If we represent the two quantities by the same letters as before, we shall have a + b, their sum, to be multiplied by a- - b, their difference. To express this formula in words: The product of the sum of two numbers, multiplied by their difference, is equal to the difference of their second powers. 10. Multiply 9 +7 by 97, according to this rule. 12. Multiply 24+ 15 by 24 — 15. 13. Given 9310 and 9298, and required the difference of their squares. The answer to this question can be far more easily obtained by means of the above formula, than by actually involving each of these numbers, and subtracting the second power of the one from the second power of the other. For, if we multiply their sum by their difference, the product will be the difference of their second powers. The last question is solved by means of the formula, in the following manner : 9310 9298 18608, the sum of the numbers. 9310 9298= 12, their difference. 223296, the difference of their squares. This principle should be remembered; as calculations of this sort may be very much abridged by an application of it, when the given numbers are large. 14. Given 23136 and 21947, and required the difference of their second powers. 15. Given 15925 and 14987, and required the difference of their second powers. 16. Given 987264 and 978351, and required the difference of their second powers. 17. Given 999999 and 999993, and required the difference of their second powers. 18. What is the second power of 28? ANS. 784. Let a = 20, and b = 8; then a + b = 28. a+b a+b a2 + ab + ab + b2 a2+2ab+b2. Hence it appears, that the second power of the sum of two quantities is equal to the sum of their second powers, increased by twice their product. That is, the second power of the difference of two quantities is equal to the sum of their second powers, diminished by twice their product. 20. An artist sold to A and B two pictures for 573 dollars; and B's picture cost 57 dollars more than A's. What was the price of each? Let the price of A's picture. Then x + 57 = the price of B's. 2 x 57357 516, by transposition, x+57 315. ANS. A's, $258; B's, $315. In this question, we are required to divide 573 into two such parts, that the greater shall exceed the less by 57. In other words, having the sum and the difference of two numbers given, we are required to find those numbers. Let the sum of the numbers be represented by a, and their difference by b; and let the smaller number. Then, as the difference of the numbers is b, we have x + b = the larger number. To express this formula in words: If from the sum of two numbers we subtract their difference, half of the remainder will be equal to the smaller number. Therefore, x + b =a=b+b, by adding b to each Or x + b = = b+22 α b+2b ; x + b = a+", the greater number. [member. That is, if to the sum of two numbers we add their difference, half of the amount will be equal to the greater number. 21. The sum of two numbers is 175, and their difference 81. What are the numbers? Here, a 175, and b = 81. 22. The salaries of two men, A and B, amount to $3529 per annum; and A receives $721 more than B. What is the salary of each? 23. A gentleman paid $387 for a horse and chaise: and the chaise cost $75 more than the horse What was the price of each? 24. A man left an estate of $9134, to be so divided between his widow and son, that the former should receive $1486 more than the latter. share of each? What was the 25. A's farm contains 12 acres more than B's; and the two farms together contain 162 acres. Required the number of acres in each. 26. Says A to B, "If you will give me 16 dollars, I shall have as much money as you; and we both have 130 dollars." How much money has each ? 27. A cistern has two cocks, which together will discharge 203 gallons of water in an hour, and one cock will discharge 13 gallons more than the other. How much will each discharge if running alone? 28. How much tea, at 5s. per pound, must be given in barter for 75 gallons of wine at 8s. per gallon? Let the pounds of tea. Then 5 x 75 x 8 = 600; and x 120. ANS. 120 lbs. In general terms. How much tea, at a per pound, must be given in barter for b gallons of wine at e per gallon? To express Let the pounds of tea. Then a xb c, this formula for Barter in words: Multiply the given quantity by its price, and divide the product by the price of the quantity required. 29. How much oil, at 4s. a gallon, must I give in barter for 58 barrels of flour at 5 dollars a barrel? |