is ten times the value of the second; the fourth, that of the third; and so of the rest. EXAMPLE. In the third line of the Numeration Table (counting from the top) I find 3 in the place of hundreds, which is three hundred; 2, in the place of tens, which is twenty; and 1, in the place of units, which is one; therefore, the whole value of that line is three hundred and twenty-one. APPLICATION. Write, in figures, the following numbers. 1st. Six hundred and twenty-five. 2d. Three thousand, one hundred and ten. sd. Forty-five thousand, two hundred and sixteen. 4th. Thirty-three thousand, two hundred and two. 5th. One hundred and twelve thousand, five hundred. 6th. One hundred thousand, and twenty nine. 7th. Two million, three hundred and twenty thousand, five hundred and eleven. 8th. Sixty-nine million, eight hundred and two thousand, three hundred and five. 9th. Seventy-two million, thirteen thousand and nineteen. Write in words each line of the Numeration Table, respectively, beginning at the top. SIMPLE ADDITION teaches to collect into one sum, several numbers, which consist of one denomination only. RULE 1st. Place units under units; tens under tens; hundreds under hundreds, &c. RULE 2d. Begin with the right hand column, or line of units, when two or more numbers are to be added together. RULE 3d. Carry one for every ten * that is, in adding the first column of any sum, if it exceed ten, twenty, thirty, forty,|| &c. set down what there are over ten, or tens, and carry as many to the second column, as there were tens in the first ;thus proceed with each column, till the last is added, under which set down the whole amount. * The reason of this is, because the value of each figure from right to left increases in a ten fold proportion; that is, ten units, or ones, make ten; ten tent, an bundred; and in bundred, one thousand, &c. PROOF. Add each column as before, omitting the top line; set this amount under the first; then, if the amount of this and the top line be equal to the total sum, the work is right. In the first example, I say, five and two are seven and three are ten and six are sixteen: I then set down what there are over ten, which are six, then proceed with the second column thus,-one that I carry to one are two, and nine are eleven, and seven are cighteen; I set down eight, and carry one to three, which are four, and one are five, and four are nine, and three are twelve; I then set down the whole.-In the next place, I add each column as before, omitting the top line, and set the amount under that of the whole sum; lastly, I add the sum of all except the top line," to the top line, and find the amount is equal to the whole sum; therefore, I conclude the work is right. Practical Questions. 1. A man has four farms. The first is worth two thousand, seven hundred, and twenty-five dollars :-the second is worth three thousand, eight hundred and nineteen dollars ;the third is worth one thousand, six hundred and ten dollars; the fourth is worth five hundred and twelve dollars; what are they all worth? Ans. 8,666 dols. 2. A man has four horses. The first is worth eighty-four dollars; the second is worth forty-five dollars; the third is worth as much as the second; and the fourth is worth as much as the first; what are they all worth? Ans. 258 dols. 3. A man possessed a tract of land, which contained fortynine thousand, eight hundred and thirty-five acres; now suppose he had six tracts of equal dimensions, how many acres did the whole contain? Ans. 299,010 acres. 4. Suppose one ox weigh one thousand and forty-five pounds; another, eight hundred and twelve pounds; and a third, nine hundred and one pounds; what is their whole weight? Ans. 2,758 pounds. 5. The hind quarters of a cow weigh one hundred and three pounds each; the fore quarters weigh ninety-seven each; the hide, sixty-three, and the tallow, fifty-six; what is the weight of the cow? Ans. 519 pounds. Questions to be answered mentally. 1. 65 are how many? 4. 2111 are how many? 5. A person paid at one time, 16 cents, and at another 8; what was the amount paid? 6. A man bought a barrel of flour for 6 dollars, a bushel of nuts for I dollar, and a box of raisins for 5 dollars; what did the whole cost him? 7. A, gave some money to B, to purchase some articles; B spent 50 dollars, and had 12 dollars left; how much did A give B? 8. A person said he was 18 years old when his father died, which was 27 years ago. Re. quired his present age? 9. A person having 87 dollars, found that if he had 13 more, he could purchase a horse. What was the price of the horse? 10. George Washington was born in the year 1732, which was 44 years before the declaration of Independence. In what year was it declared ? 