2. Find the least common multiple between 12, 25, 30, and 42? Ans. 2100. 3. Find the least common multiple of 12, 16, 20, and 30? Ans. 240. ་ 4. What is the least common multiple of 25, 35, 60 and 72? Ans. 12600. 5. What is the least common multiple that will measure 3, 4, 8, and 12? Ans. 24. 6. What number is the least, that 7, 8, 16, and 28, will measure? CASE IV. Ans. 112. To reduce fractional parts of a dollar to cents. RULE.-Multiply the numerator by 100 (because 100 cents is a dollar) and divide by the denominator. 1. Bring of a dollar to cents? 100 8)500 62 or 62 cts. may be reduced to because 4 will divide the numerator and denominator without a remainder, thus: 4) = reduced to its lowest terms. How many cents in of a dollar? Ans. $.50. Ans. $.871. Ans. $.37. Ans. $.12. Ans. $.75. of a dollar? Ans. $.90. of a dollar? Ans. $.60. of a dollar? Ans. $.80. To reduce fractions to their lowest denominations, and also into cents. 1. Reduce $1 to its lowest terms? Ans. $ or 19 2. Reduce $3 to its lowest terms? Ans. $ cts. or 75 cts. RATIO OF FRACTIONAL PARTS OF A DOLLAR. 1. What is the ratio between 1 and 1? Ans. 2. From the preliminary examples it is evident that 2 quarters are = to a half; therefore, the ratio is as 1 to 2 as required. 2. What is the ratio between 1 and 2? Ans. 3. USEFUL THEOREMS IN FRACTIONS. Note to Teachers.-The learner should be required to recite these theorems and to apply them practically. THEOREM I. TO ADD or SUBTRACT fractions which have the same common denominator, the sUM or DIFFERENCE of their numerators must be taken, and the common denominator written under the result. THEOREM II. To reduce fractions to the same denominator, the two terms of each of them must be multiplied, by the denominator of the other. THEOREM III. A fraction can be multiplied in two ways; namely, by multiplying its numerator or dividing its denominator. Thus, multiply' by 5 which reduced is or == 3 35 (by dividing the denominator 30 by 5) = 1. THEOREM IV. A fraction can be divided in two ways, by dividing its numerator or multiplying its denominator: thus, divide by 4 by dividing the numerator we get and by multiplying the denominator x 4 = = which is exactly the same. THEOREM V. Multiplication alone, according as it is performed on the numerator or denominator, is sufficient for the multiplication and division of fractions; that is, when you multiply the numerator you INCREASE, and when you multiply the denominator you DECREASE. CASE I. By multiplying the numerator the fraction is CASE II. By dividing the numerator the fraction is THEOREM VI. divided. To multiply a whole number by a fraction. RULE.--Multiply the number by the numerator and divide by the denominator: or divide the number by the denominator and multiply the quotient by the numerator. EXAMPLE.-Multiply 20 by first, 20 x 30 = 15 20 5 x 3 = 15. COROLLARY. Every common divisor of two numbers must also divide the remainder resulting from the division of the greater of the two by the less. ADDITION. CASE VI. When the numerators are alike and not more than a unit. RULE.-Multiply the numerator and denominator of the fraction having the least denominator by the common measure of the fractions. RULE. Add the denominators together for a new numerator, and multiply them together for a new denomi nator. CASE VII. When the numerators are alike and more than a unit. RULE. Add the denominators together, and multiply their sum by the common numerator, and the product will be a new numerator; also, the product of the denominators will be a common denominator. Add and, here 4 and 7 make 11, which multiply by the numerator 3, which is common to both. × 7, here and THIRD METHOD. by multiplying the numer ators alternately by the denominators. Add and 1, Add and f, Add and TT, = Add and · Here the ratio between the denomi nators is as 1 to 7; therefore, × 7 or answer. = and make TT RULE.-Find a common denominator by reducing the fractions to the lowest terms. Multiply all the divisors together 4 x 5 x 2 = 40 common denominator. CASE IX. To add mixed fractions whose numerators and denominators are unlike. Add $153 $35 () The operation can be performed thus, by cross multiplication, 24 + 2018 reduced, from whence the following Rule is deduced: multiply each numerator by all the denominators, except its own, for a new numerator, and all the denominators together for a new denominator. Add 1,,, and together, (say dollars.) |