When the price is dollars and cents-Its half is the answer, in mills. And multiplying the whole number of pounds in a given quantity, by one half of the price of a ton, you will find the price of the given quantity. EXAMPLES. 1. Required the price of 100 lb. of hay, at $5 a ton. Thus, $5, with a cipher annexed, are 50,÷2-25 cts. 2. Required the price of 100 lb. of pork, at $100.56 a ton. Thus, 100.562-5028 mills, Ans. 3. Required the price of 1978 lb. of hay, at $12 a ton. Thus, 1.978×6=$11.868. Or, 1978-2000,×12—$11.868, Ans. 2. To find the cost of a pound, when cuts. qrs. and lbs. are given. RULE.-Find the price of a cwt., which divide by the pounds in a cwt. 3. When the price is cents, and the quantity whole numbers. RULE.-Write down the quantity, as so many dollars. See what part the cents in the price is of a dollar; divide by that part, the quotient is the answer. If the price is dollars and cents-Multiply the quantity by the dollars, working for the cents as before. EXAMPLES. 1. If 3 cwt. 2 qr. 14 lb. of iron cost $16. 67 cts. 5 m., required the price of a pound. cwt. qr. lb. cwt. 3 2 14-3.625) 16.675 (4.6, 112041, Ans. 2. Required the price of 622 yds. of India cotton, at 25 cts. a yard. Thus, 25 cts.} of $1. $662÷$165. 50 cts., Ans. 3. Required the cost of 22000 of boards, at $9. 25 cts, a thousand. $22X9 $198. Then, 25 cts.)22($5. 50 cts., +$198-$203. 50 cts., Ans. 4. What will 622 yds. of linen cost, at 30 cts. a yd.? Thus, 20 cts. })$622.00 10 cts. of 20) 124.40, price at 20 cts. NOTE.-You proceed as in the foregoing example when the price is no even part of $1. 5. What will 642 yds. of cloth cost, at 17 ets. a yd.? 4. When there are several denominations in the quantity, and the price is dollars and cents. RULE.-Multiply the dollars in the price into the whole numbers of the quantity; work for the cents in the price as in the preceding cases; and for the parts in the quantity, divide by the even parts of the price of a whole number, add the quotients and products; the sum is the answer. Questions. What is Practice?-What is the first case?-How do you proceed? When the price is dollars and cents, how?-When do you multiply the quantity, in pounds, by half the price, to find what?-What is the second case? Then how should you proceed?-What is the third case?-How? Why?-How do you proceed when the price is no even part of $1?-What is the fourth case?-What is the rule? Example.-What will 112 cwt. 3 qr. 14 lbs. of sugar cost, at 12. 25 cts. a hundred? Each of which being forms for solving others, that are similar. 1. Divide $373. 50 cts. in shares, among A, B, and C, in the manner, that B may have twice as much as A, and C twice B's. 2. Divide $600 among 5 persons, in the manner that B may have two more than A, C two more than B, &c. Thus, $6005 $120, the mean or middle share. Here is an odd number of persons, in which case, subtract the ratio from the mean; from that, subtract the ratio again, down to the number less than the mean, and add the ratio to those shares to be larger. Ans., C's, (3d, or mean,) $120; B's, $118; A's, $116, those less than C's. And, D's, $122; E's, $124, those larger than C's. 3. Divide $1600 among four persons, that the second may have one more than the first, the third one more than the second, &c. OPERATION. 16004-400, the middle share. Then, B's, $399.50-100 cts., ratio, = $398.50 cts., Ans. 400.00 .50 cts., ratio, $399.50 cts., B's. $400.50 cts., C's. $401.50 cts., D's. 400.00+.50 cts.,ratio, NOTE.-It may be noticed that we constitute two means, from the one found, as above, and then proceed as before. In this case there is an even number of persons. 4. What is the value of a piece of land, valued by A, at $10; by B, at $11.50; by C, at $12.30; and by D, at $13.40 an acre? Thus, A+B+C+D=4 appraisers. 10+11.50+12.30+13.40,4$11.80 cts, Ans. And, 5. E and F proposed to swap watches-agreeing to refer it to A, B, C, and D to say how they should exchange: A marked that E should have $4; B said E should have $5; C said E should have $2; but D said F should have $3.50 cts.; which receives the boot, and how much? Thus, 4+5+2-11, -3.50-7.50, 4-$1.87 cts. to E. Questions. Of examples being forms for solving others, which are similar, what case is given?-What rule?-To the second example, what is required?-How do you proceed?-When do you proceed thus?-When is the third example?-How do you proceed? For what is the fourth example? How do you proceed?