Arithmetic, Part 1 · Section 9 – Section 10

Arithmetic · Part 1 · Section 9 – Section 10

Subtraction.

SUbtrrction is the taking of one or more ſmall ſums out of a greater, as 7 s. out of 12 s. or 37 li. out of 100 li. or 137 foot out of 983 foot, and the like.

As in Addition the ſums to be Added may be either of one or of diverſe denminations, ſo likewiſe they may be in Subtraction, and the manner of placing them is the ſame, for you muſt ſet Unites under Unites. Tens under Tens, Hundreds under Hundreds, &c.

Example 1

Of Subtraction of numbers of one denomination, let it be required to ſubtract 234, out of 986, place the numbers one unde the other as you ſee done in the Margine, draw a line under them and begin with the firſt figure towards your right hand, which is 4, ſaying take 4 from 6, and there remains 2, place 2 under the line, and go to the next figure which is 3, ſaying, take 3 from 8, and there remains 5, place 5 under the line, and go to the next figure which is 2, ſaying take 2 out of 9 and there remains 7, place 7 under the line, and your ſubtraction is ended, and it is evident by the work, p. 25that if you take 234 out of 986 there will remain 75…, which you may thus prove, for if you adde the 234 to 752, you ſhall find the ſum of that addition to be 986, which is equall to the whole ſum from which 234 was ſubtracted,

Example. 2.

Let it be required to ſubtact 2976 out of 96527, place the numbers one under another as in the margine you fee done, then draw a line under them, and beginning with the firſt figure towards your left hand, ſay take 6 out of 7, and there remains 1, place 1 under the line and prodeed to the next figure, ſaying 7 out of 2 I can not (wherefore you muſt alwayes adde 10 to the number above which in this example is 2, and it makes it 12,) therefore take 7 out of 12, and there remains 5, place 5 under the line, and (becauſe you added 10 to the 2 to make it 12, you muſt) carry a unite to the next figure, ſaying, one which I carryed and 9 is 10, take ten out of 5, which I cannot, therefore I muſt adde 10 to 5 and it makes 15, and ſay 10 out of 15, and there remains 5, place 5 under the line, and becauſe you added 10 to 5 to make it 15, you muſt therefore carry a unite to the next figure ſaying, one which I carryed, and 2 is 3, take 3 out of 6 and there remains 3, place 3 under the line, and becauſe there is no more figures to be ſubtracted from the number above, you muſt, ſay nothing fom 9 and there remains 9, ſet the 9 under the line, and your Subtraction is ended.

p. 26Other Examples for Practice.

Subtraction of Numbers of diverſe Denominations, 1 Of Engliſh Money.

In Subtraction of numbers of diverſe denominations, you muſt obſerve the ſame order as in Addition, namely, to place every number in due order, with reſpect to its denomination, as pounds under pounds, ſhillings under ſhillings, &c. the greater number alwayes uppermoſt, and drawing a line under them, begin with the leaſt denomination firſt, ſubtracting it from the line above, and ſetting the remainder under the line as in whole numbers, but if the pence or ſhillings in the upper row be ſmaller then thoſe in the neather row, you muſt adde 12 d. or 20 s. to the ſmaller number, that ſo ſubtraction may be made, as by the examples following will appear.

Example 1.

Let it be required to ſubtract 38 li. 12 s. 8 d. out of 269 l. 18 s. 10 d. Place your numbers as in the margine, then beginning with the leaſt denomination firſt, p. 27(which in this Example is pence) ſay 8 d. from 10 d., and there remains 2 d. ſet the 2 d. under the line, and proceed to the next denomination which is ſhillings, ſaying take 12 s. out of 18 s. and there remains 6 s. place 6 s, under the line, and go to the pounds, ſaying 8 out of 9 and there remains 1, place 1 under the line, and ſay 3 out of 6, and there remains 3, then (becauſe there is no more figures to be ſubtracted) ſay nothing out of 2 and there remains 2, which ſet under the line, ſo is your ſubtraction ended, and the remainder is 231 li. 6 s. 2 d.

Example 2.

