1 2. What mixed number is equivalent to 131? 3. What mixed number is equivalent to 12345 ? Ans. 647. 8. What mixed number is equivalent to 457354? Ans. 12941129411. 37 9. What mixed number is equivalent to 227322? 433 337 Ans. 5243. 6 10. What mixed number is equivalent to 22912? 4583 5 05 Ans. 491. 11. What mixed number is equivalent to 3573? 4927 Ans. 7. 19. To reduce a mixed number to its equivalent improper fraction, we have this RULE. Multiply the integral part of the mixed number by the denominator of the fractional part; to the product add the numerator of the fractional part; the sum will be the numerator of the improper fraction, under which place the denominator of the fractional part. This rule is obviously correct, since it is the reverse of the rule under Art. 18, where a reverse operation was required to be performed. EXAMPLES. 1. Reduce 13 to an improper fraction. Multiplying the integer 13 by the denominator 7, we obtain 91; to which, adding the numerator 6, we get 97 for the numerator of the improper fraction; .. the improper fraction equivalent to 139 is 27. 2. What improper fraction is equivalent to 1278? Ans. 3835. 3. What improper fraction is equivalent to 189104 ? Ans. 132874. 4. What improper fraction is equivalent to 4925361? Ans. 1972. 13 9. What improper fraction is equivalent to the mixed number 683375? Ans. 205053072 44643 10. What improper fraction is equivalent to the mixed number 11223344,5? Ans. 123456789. 20. Reduce the compound fraction of to its equivalent simple fraction. of can be obtained by dividing the value of the fraction by 4, which (by Prop. II., Art. 16,) can be effected by multiplying the denominator by 4; Again, of is obviously three times as great as of;.. to obtain of, we must multiply 7 4 x 11 by 3, which (by Prop. I., Art. 16,) can be done by multiplying the numerator by 3; hence, we have of 3 4 Hence, to reduce compound fractions to their equivalent simple ones, we have this RULE. Consider the word or, which connects the fractional parts as equivalent to the sign of multiplication. Then multiply all the numerators together for a new numerator, and all the denominators together for a new denominator, always observing to reject or cancel such factors as are common to the numerators and denominators, which is the same as dividing both numerator and denominator by the same quantity, and (by Rule under Art. 17,) does not change the value of the fraction. EXAMPLES. 1. Reduce ofofofto its equivalent simple fraction. Substituting the sign of multiplication for the word of, we get ××× First canceling the 8 of the numerator against the 2 and 4 of the denominator, by drawing a line across them, we get $ 1 3 5 X X 5 12 Again, canceling the 3 and 5 of the numerator against the 15 of the denominator, we finally obtain 2. Reduce of of of of to its simplest form. First, canceling the 7 and 5 of the numerator against 3 147 4 the 35 of the denominator, we get XX X X $ 8 9 11 Again, canceling the 7 of the denominator against a part of the 14 of the numerator, and the 3 of the numerator against a part of the 9 of the denominator, we obtain Finally, canceling the 2 and 4 of the numerator against 8 of the denominator, we get NOTE. We have written our fractions several times, in order the more clearly to exhibit the process of canceling. But in practice, it will not be necessary to write the fraction more than once. It will make no difference which of the factors are first canceled. When all the common factors have, in this way, been stricken out, the fraction will then appear in its lowest terms. The student will find it to his interest to perform many examples of this kind, as this principle of canceling will be extensively employed in the succeeding parts of this work. 3 36 10 3. Reduce of of 2 of 33 to its simplest form. Ans. 4. Reduce of 1 of 1 of 2 of 2 of of to its simplest form. Ans. 15. 5. Reduce of of 18 of 14 of to its simplest form. Ans. T 6. Reduce of 3 of 3 of 3 to its simplest form. Ans. 6292 32085 7. Reduce of of of of to its simplest form. Ans. 736 8. Reduce of of of of of of of to its simplest form. Ans. . 9. Reduce of 1 of 9 of 1 to its simplest form. 10. Reduce of of of to its simplest form. 21. To reduce fractions to a common denominator, we have this RULE. Reduce mixed numbers to improper fractions-compound fractions to their simplest form. Then multiply each numerator by all the denominators, except its own, for a new numerator, and all the denominators together for a common denominator. It is obvious that this process will give the same denominator to each fraction, viz: the product of all the denominators. It is also obvious that the values of the fractions will not be changed, since both numerator and denominator |