§ XCVII. 1. I wish to mix 2 gallons of brandy, at $1.50 pr. gal. with rum at 80 ets. pr..gal., so that the mixture may be worth $1.00 pr. gål. What quantity of rum must I take? 100 80. 50 Here I obtain 20 gals. of brandy by link. ing. This, by the terms of the question, 20 must be diminished down to 2, and there. fore, it is necessary that the 50 gals. of rum should be diminished in the same ratio. *. 1.50 (§ Xci.) 20: 2::50:50×2÷20=5 gals. rum Ans. The proportion then seems to be, AS THE RELATIVE QUANTITY OF THE LIMITED INGREDIENT, IS TO THE GIVEN QUANTITY; SO IS THE RELATIVE QUANTITY OF EACH OTHER INGREDIENT TO THE ABSOLUTE QUANTITY OF THAT INGREDIENT REQUIR ED. And, hence, the rule, MULTIPLY EACH RELATIVE QUANTITY BY THE GIVEN QUANTITY OF THE LIMITED INGREDIENT; AND DIVIDE THE PRODUCT BY THE RELATIVE QUANTITY OF THE SAME INGREDIENT. THE SEVERAL QUOTIENTS WILL EXPRESS THE QUANTITIES REQUIRED. 2. A merchant has spices at 32 cts. 40 cts. and 64 cts. pr. lb. He wishes to mix 5 lbs. of the first with the others, so that the com. pound may be worth 48 cts. How much of each must he use? Ans. 5 lb. of the 2d, and 7 lb. 8 oz. of the third. 3. A farmer wishes to mix 16 bu. of rye worth 50 cts. pr. bu. with corn at 40 cts. and oats at 30 cts. pr. bu. so that the mixture may be worth 37 cts. pr. bu. What quantities must he take of each ? When the whole compound is limited; that is, when it is desired that the whole mixture should amount to a certain quantity, the process is somewhat similar. For, since each individual quantity found by linking, is a relative quantity, it is evident, that the sum of the whole can only be a relative sum. Hence, if this sum be increased or diminished, the quantity of each ingredient must be increased or diminished in the same ratio. This is done by Simple Proportion, or by analysis. 4. A flour merchant, having flour at $4, $6, and $8, pr. bar. sold 120 bar. at the average price of $6.50 pr. bar. How many barrels of each kind must he have sold, in order not to have gained nor lost? NOTE. This case is evidently the same as if the flour had been mixed, as far as calculation is concerned. Here, by linking, I obtain 150 & 150 & 300, the several quantities of the Simples. The sum is 600. Now, by the terms of the question, this must be diminished down to 120, and of course, the quantity of each Simple, must be diminished in the same ratio. (§ XCI.) 600 120 150 120X150÷600=30 at $4, and also at $6, since its relative quantity is likewise 150. Also 600 120 300: 120X300÷600-60 at $8. The proportion, then, is, AS THE SUM OF THE RELATIVE QUANTITIES, IS TO THE GIVEN QUANTITY; SO IS EACH RELATIVE QUANTITY, TO THE ABSOLUTE QUANTITY OF THAT INGREDIENT REQUIRED. And the rule, MULTIPLY EACH RELATIVE QUANTITY BY THE GIVEN QUANTITY, AND DIVIDE THE PRODUCT BY THE SUM OF THE RELATIVE QUANTITIES. THE SEVERAL PRODUCTS WILL EXPRESS THE QUANTITIES REQUIRED. It may be mentioned, though perhaps the pupil has always made the discovery himself, that this process is precisely the same with that employed in Simple Fellowship. 5. How much gold, that is 15, 17, 18, and 22 carats fine, must be mixed together to form a compound of 40 lb. 20 carats fine? Ans. 5 lb. of 15, 17, and 18, and 25 lb. of 22. 6. A man would mix 100 lbs. of sugar, at 8 cts. 10 cts. and at 14 cts. pr. lb. so that the compound may be worth 12 cts. pr. lb. What quantity of each must he use ? Ans. 20 lb. at 8, and at 10 cts., and 60 lb. at 14 cts. 7. A grocer has currants at 5 cts. 7 cts. 10 cts. and 12 cts., and he wishes to mix them so as to sell them at 8 cts. pr. lb. How many lb. of each sort must he take? This and some of the following admit of a great variety of answers. 8. A farmer mixed meal of the value of 25 cts. pr. bu. with other kinds of the values of 30 cts., 35 cts. and 18 cts. The compound contained 4 bu. and was worth 28 cts. pr. bu. What quantity did he take of each? 9. If a grocer make a mixture of teas of the following prices, pr. lb. viz. $1.25, $1.40, $1.63, and $1.75, so that the mixture may be sold at $1.50; how much does he take of each kind, supposing the whole mixture to contain 120 lb. ? 10. If a grocer fill 3 wine hogsheads with water and liquors at $1.20, $1.30, and $1.40 a gallon; how much does he take of each, supposing the compound worth $1.35 pr. gal. ? 