11. A labourer found that 25 days of his engagement had elapsed, and that his time would not expire till ten days longer. How long did he agree to serve ? 12. A landlord received at one time 156 dol. lars, at another 128 dollars, and at another 56 dollars, which was payment for one year's rent. What did he receive per year? 13. The distance from Boston to Worcester is 40 miles; from Worcester to New Haven, 94; from New Haven to New York, 76; What is the distance from Boston to N. York? 14. A person received a certain sum of money to purchase an estate. After having paid 2500 dollars for it, and also 50 dollars for expenses, he had 185 dollars left. Required the sum that he first received? SIMPLE SUBTRACTION Teaches to find the difference of two numbers, which are of one name or denomination, by taking the less from the greater. RULE 1st. Place the greater number uppermost, and the less directly under it, setting units under units, tens under tens, &c. RULE 2d. Having properly stated the question, draw a line underneath; then, beginning with units, subtract or take the less number from the greater, and set down the remainder, or difference. RULE 3d. Borrow ten; that is, whenever the lower figure happens to be greater than the upper, add ten to the upper figure, and subtract the lower figure therefrom, and set down the remainder, always remembering, when you borrow in one place to carry one to the next. PROOF. Add the difference of two numbers to the least num ber; if the amount be equal to the greatest number, the work ises in right. and u In the first example I say, three from five, there remain two, which I set down; then, two from six there remain four; then four from three, I cannot, but four from thirteen, there remain nine; then, one to carry to five are six, six from two I cannot, but six from twelve, six remain; then one to carry to one are two; two from five three remain. I then draw a line, and add the remainder, or difference of the two numbers in the question, to the least number or numbers subtracted; the amount of which, being equal to the greatest number, shows the work to be right. Practical Questions. 1st. Subtract two thousand, one hundred, and nineteen, from five thousand, two hundred and twelve. Ans. 3093. 2d. A man is worth five thousand, eight hundred pounds; but he owes three hundred and forty-eight pounds; how much will be be worth when his debt is paid? Ans. £5452. 3d. What is the difference between nine, and ninety-nine million? Ans. 98999991. 4th. If one be taken from ten thousand, what will then remain ? Ans. 9999. 5th. A gentleman possessed fifteen thousand, eight hundred, and forty acres of land; but he sold two thousand, three hundred and fifty to one man; four thousand five hundred to another; and three thousand, two hundred, and twenty-five to a third ;—how much had he left? Ans. 5765. SIMPLE MULTIPLICATION. MULTIPLICATION is the increasing of any number, by so many of itself, as there are units in that number by which it is increased, or multiplied. Simple Multiplication teaches to multiply any two numbers into each other, which are of one denomination. In Multiplication, there are three things to be considered, viz. 1st. The multiplicand, or number to be multiplied. 2d. The multiplier, or number to multiply by. 3d. The product, which is the result of two numbers when multiplied together, and is the answer to the question. MULTIPLICATION TABLE. RULE. Place the Multiplicand (which is commonly the largest number) uppermost, and the multiplier underneath; setting units under units, tens under tens, &c. then draw a line and proceed to multiply, beginning with units, and set down the product, observing to carry one for every ten, as in Simple Addition. PROOF. Multiplication may be proved, 1st. by inverting the order of the multiplier and multiplicand; that is, make the multiplicand the multiplier, and proceed to multiply in the usual way; the product being like to the product before the question was inverted, shows the work to be right. 2d, by dividing the product by the multiplier; the quotient being equal to the multiplicand shows the work to be right.* But the most ready method of proving multiplication, is by casting out the nines,† thus: 1st. cast the nines out of the multiplicand, and set the overplus at the right hand of a cross, as you see in the example. 2d, cast the nines out of the multiplier, and set the remainder at the left hand As it is supposed the learner is not acquainted with Divison, he cannot at first put this method in practice. A question may prove by this method, when the work is not right; but this will not happens in unless there are two wrong figures in the work, one of which must be just as much too largend 14 as the other is too small, which is rarely the case. |