what was the increase of population in New England between 1810 and 1820 ? Ans. 187,881 increase. 10. At the census in 1810, the number of inhabitants in the State of New-York, was 958,066; and on taking the census in 1820, the number of inhabitants had increased to 1,372,812; then what was the increase between 1810 and 18.20 ? Ans. 414,746 increase. 11. According to the two preceding sums, how much more was the increase of population in the State of New-York, than in the six New England States between 1810 and Ans. 226,865. 1820 ? 12. The number of square miles in New England is 65,047, and the number of square miles in New-York is 46,000; then how many more square miles are there in New England than in New-York? Ans. 19,047. 13. News papers were first printed at Paris in 1631; how many years since, up to the year 1829 ? Ans. 198 years. 14. Bought 2000 yards of shirting for 466 dollars, and sold 1476 yards for 369 dollars; how many yards have 1 left, and how many dollars do I want to make up the first cost? Ans. I have 524 yards, and want $97. 15. What number must be subtracted from 2081, that the remainder may be 1104? Ans. 977. 16. America was discovered by Christopher Columbus in 1492; how many years have since elapsed up to the year. 1829 ? Ans. 337 years.. 17. Union College was incorporated in 1794; how many years since ? 18. Yale College at New Haven was incorporated in 1700; how many years since? 19. Gen. George Washington died in 1799, aged 67. years; in what year was he born? Ans. 1732.. 20. How long since the declaration of Independence which was declared in 1776-reckoning to the year 1829 ? SUBTRACTION OF DECIMALS. Ans. 53. Subtraction of decimal or federal money, is exactly the same in the operation of the work, as subtraction of whole numbers. Because ten in an inferiour denomination, is equal to one in the next superiou*, C through all the denominations. exists as in whole numbers. Consequently the same proportion RULE.-Place whole numbers the same as if no decimals were annexed; and decimals, so that tenths stand under tenths, hundredths under hundredths, and thousandths under thousandths, &c. Or which is the same thing, place dollars under dollars, cents under cents, and mills under mills; so that the separatrix in the subtrahend shall stand directly under that of the minuend; and care must be taken to place the separatrix in the result directly under the separatrix in the given numbers. Here, as in addition of decimals, a dollar is a unit; and dimes, cents, and mills, according to their order, are tenths, hundredths and thousandths of a dollar; eagles and dollars are expressed in dollars; dimes, cents and mills are expressed in cents and mills; thus, 4 eagles, S dollars, 6 dimes, 8 cents, and 3 mills, are expressed 48 dollars, 68 cents and 3 mills, or $48, 68 cts. 3 m. A dollar being a unit, a separatrix must always be placed directly after dollars when any of the inferiour denominations follow; thus, five dollars and sixty eight cents must be expressed in figures, $5,68 cts. NOTE.-Care must be taken to place ciphers in the place of vacant. denominations, whenever they occur in either of the given sums, so that the significant figures may stand in their proper places; thus, you should write 1 dollar and 5 mills, $1,00, cts. 5 m. and a separatrix may be placed between cents and mills, as well as between dollars and cents, for the ease of expressing numbers. EXAMPLES. 1. From 4 dollars and 68] 3. From 60 dollars, 80 cts. cents, take 3 dollars and 20 and 5 mills, take 1 dollar and $1, 4 8 Ans. 2. What is the difference between 84 dollars, 91 cents 1, 0 0, 1 $5 9, 8 0, 4 Ans. and 6 mills, and 45 dollars, 64 cents and 5 mills. cents and 2 mills? 5 $0, 1 9, 5 Ans. 5. A miller bought a quantity of corn, for which he paid $480; he afterwards sold it for 580 dollars, 68 cents and 5" mills; what did he gain by the sale? Ans. $100, 68cts. 5m. 6. A merchant borrowed $600, for which he gave his note; he paid at one time $240, at another $150; how much remains due on the note? Äns. $210. 7. Joseph bought 10 oranges for 30 cents, he sold 6 for 24 cents; how many oranges has he left, and how many cents does he want to make up the first cost? Ans. he has 4 oranges left, and wants 6 cents. 8. From one dollar, take one dime and one mill. Ans. ,89 cts. 9 m. Ans. $8,49 cts. 6m. 9. From two dollars, take two dimes. 10. From two dimes, take two cents. 11. From seven dollars, take six dimes. 12. From eight dollars and eight dimes, take one dime and one mill. Ans. $8,69 cts. 9 m. 13. From ten dollars, take one dollar, fifty cents and four mills. 14. From forty dollars, take forty cents and five mills. Ans. $39,59 cts. 5m. 15. If one mill be taken from a thousand dollars what will remain ? Ans. $999,99 cts 9 m. 16. A gentleman found four bags of money to the amount of $500; the first bag contained $95, the second $130, and the third $101; I wish to know what the fourth contained? Ans. $174. 