FRACTIONS 



2290 



FRACTIONS 



Division by a Fraction. The following prob- 

 lems illustrate the principles involved in divid- 

 ing numbers by fractions. 



How many \b feet in 1 foot? There are 3. 



How many % feet in 2 feet? There are 2X3 

 or 6. 



1-^=3 



10-^=10x3 = 



In 1 there are 3 one-thirds. In 2 there are 

 2 times as many one-thirds as in 1, or 2X3 

 or 6. In 5% there are 5 l /& times as many one- 

 thirds as in 1, or 5%X3 or 16. Such problems 

 should be read, "How many %'s in 1? In 2? 

 In 10? In 5^?" Do not say, "1 divided by 

 !/& equals what? 5% divided by % equals 

 what?" until the child is very familiar with 

 division. The phrase "divide by" is a conven- 

 tional expression which clouds the meaning of 

 the question. 



Fig. 22 (below) illustrates the number of 

 %'s and %'s in %'s. 



ft. 



ft. 



I ft. 



FIG. 22 



(a) 1^-%= 



How many % mile in 1 mile? (See Fig. 23.) 



mi. 



2/5 mi. 



2/5 mi. 



FIG. 23 



There are 2% two-fifths miles in 1 mile. 

 (b) l-=-%=2% 



K7 



1/7 



2/7 



2/7 

 FIG. 24 



2/7 



(c) l-% = 3^ 



(d) !-%=!% 



(e) l-%=2% 



These processes may all be illustrated by 

 drawings like Fig. 24, which applies to (c). 



The answers above in (a), (b), (c), (d) and 

 (e) may be shown as fractions instead of as 

 mixed numbers, and we have: 



!-%=% 

 !-%=% 



1-1-94=% 



From this may be deduced the rule: 

 1 divided by a fraction is the reciprocal of that 

 fraction or the inverted form of that fraction. 



2-M{,=; 



-=-,= 6% X %= 



Any number divided by a fraction equals that 

 number times the reciprocal of the fraction. 



We may solve the problem of dividing by a 

 fraction in another way; namely, by changing 

 the dividend and divisor to a common denom- 

 inator or expressing them in terms of the same 

 fractional unit. 



2% --% = !% H-% = 4% -f-l% = 31/ > 

 % + % = tt8-^%8 = tt5 



Dividing by a Mixed Number. Division by a 

 mixed number is explained by the following 

 problems : 



This division becomes division by a fraction, 



as soon as we reduce the divisor to a fraction. 



Problems. (1) At %< a piece, how many 



papers can a boy buy for 



30 



Number of papers=:60^-%=>0x5^=150 



i 



(2) Class badges are % yard long. How 

 many badges can be supplied from a bolt (10 

 yards) of ribbon? 



Number of 



There are 26 badges and a piece of ribbon % as 

 long as a badge. 



(3) 16% yards of silk are cut into remnants 

 of 1% yards each. How many remnants are 

 there? 



Number of . 



(4) The circumference of a circle is, roughly, 

 3Vi times as long as the diameter. The boys 

 have a game ring whose circumference is 39 x %i 

 feet. What is its diameter? 



