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43'298 inhabitants ; Philadelphia, 108'161; how many more inhabitants were there in Philadelphia than in Boston ?
Ans. 64'863. 21. A gentleman purchased twenty-five thousand acres of wild land. After selling a certain quantity, he had eighteen thousand, eight hundred and sixtyone acres left. How many did he sell ?
Illustration of Rule 2.- To prove the correctness of rule 2, it will be best to make use of an example. Let it be required to subtract 478 from 592.
Operation 1. Operation 2. 5 0 0+8 0+1 2=5 9 2 400+70+ 8=4 7 8
100+1 0+ 451 1 4
By separating the minuend and subtrahend into parts, as in the first operation, and subtracting, we take 8 from the 2, and 1 ten which was contained in the 9 tens, or from 12; 70 from 80, and 400 from 500, and obtain the differences between all the parts of the minuend and all the parts of the subtrahend; and these differences, viz. : 4, 10 and 100, added together, must give the difference between the whole minuend and the whole subtrahend, or between 592 and 478. But as we do not generally separate the numbers for subtracting, it will be necessary to show that we obtain the same answer by the common method. In the second operation of the foregoing example, when we suppose 10 added to the 2, we may suppose this 10 to be taken from the 9 tens, and by subtracting 8 from the 12, we obtain 4, the same remainder as when we separated the numbers. In the next place, as we have subtracted from one of the tens in the 9 tens, we have now to subtract the 7 tens from the remaining 8 tens in the 9, which leaves 1 ten; but as we did not remove the í ten from the 9 which we supposed added to the 2, if we were to
subtract the 7 from the 9, we should subtract twice from that 1 ten which we supposed removed, therefore, to balance that ten, we must add 1 ten to the 7 tens, and the difference will be the same as though we had taken 1 ten from the 9 and not added any to the 7, for the difference between 8 and 9, is the same as the difference between 7 and 8. The difference between the 4 and 5 is 1, and the whole remainder the same as found by the first operation. As the 1, which we add to the next place in the subtrahend, is just equal to the 10 which we added to the minuend, the difference must always be the difference between the given numbers, for by adding 10 to both numbers we increase both alike, there being no difference between 10 and 10.
22. Plymouth, Mass. was first settled in the year 1620; how many years from that time to the year 1826 ?
Ans. 206. 23. Required the difference between 170'063 and 169'007.
Ans. 1'056. A short horizontal line thus is used as a sign of subtraction, and denotes that the number after it, is to be taken from the number before it; thus, 1248 =4, and is read, 12 minus 8 equals 4; or, 8 subtracted from 12 equals 4; the word minus meaning less.
The scholar is required to work the following examples, as directed by the sign of subtraction: 24. 5'924-
279= 5'645. 25. 75'428 2'096=73'332. 26. 100'000 50'009=49'991.
1'961'851 987'6545 28. 85'400'301 98'394= 29. 506'070'809_90'706'050=
The sum of the remainders in the three last examples, is 501'640'863.
30. What is the difference between seventy thousand, three hundred and twenty-six, and ten thou. sand, two hundred ?
31. How much larger is two million, two thousand and two, than one million and two ?
Ans. 1'002'000. 32. America was discovered in 1492; Jamestown, Va. was settled 1608, and Plymouth, Mass. in 1620. How many years from the discovery of America to the settlement of Jamestown and Plymouth?
To the settlement of Jamestown, 11
To the settlement of Plymouth, 128.
METHOD OF PROVING SUBTRACTION. Add the remainder and subtrahend, or less number, together, the sum will equal the minuend, or greater number, if the work be right.
Illustration. The difference between two numbers added to the less, must produce a sum equal to the greater ; thus, 11 taken from 15 leaves 4; 4 is what 11 wants of being as much as 15, consequently, if we add 4 to 11, the sum will be 15.
But more mathematically ;-let us suppose the line A B to be longer than the line CD.
To find how much longer the line A B is than CD, we must take away a part of the line A B, which shall be as long as the line CD. Suppose the part A a to be as long as CD; by removing that part of A B, there will remain the part a B, which is the difference between the lengths of AB and CD. Now if we annex the line a B to CD, we should make the line C B just as long as A B.
We may suppose these lines to be any lengths, and if we wish to find the difference of their lengths;
we must subtract the length of the shorter from that of the longer, and the difference would be the length of a B; but the length of a B added to the length of the shorter line, will always give the length of the longer, which would be the minuend,
33. 670135 minus 478103 are how many ?
Operation. 6 7 0 9 1 3 5 Here 192032 is the dif4 7 8 1 1 0 3 ference between 670'135
and 478'103, and by add1 9 29 0 3 2 ing 192'032 to 4782103,
we obtain the minuend, as Proof 6 7 0 11 3 5 was proved in the Illus
The pupil can ascertain whether he performs the following examples correctly, by proving his work.
34. 17800 is how many less than 67987 ?
37. What is the difference between 100000 and 99999, equal to ?
38. What number must be added to 1'111, that the sum may be 10'000 ?
39. To what number must i be added that the sum may million ?
Ans. 999'999. 40. To what number must 100'101 be added that the sum may be one million, ten thousand and ten?
Ans. 909'909. 41. A man owing 1076 dollars, paid all but 108 dollars; how many dollars did he pay?
Ans. 968. 42. A clergyman being asked how long it was since he began to preach, answered that he was 69 years old, and that he began to preach at the age of 25. How many years had he been preaching ?
Ans. 44 years.
43. A man owing 721 dollars, paid 509; how much did he then owe?
Ans. 212 dolls.
Questions to be worked by Addition and Subtraction.
1. A gentleman bought 2 pieces of land; one contained 96 acres, the other 103. If he sell 47 acres, how many will remain ?
Ans. 152. 2. A merchant bought a cask of molasses containing 119 gallons. He sells to one man 11 gallons ; to another, 13; to another, 14. How many gallons has he left ?
Ans. 81. 3. One man owes another fifteen hundred dollars, to be paid as follows; one hundred and ten dollars in 3 months; two hundred and thirty-one, in 6 months; a thousand and ten, in 10 months, and the remainder at the end of one year. How much is he to pay at the end of the year. Ans. 149 dollars.
Questions to be worked by Multiplication and
Subtraction. 1. What is the difference between 11 times 71 and 1,000?
Ans. 219. 2. Which is the larger number, 6 times 665, or 7 times 565 ?
Ans. 6 times 665, by 35. 3. 290 x 9023-6'504=2610'166. 4. 167 X7 X 8-3'4705882. 5. 265-77X3=564.
6. A merchant buys 175 pounds of pork, at ? cents a pound, and sells it for 8 cents ; how much does he gain?
Ans. $1,75. 7. If I purchase 1200 pounds of butter for 156 dollars, and sell it at 14 cents per pound, do I gain or Jose, and how much? Ans. I gain $12.
DIVISION OF SIMPLE NUMBERS. Division is ascertaining how many times one number can be subtracted, or taken, from another.
When we take one number from another once, to