17. A Clergyman's salary is $1500 a year, and he spends $800,50 cts.; how much of his salary does he lay up? Ans. $699,50 cts. 18. A gentleman bought a horse for $100, which proving unsound, he is willing to lose $12,373 cts.; how must he sell the horse? Ans. $87,62 cts. 19. A merchant bought a piece of broadcloth for $205, which proved to be damaged, so that he sold it for $175,121 cents; how much did the merchant lose? Ans $29,871 cts. 20. On a note of $105,37 cts. if you receive at one time $42,20 cts. at another $37,84 cts. ; what remains due? Ans. $25, 33 cts. QUESTIONS ON SIMPLE SUBTRACTION. NOTE. The teacher should not wait for the student to go over the rule with all the examples before the questions are asked that are annexed to each rule. When the student commences a rule, the teacher should commence questioning him, and explain the nature and princi❤ ples of the rule, requiring the student to give reasons for all the operations of his work. What is simple subtraction? A. Taking a less number from a greater. What are the two given numbers called? A. Minuend and subtrahend. What is the number called that arises from the operation of the work? A. Difference or remainder. What does the difference or remainder show? A. It shows how much the minuend is greater than the subtrahend. What does minuend signify? A. A number to be lessened by another. What does subtrahend mean? A. A number to be taken from a greater. How do you place the given numbers in subtraction! A. So that units stand under units, tens under tens, &c. How do you proceed in the work? A. Commence at the right hand and take the figures in the subtrahend from those directly above them in the minuend, placing the difference in each place directly under. When the lower figure in any place exceeds that directly above it, what must be done? A. Add ten to the upper figure and take the lower figure from the amount, placing the difference directly under, and then add one to the next lower figure of the subtrahend. Why does not this adding ten to the figure in the minuend and then one to the next lower figure of the subtrahend, alter the true difference between the given numbers? A. Because it is adding equals to both the given numbers,. and adding equals to both, the difference between two numbers must ever remain the same; and we add equals to both, because ten in an inferiour column, (which we add to the minuend) is only equal to the one which we add to the next superiour column of the subtrahend.How do you prove subtraction? A. By adding the difference to the subtrahend, and if the amount equals the minuend, the work is right; because it is evident, that it can take no more than the difference between two numbers to make the less equal to the greater. Can subtraction be made to prove itself? A. Yes. How is that done? A. By subtracting the difference from the minuend, and if it leave e number equal to the subtrahend the work is right, because it is evident, that if the difference between two numbers be taken from the greater, it reduces it equal to the less. What is subtraction of decimals? A. Taking something that is less from that which is greater, where one or both of the given sums contain parts of an integer. How do you place decimals for subtracting? A. Tenths under tenths, hundredths under hundredths; or in federal money, which is the same, dimes under dimes, cents under cents, &c. What are integers in federal money? A. Dollars. What are dimes? A. Tenths. What are cents? A. Hundredths. What are mills? A. Thousandths. What is the decimal expression called when dimes, cents. and mills are taken together? A. Thousandths of a dollar. How do you subtract decimals? A. The same as in whole numbers. Why do you subtract the same? A. Because ten in an inferiour column or denomination, is equal to one in the next superiour, the same as in whole numbers. Why do we call dimes and cents, cents only? A. We call them all cents, because the left hand figure in cents expresses tens in cents which is the same as dimes. When your subtrahend contains denominations which are not named in the minuend, what do you do? A. Supply the vacant denominations of the minuend with ciphers, then place the subtrahend under, so that cents may occupy the place of cents, and mills the place of mills. If you are required to subtract dimes from dollars, what do you first do? A. Join two ciphers to the right hand of the minuend, placing a separatrix between them and dollars; then place your dimes under the left hand cipher joined to the minuend, and to the dimes join a cipher, and they will stand as cents, and then subtract. How do you point off in subtraction of decimals? A. Place the separatrix in the result, directly below the separatrix in the given numbers. Why is federal money introduced under the head of decimals? A. Because dimes, cents and mills form a decimal expression, of which a dollar is the integer. SIMPLE MULTIPLICATION, Is repeating one of two numbers as often as the other contains a unit; or, it is the shortest method of performing addition, where the same number is to be repeated a given number of times. The two given numbers are called multiplicand and multiplier. The multiplicand is the number to be repeated. The multiplier is the repeater, or number by which you multiply. The number produced by the operation of the work, is called the product. This is the most useful rule in practical arithmetick. When the price of one is given; by this rule, we obtain the price of any number, or quantity; when length and breadth are given; by it, we find the area or surface; in reduction, it affords the greatest facility in reducing higher denominations to lower, and its principles are advantageously applied in all practical business of buying and selling. NOTE.-The multiplicand and multiplier taken together, are called factors or substitutes. Multiplication is denoted by this character, X; thus,6X 3=18, which signifies, that the product of 6 multiplied by 3, is 18. NOTE. No pains should be spared by the student in making himself master of the following table. The task is easy if persevered in; but when the student suffers himself to pass over it superficially, no time afterwards spent in work is scarcely sufficient to make it familiar to his mind. A good knowledge of it will greatly facilitate his progress, and save him the trouble of repeatedly reviewing the same work to detect mistakes. |