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7. The lesser of two numbers is 1738, their difference is 745 ; what is the greater number?

Ans. 2483.

8. A gentleman's granary, consisted of 300 bushels of wheat: 70 busbels rye; 400 barley; and 100 oats: how many bushels did his granary contain ?

Ans. 870 bush.

9. How many years from the time of the christian æra to the creation of the world; allowing prior to the christian æa from the death of Julius Cæsar, to be 44 years; to the destruction of Carthage 102; to the expulsion of the kings of Rome 363 years; to the building of Rome 244 years; to the building of Solomon's temple 251 years; to the siege of Troy 180 years; to the departure of the Israelites from Egypt 307 years; and from the creation to their departure 2513 years? Ans. 4004.

10. At the late census, taken A. D. 1810, the number of inhabitants in the several New-England states was as follows, viz: Maine 22 705; New Hampshire 214460; Vermont 217895; Massachusetts 47 040; Rhode-Island 76931; Connecticut 261942; what was the number of inhabitants at that time in New England?

Ans. 1471973.

11. A farmer left to bis only daughter 2475 dollars more than he left to his wife and son, which was as follows, viz: his wife's jointure was 25000 dollars; his son's part 25 hundred and 25 dollars; what was his daughter's portion; and what was the amount of his whole estate ?

Ans. Daughter's 30000.
Whole estate 57525.

C

SUBTRACTION

Teaches to take a less number from a greater, and thereby shew the remainder or difference; which will be a third number, shewing the inequality, excess or difference between the given numbers; and it is both simple and compound.

SIMPLE SUBTRACTION

Teaches to find the difference between any two numbers, which are of a like kind.

RULE.

Place the greater number uppermost, and the less underneath, so that units may stand under units, tens under tens, &c. from the right hand to the left. Then drawing a line underneath, begin with the units and subtract the lower from the upper figure, and set down the remainder; but if the lower figure be greater than the upper, borrow ten, and subtract the lower from the sum of both, and set down the remainder, but in this case you must add one to the ten's place of the lower line, for that which you borrowed; and proceed in like manmer through the whole.

PROOF.

Add the remainder and the less line together, which sum, if the work be right, will be equal to the greater line; or subtract the remainder from the greater line, and the difference will be equal to the less.

NOTE. The above methods will hold good in either simple or compound subtraction.

Demonstration 1. When all the figures of the less number are less than their correspondent figures in the greater, the difference of the figures, in the several like places, must, when all taken together, make the true difference sought; because, as the sum of the parts is equal to the whole, so must the sum of the differences, of all the similar parts, be equal to the difference of the whole.

2. When any figure in the greater number is less than its correspondent figure in the less, the ten, which is added by the rule, is the value of an unit in the next higher place, by the nature of notation; and the one which is added to the next place of the less number, is to diminish the correspondent place of the greater accordingly; which is only taking from one place, and adding as much to another, whereby

the total is never changed; and by this mean, the greater is resolved into such parts as are each, greater than, or equal to the similar parts of the less; and the difference of the correspondent figures, taken together, will evidently make up the difference of the whole.

The truth of the method of proof is evident; for the difference of two numbers added to the less, is manifestly equal to the greater.

ANOTHER METHOD TO PROVE SUBTRACTION.

1. Cast the nines out of the minuend, and note down the excess above the nines on a cross; then cast the nines out of the subtrahend and note the excess as before; lastly, cast the nines out of the remainder, and add the excess last found, and the excess of the subtrahend together, and if the sum of both be equal to the excess found in the minuend, the work is allowed to be right.

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Explanation of the second Example. The numbers being placed with the larger uppermost as the rule directs, I begin with the unit or right hand figure in the subtrahend, and say 7 from 6 I cannot, I therefore borrow, or suppose ten be added to the upper figure (6) which make 16, then I say, 7 from 16, and there remain 9, which I set down; this assumption changes the three next left hand figures 400 into 399; therefore I go on with the next figures in the subtrahend, and say 6 from 9 leave 3; 5 from 9, 4; 4 from 13, 9; 8 from 14, 6; 5 from 5,0; and the subtraction is completed-leaving for a remainder, sixty nine thousand, four hundred and thirty nine.

NOTE. The reason for borrowing ten is evident, if we consider, that when two numbers are equally increased by adding the same to both, their difference will be equal. Thus the difference between 6 and 8 is 2; add the number 10 to each of these figures (6 and 8) they become 16 and 18, still the difference is 2, the same as before.Therefore when we add or suppose to be added, 10 to the minuend, and we likewise add one to the next higher place of the subtrahend, it is just equal in value to 10 of the lower place.

Now to prove by casting out the digits. I begin with the uppermost line, as the rule directs, which I call the minuend, at the left hand

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figure, and say 8 and 6 are 14; and the excess above 9 is 5, 5 and 4 are 9, the excess is 0, but leaves 5, which I set down (as appears in the first example) at the top of the cross, I proceed in like manner, to cast out the nines of the subtrahend, and find the excess in the last figure in the unit's place to be 0, which I set down at the left hand of the cross; next I cast out the nines in the remainder and find there is 5 left, which I place at the right hand of the cross; lastly, I add this last found 5 to the 0, at the left hand, the sum is 5, which I set down at the bottom, and find it to be equal to 5, at top; the work is therefore allowed to be right.

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9. From 31489076 take 278901 Rem. 31210175 10. From 27890516 take 172805 Rem. 27717711 11. From 10031964 take 936743 Rem. 9095221 12. From 7749450 take 2732401 Rem. 5017049 13. From 1385267 take 274675 Rem. 1110592

The elapsed time since any remarkable event, may be found by subtracting the date of the event from the present year; and likewise may be found the time from the occurrence of any remarkable event to that of another, by subtracting the date of the first occurrence from that of the last, as will appear in the following exercises.

USE OF SUBTRACTION.

1. How long from the discovery of America by Columbus, which was in the year of our Lord 1492, to the commencement of the Spanish war with France, which happened in the year of our Lord 1806 ?

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2. How long since the commencement of the war 1775, which gained the American Independence?

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3. Martin Luther opposed the Pope 1520, how long is it since?

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4. King Henry the eighth, of Old England, wrote a treatise on the seven sacraments, in opposition to Martin Luther; how many years is it since, allowing he wrote in the year 1521 ?

5. How long since the termination of the war which gained the American Independence in 1783, to the commencement of the late war, between Great Britain and the United States in 1812 ?

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6. What number is it, to which if you add 72939, will make 78360 ?

Ans. 5421. The difference of two numbers is 40; the greater is 1804, what is the lesser ?

Ans. 1764.

8.

What number is that which taken from 100 leaves 40 ?

9.

Ans. 60. A piece of broad cloth that measured 324 yards; there were sold 128 yards; how many yards should remain of the piece? Ans. 196. 10. There are two numbers, whose dierence is 172; the greater number is 650; I demand the lesser ?

11.

Ans. 487.

What number is that which added to 72 will make 102 ?

Ans. 30. A man being asked how old he was, answered, if you add 32 years to my age, the sum total will be 100 ; what was his age ?

12.

Ans. 68. 13. Suppose an army, composed of 19000 men, was stationed in a garrison; but in consequence of a general peace, 1734 men were discharged; I demand how many men remained in the garrison?

Ans. 17266.

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