Let it be required to ſubtract 2628 li. 16 s. 10 d. out of 9320 li. 10 s. 7 d. place the numbers in order, and beginning with the pence, ſay 10 d. out of 7 d. I cannot, (therefore I muſt adde 12 d. (which is one ſhillings) to 7 d. and it makes 19 d.) but 10 d. out of 19 d. and there remains 9 d. ſet the 9 d. under the line, and (becauſe I added 12 d. to 7 d.) I muſt therefore carry one to the place of ſhillings, ſaying 1 s. which I carryed, and 16 s. is 17. then 17 s. from 10 s. I cannot take, therefore, I muſt adde 20 s. (which is one pound) to 10 s. and it makes 30 s. and 17 s. out of 30 s. and there remains 13, ſet 13 under the line, and carry one to the place of pounds, ſaying one which I carried and 8 is 9, take 9 of o I cannot, but 9 out of 10, and there p. 28remains 1, ſet 1 under the line, and carry a unite to the next place, ſaying 1 which I carried and 2 is 3, take 3 out of 2 I cannot, but 3 out of 12, and there remains 9, place 9 under the line, and carry 1 to the next place, ſaying 1 which I carry and 6 is 7, take 7 out of 3 I cannot, but 7 out of 13, and there remains 6, place 6 under the line, and carry one to the next row, ſaying 1 and 2 is 3, take 3 from 9 and there remains 6, place 6 under the line, ſo is your Subtraction ended, and the remainder is 6691 li. 13 s. 9 d.

Example 3. Suppoſe a man had lent to another man 1000 pound, and that the borrower had paid thereof at one time 127 li. at another time 430 li. 10 s. and at a third payment 50 li. and the creditor would know how much he hath received, and how much is owing of his debtor.

Place the numbers as here you ſee,

firſt the ſum of money lent, and draw a line under it, then ſet the ſums paid at ſeveral times one under another, and draw a line under them: Then adde all the ſums which have been paid at ſeveral times together, which make 607 li. 10 s. which is the ſum which the debtor hath paid in p. 29all, then ſubtract this 607 li. 10 s. from 1000 li. and there will remain 392 li. 10 s. and ſo much is ſtill owing to the Creditor.

Other Examples for Practice.

The Proof of Subtraction.

The Proof of Subtraction is performed by Addition, for adding the number to be ſubtracted, to the remainder, the Sum of them muſt be equal to the number given, if you have truly wrought. As in the firſt example of numbers of one denomination,

p. 30Adde the number to be ſubtracted 234, to the remainer 752, the ſum of them is 986, equal to the number given.

Examples for Practice proved.

Other Examples in Weight and Meaſure.

1 Exa. in Troy weight

2 Example in Avoirdupois great weight.

3 Example in Avoirdupois little weight.

4 Example in Time.

p. 31Queſtions performed by Addition and Subtraction.

Queſtion 1. What number is that which added to 376 ſhall make 1000? Subtract 376 from 1000, the remainder is 624, the number ſought.

Queſtion. 2. What number of pounds, ſhillings and pence muſt be added to 36 l. 17 s. 3 d. to make that ſum up 100 li. ſubtract 36 li. 17 s. 3 d. from 100 li. the remainer is 63 li. 2 s. 9 d. which added to 36 li. 17 s. 3 d. makes 100 li.

Queſt. 3. In the year of our Lord 1440, the famous art or myſtery of Printing was invented, I would know how long it is ſince that time to this yeer of our Lord 1655. From 1655, ſubtract 1440, the remainer is 215. and ſo many years are expired ſince printing was invented.

Queſtion 4. An Armie conſiſting of 13721 horſe, and 26850 foot, in an engagement there were ſlain 3760 horſ, & 7523 foot, the queſtion is how many were ſlain in all, & how many horſe and how many foot eſcaped. From the 1372 horſe which went out, ſubtract the 3760 that were ſlain, there remains 9961, and ſo many horſe eſcaped, Alſo from the 26850 foot which went out, ſubtract the 7523 which were ſlain, and there remains 19327, the number of foot p. 32which eſcaped, and by adding the 3760 horſe which were ſlain, to the 7523 foot that were ſlain, their Total is 11283, and ſo many were ſlain in all.