-For what is the last example?-How do you proceed? DOMINICAL LETTERS: These are the first seven characters of the English alphabet, as the representatives of the days in a week, one to each. Because a given day of a known month will be on the day of the week of its day after, for a next year; hence to find the Dominical letter for any year, is to find which letter represents the first Sabbath in January, it being the letter for that year, leap year excepted, which has two, the first serving to the 24th of February, and the other for the rest of the year. The last is found as for a common year, and the first letter following it, will be the Dominical letter for January, and to February 24th. Hence, knowing the Dominical letter for the first Sabbath, we reckon in a retrograde order to find A, which is always prefixed to January 1, and thus we find the day of the week, on which the required year comes in. Consequently, 1. To find the Dominical letter for any year. RULE.-Divide the given year and its fourth part, by 7. Subtract the remainder, if any, from 7, when the given date is prior to A. D. 1800. If later, subtract it from 8, and the remainder will be the Dominical letter, reckoning from A toward G. Example. Find the letter for A. D. 1778. Thus, 17784-444, +1778-2222,7-317, and 3 over. Then, 7—3—4—D, Ans. But to the 24th of February, it would have been E, had it been leap year. 2. To find the day of the week, on which any given day of a known month will occur. RULE.-Find the day of the week answering to Jan. 1 of the given date. Add together the days contained in each month, from the first day of that year to the proposed day of the month, inclusively. Divide this sum by 7, and if anything remain, count that number forward, beginning with the first day of January. But if nothing remain, the day of the week, preceding Jan. 1, answers to the proposed day. Questions.-What sometimes represent the days of the week?-When?-Why? What represents Jan. 1?-What does the dominical letter represent?-Except when?-How is it then found?-How does it serve?-How do you reckon to A?-When do you reckon from A, toward G?-Repeat the rule to the first case. Repeat the preceding rule. Example.-A. D. 1786, May 5, is what day of the week? And, Jan. 31+28+31+30+5=125 days from Jan. 1 to May 5. 7) 125(17, and 6 remainder, that is, 6 d. from Jan. 1., is Friday; hence, May 5 is Friday, Ans. SUNDRY RULES: For the solution of questions that have hitherto been solved by an Algebraic process. In these rules some help may be derived from the following: AXIOMS. 1. "The product of an even and an odd number, or of two even numbers, is even. 2. The product of any two odd numbers, is an odd number. 3. If an odd number measure an even number, it will measure the half of it. 4. Two quantities respectively equal to a third, are equal to each other. 5. The product arising from two different prime numbers, cannot be a square. 6. The product of any two different numbers prime to each other, cannot make a square, unless each of those numbers be a square, 7. If equal quantities be added to, subtracted from, multiplied or divided by equal quantities, the wholes, remainders, products, and quotients will be respectively equal." NOTE. It is believed by a little consideration, the following rules will appear obvious. 1. The sum of two numbers and their quotient given, to find those numbers. RULE. Add 1 to the quotient, and divide the sum of the two numbers by this sum, which will give the less number. Subtract the less number from the sum given, and you have the greater. Example.-Divide 100 into two such parts, that if the greater be divided by the less, the quotient may be 30. er part. 30+1) 100 (37, less part. Then, 100-3,7-9631, great 2. The difference of two numbers and the quotient given, to find those numbers. RULE.-Divide the difference of the two numbers by the quotient, less 1, and you will have the less number. Add the less number to the difference, and this sum is the greater. EXAMPLES. 1. A greyhound, in pursuit of a hare, run three times as fast as the hare; and when he overtook her, he had run 30 rods more than the hare. Required the rods that each run. Thus, 30÷3-1*--15 rods, the hare run. Then, 15+30=45 rods, the hound run. 2. A and B started at opposite points, to skate to the other's starting point; distance 8 miles. A, by having the advantage of * The line over 3-1, shows the difference, 2, to be the divisor. |