PREPARE the fractions, if necessary, by reduction; invert the divisor, and proceed as in multiplication. The Rule of Three in Vulgar Fractions. THE operation of the Rule of Three in Vulgar Fractions, whether direct, inverse, or compound, is performed in the same manner and agreeably to the principles laid down in whole numbers under these rules. When the question is in direct proportion, prepare the terms by reduction, and invert the first term; then proceed as in multiplication of fractions. EXAMPLE. 1. If of a yard of cloth cost of a dollar, what will of a yard come to? 2. If 1 x 3 x 8 24 of a ton of iron cost 164 dollars, what will of a ton come to? Ans. 211 dolls. 284 cts. 3. A person having of a coal mine, sells of his share for 171 dollars, what is the value of the whole mine at the same rate? Ans. 380 dollars. 4. At of a dollar per yard, what will 42 yards come to? 20 Ans. 35 dollars. 5. A gentleman owning of a vessel, sells of his share for 312 dollars, what is the whole vessel worth? Ans. 1170 dollars. 6. If 13 bushel of apples cost 793 cents, what will 33 bushels cost at the same rate? Ans. 202,3 cents. 7. If of a ship be worth 175 dollars 35 cents, what part of her may be purchased for 601 dollars 20 cents? Ans. SECTION 7. INVERSE PROPORTION. RULE. PREPARE the question as in direct proportion, invert the third term, and proceed as in multiplication of fractions. EXAMPLES. 1. How much shalloon yard wide, will line 4 yards of cloth 1 yard wide? 2. If 6 hundred weight be carried 22 miles for 25 dollars, how far may 1 hundred weight be carried for the same money? Ans. 145 miles. 3. If 12 men can finish a piece of work in 373 days; how long will it take 16 men to do the same work? Ans. 281 days. 4. A lends to B 100 dollars for 6 months; what sum should B lend to A for 35 years, to requite his kindness? Ans. 14122 dollars. 5. How many feet long must a board be, that is of a root wide, to equal one that is 201⁄21⁄2 feet long, and 2 of a foot wide? Ans. 174 feet long. 6. In exchanging 20 yards of cloth of 14 yard wide, for some of the same quality of yard wide, what quantity of the latter makes an equal barter? Ans. 341 yards. PART VII. EXTRACTION OF THE ROOTS, AND COMPARATIVE SECTION 1. Involution, or the Raising of Powers. INVOLUTION is the multiplying of a given number by itself continually, any certain number of times. The product of any number so multiplied into itself, is termed the power of that number. Thus 2x2=4= the second power or square of 2. 2×2×2=8= the third power or cube of 2. 2×2×2×2=16= the fourth power of 2, &c. The number denoting the power to which any given sum is raised, is called the index or exponent of that power. If two or more powers are multiplied together, the ir product will be that power, whose index is the sum of the exponents of the factors. Thus 2x2=4, the 2d power of 2; 4×4=16, the 4th power of 2; 16×16=256, the 8th power of 2, &c. 2561 512 6561 19683 262144 24 8 16| 321 641 128 9th power EXAMPLE. 1. What is the 3d power of 15? 15 x 15 x 15=3375 2 What is the 4th power of 35? 5. What is the 4th power of ? Ans. Ans. 1500625. Ans. 1,092727. Ans. ,000000707281. 8 1 25 SECTION 2. Of Evolution, or the Extracting of Roots. EVOLUTION is the reverse of involution. For as 3x 3=9 ×3=27, the power; so 27÷3=9÷3=3, the root of that power. Hence the root of any number, or power, is such a number as being multiplied into itself a certain number of times, will produce that power. Thus, 4 is the square root of 16, for 4×4-16; and 5 is the cube root of 125, for 5x5x5=125. SECTION 3. THE SQUARE ROOT. ANY number multiplied once into itself is called the square of that number. Hence, to extract the square root of any number, is to find such a number as being multiplied by itself, will be equal to the given number. RULE. 1. Point off the given sum into periods of two figures each, beginning at the right hand. |