11. In a brewery there are in an upper room 3 vats, each capable of containing 120 gals., and in a lower room 1 vat capable of holding 192 gals. The lower vat is empty and the three upper ones are filled with beer, worth respectively 12 cts. 20 cts. and 30 cts. pr. gal. Pipes are set running at the same moment, from each upper vat to the lower, and in 1 hour, exactly, it is filled. The owner then finds, on mixing what remains in the upper vats, that the compound is worth 25 cts. pr. gallon. What is the value of the mixture in the lower vat, and in what time would each pipe separately have filled it ? A. Price, $0.168.-Times, for first two, 1 h. 8 m.-for last, 16 h. 12. Suppose there are vats situated as the above, the upper ones being of the same dimensions, and containing beer of the values 14 cts. 18 cts. and 32 cts. pr. gal. Suppose the lower one filled as before, in an hour, and that there remains of the first kind of beer, 40 gals., the price of the mixture in the room above, being 25 cts., as before. Required the dimensions of the lower vat, the value pr. gal. of the mixture contained in it, and the time in which each pipe will fill it. INVOLUTION. MENTAL EXECRISES. XCVIII. 1. How much is 3 times 3? 2 times 2? 4 times 4? 2. How much is 5 times 5? 6 times 6? 7 times 7? 8 times 8? 9 times 9? 10 times 10? 11 times 11? 12 times 12 ? 3. If you multiply 2 into 2, and that product, again by 2, what result do you obtain ? 4. How much is 3 multiplied by 3, and the product again by 3 ? 5. How much is 4X4×4? 6. How much is 2×2×2×2 ? 7. How much is 3×3×3×3 ? 8. How much is 2×2×2×2×2? 9. How much is 2×2×2×2×2×2? Here you perceive that you have been making continued multiplications of numbers into themselves, or, into the same numbers. THE PRODUCT, WHICH ARISES FROM MULTIPLYING A NUMBER, ONE OR MORE TIMES BY ITSELF, IS CALLED A POWER. THE PROCESS OF FINDING POWERS IS CALLED INVOLUTION. Involution, of course, consists in the continued multiplication of a number by itself. This number, and in fact, any number is called the FIRST POWER, of itself. It is likewise called the Roor of the other powers; since they seem to grow up out of it. Different powers have different names. Thus we have the second, third, fourth, fifth, &c. powers. 2X2-4 is the second power of 2, because 2 is used as a factor, twice, in finding it. 3X3X3-27 is the third power of 3, because 3 is used as a factor three times in finding it. 3X3X3X3=81 is the fourth power of three, because 3 is used as a factor, four times, in finding it. Hence, A POWER TAKES ITS NAME, OR NUMBER, FROM THE NUMBER OF TIMES THE ROOT IS CONTAINED IN IT AS A FACTOR. The pupil must carefully distinguish between this number, and the number of multiplications, necessary in finding a power. For in finding the second power of 2, for instance, we multiply 2 into 2 and call the product the second power, because 2 is used twice as a factor in obtaining it. But it will be seen that but one multiplication takes place. So also the expressions 3X3, 5X5X5, 7X7X7X7 denote, respectively the second power of 3, the third power of 5, and the fourth power of 7. The numbers, themselves, of course, are the factors, and the crosses represent the number of multiplications. In every instance the latter number is a unit less than the former. Hence, THE NUMBER OF A POWER IS ALWAYS A UNIT GREATER THAN THE NUMBER OF MULTIPLICATIONS NECESSARY TO PRODUCE IT. A short mode of expressing powers is often employed for the sake of convenience. For instance, instead of writing out the words, fourth power of 7, in full, we write the root, 7, simply, and the number of the power, at the right of it a little elevated, thus, So, the third power of 8 is written, The eighth power of 5, This mode of writing is called indicating the power. 