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It was shown in Arithmetic, Art. XXII, that a common denominator may frequently be found much smaller than that produced by the above rule. This is much more easily done in algebra than in arithmetic. -
9. A *— *, and s. dd together 5 co’ 5 o' and cg Here the denominators will be alike, if each be multiplied by all the factors in the others not common to itself. If the first be multiplied by e.g., the second by cog, and the third by b c e, each becomes b c e g. Then each numerator must be multiplied by the same quantity by which its denominator was multiplied, that the value of the fractions may not be altered.
But if they are reduced to a common denominator, the numerators may be subtracted.
Ans. 3 arc – 2 e
- 2 b co t 2 a b 5 m n 22. Subtract ** from ***. u 3 cod; r 2 coa: 3 c d c de 23. Subtract – o – from Ulotrac 7 b” m” a. ro 21 b moo 24. Subtract od from #. na'y' 2m y 5 a y 2 ry 5. tract — & fi 25. Subtrac 2 m” s” t r0m 3 m"s to 17 ra.” 15 a. 26. Subtract of Zoo from ** * 9. uloorac 3 m b ro 5 m" bo 13 m" 11 r s 27. Subtract # from 3 mo' 28. From 13 a c + b c subtract 8 a.c. 2 b m * 29. From ***-** subtract 1944.
2 a” m a. 14 a m
which is the answer.
When the fraction was subtracted, the
XIX. Division of whole numbers by Fractions, and Fractions by
How many times is ; contained in 7 ° ..Ans. } is contained in 7, 35 times, and # is contained 4 as many times; that is, *, or 114 times. 2. How many times is ; contained in a f
.Ans...} is contained in a, 8 a times, and # is contained ; as many times; that is, *.
3. How many times is # contained in c :
Jáns. # is contained b c times in c, and # is contained l - 0.
- ... be as many times; that is, 7"
Arith. Art. XXIII.
4. Of what number is c the + part 2
Ans. If c is the # part of some number, # will be #
part of the same number, and 5 is ; part of bc. & 0. Arith. Art. XXIV. Hence, to divide a whole number by as. o it by the
denominator of the fraction, and divide the product by the numerator.
How many times is ; contained in #.
Solution. Reducing them to a common denominator, # is ##, and is #: # is contained in ## as many times as 24 is contained in 35; that is, # or 14+. •Ans. 144.
6. How many times is # contained in # P
Solution. Reducing them to a common denominator, #. Solution. If # is the + part of some number, l part of c. 0.
is # and # is # # is contained in # as many times as a discontained in be that is, *% 4n. +5. a d a d
7. Of what number is # the # part f
Hence, to divide a fraction by a fraction, multiply the numerator of the dividend by the denominator of the divisor, and the denominator of the dividend by the numerator of the divisor.
Or more generally, when the divisor is a fraction, multiply the dividend (whether whole number or fraction) by the divisor
inverted. Arith. Arts. XXIII. and XXIV. 8. Divide 3 a b by #. 9. Divide is a by #. - - 2 C. 10. Divide 17 a m by Tb 11. Divide a c t by 3 bic * 2 a. - - - - - - 2 a” m 12. Divide 3 a a by Ta roy 3 a
13. Divide 2 a c – b c by
7 go c 15. Divide 11 a r" — 3 a. by ###: 16. Divide * by *.*. 17. Divide # ; by É. - s 5 ...? 18. Divide . ; by . #.