the least denomination, subtract the lower numbers from those above, putting each remainder under its respective denomination.
When the lower number of any denomination is greater than the upper, increase the upper by as many as make one of the next superior denomination, subtract the lower number from this sum, set down the remainder, and carry 1 to the next lower number; subtract, and proceed in this manner until the work is finished “.
• When the inferior denominations in the upper line are respectively greater than those in the lower, the reason of the rule is plain; when they are less, the borrowing and carrying depend on the same principles as common subtrac
Method of Proof. Add the remainder (or number arising from the operation) and the lesser number together, and if the sum be like the greater, the work is right.
ExAMPLEs. 1. From 1231.4s. 5d., take 102l. 19s. 6dź. Erplanation. OPERATIon. Here I begin at the farthings, and say, 3 from L. . s. d. 1 I cannot, I therefore borrow 4 farthings,
123 4 54 which added to the 1 make 5, then 3 from 5, # and 2 remain; put down #, and carry 1 to the 102 19 6+ 6 makes 7 ; then I say, 7 from 5 I cannot, borRem. 20 4 101 row 12 to the 5 which make 17, then 7 from —: 17, and 10 remain; put down 10, and carry l Proof 123 4 5+ to 19 make 20; then 20 from 4 I cannot, thereT fore I borrow 20 to the 4, making 24, then 20 from 24, and 4 remain; put down 4, and carry 1 to the 2: the rest is merely simple subtraction. The proof arises from adding the remainder and the line next above it together by Compound Addition.
tion does; so in Ex. 1. we borrow 4 (or 1 penny) in farthings, to compensate which we carry 1 (penny) to the pence; we borrow 12 (or 1 shilling) in pence, to compensate which we carry 1 (shilling) to the shillings; and in shillings we borrow 20, (or 1 pound,) carrying 1 afterwards to the pounds. That these carryings always exactly compensate for the number borrowed appears plain, for carrying 1 to the lower number is in effect the same as taking from the upper the 1 borrowed. The same reasoning may be applied to every kind of examples that can be proposed under this rule.