MATHEMATICS 3 



A proper fraction is one in which the numerator is less than 

 the denominator, as 3. 



A mixed number is one consisting of a whole number and a 

 fraction, as 7f. 



An improper fraction is one in which the numerator is equal 

 to, or greater than, the denominator, as . This is reduced 

 to a mixed number by dividing 17 by 8, giving 2 5. 



A mixed number is reduced to a fraction by multiplying the 

 whole number by the denominator, adding the numerator 

 and placing the sum over the denominator; thus, If becomes, 



(IX 8) +7 15 



by reduction, = . 



8 8 



Addition of Fractions or Mixed Numbers. If fractions 

 only, reduce them to a common denominator, add partial 

 results, and reduce sum to a whole or mixed number. If 

 mixed numbers are to be added, add the sum of the fractions 

 to that of the whole numbers; thus, lf+2J = (l + 2) + (i+S) 

 = 41, 



Subtraction of Two Fractions or Mixed Numbers. If they 

 are fractions only, reduce them to a common denominator, 

 take less from greater, and reduce result; as, | in. ^ in. 



1-4-9 



= & in. If they are mixed numbers, subtract frac- 

 16 



tions and whole numbers separately, placing remainders 

 beside one another; thus, 3i-2J = (3-2) + (f-f) = If. With 

 fractions like the following, proceed as indicated: STS 1M 

 = (2+T+&)-lH = 2-lH = 1 = H; 7-4i=(6+*)-4f 

 -2i 



Multiplication of Fractions. Multiply the numerators 

 together, and likewise the denominators, and divide the for- 



135 1X3X5 15 



mer product by the latter; thus, ~X~X- = = . If 

 248 2X4X8 64 



mixed numbers are to be multiplied, reduce them to fractions 

 and proceed as shown above; thus, l|X3J-tXV-V*4|. 



Division of Fractions. Invert the divisor, that is, exchange 

 places of numerator and denominator, and multiply the divi- 

 dend by it, reducing the result to lowest terms or to a mixed 

 number, as may be found necessary; thus, 3-^j = jX$ = 3f = 3 