74 83 58 &c. The small figure at the right is called the INDEX, or EXPONENT, of the pow er. Thus is the index of the eighth power of 5, in the last example; and shows, that, in order to obtain the power, 5 must be employed 8 times as a factor. Hence, THE INDEX, OR EXPONENT OF A POWER IS THE NUMBER, FROM WHICH THE POWER DERIVES ITS NAME, AND SHOWS HOW MANY TIMES THE ROOT MUST BE USED AS A FACTOR TO PRODUCE THAT POWER. The second power is likewise called the SQUARE; because, Al if the root be one side of a figure of that form, the power will be the surface contained within it. Thus the figure A B C D represents a square, whose side is 3. The surface, it will be seen, is 9, which is 3X3 or 32. D The third power, is, also, called the CUBE; because, if the root be one side of a solid of that description, the power will express the solid contents. Thus, in the figure we have a cube, whose side is 2. It will be readily seen, that the solidity is 8=2X2X2 23. The fourth power is likewise called the BIQUADRATE; which means twice-squared, and is applied to this power, because it is B the same as the square of the square; the original root being a factor 4 times in each case. The fifth power is called the SURIOLID; which means beyond a solid, and is perhaps applied in distinction from the cube, which has a solid to represent it. The sixth power is called the sQUARE-CUBED; which term explains itself. All these powers are likewise called by their numbers; and except the square and cube, the above names are little used. We have seen that powers may be indicated by exponents. When a power is found by actual multiplication, involution is said to be performed, and the number is said to be involved. Hence, to involve a number, MULTIPLY IT BY ITSELF UNTIL IT IS EMPLOYED AS A FACTOR, AS OFTEN AS THERE ARE UNITS IN THE INDEX OF THE POWER, TO WHICH IT IS TO BE RAISED. Involution may be abbreviated, by multiplying together any two or more pow ers of the same root, already found, whose indices, together, make up the required index. For the sum of the indices, shows how often the root is contained as a factor in the product. EXAMPLES FOR PRACTICE. Ans. 13X13=169. 10. What is the square of 13? 27. A. 128. Hence, a Fraction is involved, by involving both numerator and denominator. 81 19. What is the fourth power of? A. 25. 20. What is the square of 5? A. 30. 21. What is the square of 30? A. 915 22. Perform the involution of 85. A. 32,768. 1331 23. Involve, and to the third power each. A. ; 111: 418. 39319; 1728 A. 390,625. A. 2.985984. The 6th power of 41. Involve 31111. 27. Find the 11th power of 67. power of 821. Involve 723. EVOLUTION. MENTAL EXERCISES. § XCIX. 1. 4 is a square or 2d power; what is its root? that is, what number, multiplied by itself, will produce 4? 2. 9 is a square; what is its root? that is, what number, multiplied by itself, will produce 9 ? 3. 16 is a square; what is its root? 36 is a square; what is its root? 4. 8 is a cube, or 3d power; what is its root? that is, what number, taken 3 times as a factor, will produce 8 ? 5. 27 is a cube; what is its root? 81 is a square; what is its root? 6. is a square; what is its root? is a 4th power; what is its root? Hence, the root of a Fraction is found by taking the roots both of the numerator and denominator. 7. is a square; what is its root? 64 is a cube; what is its root? 8. is a cube; what is its root? 25 is a square; what is its root ? 9. Find the square root of 64. The square root of 49. 10. Find the square root of 4. Of 28. Of 94. 8.1 11. Find the 4th root of 16. Of 81. Of 100. 12. Find the square root of. Of 25%. Of 10 Of T 144 The requisition, in the above examples, will be seen to be exactly the reverse of that, contained in the questions in the last §. There, a root was given to find a power. Here, a power is given to find a root. It may be well to give a more particular definition of a root, than we have yet done. A ROOT OF ANY NUMBER IS A FACTOR, WHICH, MULTIPLIED ONE OR MORE TIMES INTO ITSELF, WILL PRODUCE THAT NUMBER. 2 THE PROCESS OF FINDING ROOTS IS CALLED EVOLUTION. It consists of course is ascertaining the factor, which multiplied into itself, one or more times, will produce the given number. Of course it is the opposite of Involution. It may briefly be defined, the resolving of a number into two or more EQUAL FACTORS. Different roots have different names, corresponding to those of the different powers. 2 is the second or square root of 4, because 2 must be taken twice as a factor to produce 4. Thus 2X24. 3 is the third or cube root of 27 because it must be used as a factor three times to produce 27. Thus, 3X3X3=27. 3 is also the fourth or biquadrate root of 81 because it must be used four times as a factor to produce 81. Thus, 3X3X3X3=81. Hence, |