FRACTIONS 



2285 



FRACTIONS 



bu. and 



Unit of Measure. $% is 3 (one-fifth of a. 

 dollar) ; read, 3 times one-fifth of a dollar ; 



7 /i2 ft. is 7 (one-twelfth of a foot) ; 



% yd. is 2 (one-third of a yard) ; 



% bu. is 3 (one-fourth of a bushel) ; 



% Ib. is 7 (one-eighth of a pound). 

 In the above, $%, ^2 ft-, % yd., 

 % Ib. are each the unit of measure, while 3, 7, 

 2, 3, and 7, the numerators, designate how 

 many units of measure there are in the fraction 

 under consideration. The unit of measure is 

 called also the fractional unit. 



Note the unit of measure, or fractional unit, 

 in the following fractions: 



Fractions Unit of measure 

 % Ib. % Ib. 



% yd. % yd. 



% Ib. % Ib. 



W, $% 



% mi. 



Fractions that have the same unit of measure 

 are added very readily, as, % in.+% in.=% in. ; 

 % mi.+% mi.=% mi. Fractions which have 

 not the same unit of measure must be changed, 

 and expressed in terms of a common unit be- 

 fore they can be added, as, $%+$ 7 / 8 =$%+$ 7 /8= 



This is sometimes stated as follows: Frac- 

 tions of different denominations must be re- 

 duced to the same denomination, or to a 

 common denominator, before they can be 

 added. In many cases we can see readily what 

 this new denominator must be, as in adding 

 }4 and & % and & and %, % and %, % and % 

 and so on. Often we cannot see readily what 

 this new denominator must be, but from what 

 we have done above, in reducing fractions, we 

 see that it must be a number which is divisible 

 by each of the denominators of the fractions 

 to be added. It is called a multiple of the 

 denominators. (For further discussion of mul- 

 tiple, refer to LEAST COMMON MULTIPLE and 

 study carefully in connection with this sub- 

 ject.) 



In adding fractions we may use any common 

 denominator, but it is desirable to use the 

 smallest number that will serve the purpose; 

 this is called the least common denominator; 

 for example, 



(a) %4+%+%=i2% 76 +50# )76+ 38 %76 -ioo %76 



(b) 

 (C) 



are larger than in (c), where the least common 

 denominator is used. We see that (c) is the 

 simplest solution. 



Add %, %, %. The- least common denom- 

 inator is 120. 



l=129iao 



%=% of 129120 =% 



= 2 %20 



- 187 /120 



We may also look at the reduction in this 

 way: By what number must both terms of % 

 be multiplied so that it will be expressed in 

 120ths? By what number must both terms of 

 % be multiplied, and so on, as seen below: 



"/! 

 "/120 



1.X 15 

 8X15 

 3X24 

 5X24 

 5.X 20 

 6X20 



= 1 %20 



8 must be multiplied by 15 to produce 120, and 

 so both terms of the fraction must be multi- 

 plied by 15 to reduce it to 120ths. The same 

 analysis holds for % and %. 

 Add 2%3, % 2> %6 . 



(a) Find the least common denominator, which 

 is the least common multiple of 63, 72 and 36, or 

 504. 



(b) 



63 X 



4UA 1ftn / 

 ^^ o= 16 %04 



72X 7= 4%04 

 5X14 



Mixed Numbers. Dan has $6 in his bank, 

 and $.75 in his pocket. He has altogether 

 $6.75 or $6%. John measured his father. He 

 found him to be 5 feet and 9 inches tall. He 

 is 5% feet tall. $6% and 5% feet, and all 

 such numbers made up of a whole number and 

 a fraction, are called mixed numbers (see Fig. 

 10). 



Addition of 

 Mixed Numbers. 

 Sarah has 4% yd. 

 of ribbon in one 

 piece and 6^ yd. 



tsqyd. 



5 Z/3 sq.yd 

 FIG. 10 



in another. How much has she in all? 



4% yd. + 61/2 yd. = 4% yd. + 6% yd. = 10% yd. 

 Add 7%, 9%, 16%. 



In (a) and (b) the common denominators 



