Fractions. Reduce to equivalent fractions having their 1. c. d. (a) Find the sum of the twenty-five fractions.* 179. TO ADD COMMON FRACTIONS. RULE. Reduce the fractions if necessary to equivalent fractions having a common denominator, add their numerators, and write their sum over the common denominator. EXAMPLE. Add 1, 17, and 3. (1) The 1. c. m. of 45, 30, and 60 is 180. (a) Find the sum of the ten sums.* (For a continuation of this work, see page 231.) *This may be omitted until the subject of fractions is reviewed. Algebraic Fractions. α X 6 180. The expressions b' 4' cd' are algebraic fractions. The above expressions are read, a divided by b, x divided by 4, 6 divided by cd. Observe that to reduce a fraction to its lowest terms we have only to strike out the factors that are common to its numerator and denominator. Algebraic Fractions. 182. Reduce to higher terms: 2 a bc abc to a fraction whose denominator is abc. 1. Change 2a xa 2a2 3xy = Let a = 2, b = 3, and c = 5, and verify the reduction. 3х ху 2ay × y to a fraction whose denominator is 2ay". Give any values you please to a, x, and y, 2ay and verify the reduction. 183. Reduce to equivalent fractions having a common denominator: Since the common denominator must be exactly divisible by each of the given denominators, it must contain all the prime factors* found in either of the given denominators. The new denominator must therefore be a xa x b × d = a2bd; a2bd ÷ ab = ad; a2bd÷ a2d = b. Give any values you please to a, b, d, x, and y, and verify. The common denominator must contain the factors a, b, b, c, c. Reduce and verify. *Since the numerical values of the letters are unknown, each must be regarded as prime to all the others. The prime factors, then, in the first denominator are a and b; in the second, a, a, and d. Geometry. 184. QUADRILATERALS. 1. All the geometrical figures on this page are quadrilaterals; that is, each has four sides. 2. The first four figures are parallelograms; that is, the opposite sides of each figure are parallel. 3. The first two figures are rectangular; that is, their angles are right angles. 4. The first and third are equilateral; that is, the sides are equal. 5. There is one equilateral rectangular parallelogram. Which is it? 6. There is one equilateral parallelogram that is not rectangular. Which is it? 7. There is one rectangular parallelogram that is not equilateral. Which is it? 8. The sum of the angles of each figure on the page is equal to right angles. 9. Tell as nearly as you can the size of each angle of each figure. 185. Miscellaneous Review. 1. The difference of two numbers is 37411; the smaller number is 24317. What is the larger number? 2. The difference of two numbers is a; the smaller number is b. What is the larger number? 3. James had a certain number of dollars and John had three times as many; together they had 196 dollars. How many had each? (x+3x=196.) 4. William had a certain number of marbles; Henry had twice as many as William, and George had twice as many as Henry; together they had 161. How many had each? (x+2x+4x= 161.) 5. Divide 140 dollars between two men, giving to one man 30 dollars more than to the other. (x+x+30= 140.) 6. By what integral numbers is 30 (2 × 3 × 5) exactly divisible besides itself and 1? 7. By what is abc (a xbx c) exactly divisible besides itself and 1? (1) How many times is a contained in abc ? Observe that a number composed of three different prime factors has exact integral divisors. |