ARITHMETIC. 



f 



= ** 



same denominators. Thus let it be 

 required to add |, |, & together. 

 Arranging these fractions as in the 

 margin, and bringing them to others 

 with the same denominator, we find ^ = ** 

 them transformed into J, H, and ff 



J, and the sum is f, or 2^, or 

 2*. 



Again, let it be required to add 5$, 2, 

 Arranging these numbers as in the 

 margin, we first add the fractional 

 part of each, and we do so as before, 

 by bringing the fractions all to the 

 same denominator ; we will then have 

 for the numbers to be added 5? 8 T > 2ft, 

 3^1. Adding the fractions in each, we 

 have for their sum ff, or i, or if ; the sum of the 



ole numbers is 10 : therefore the whole sum 



be i if. 



Let it now be required to subtract f from ff- 

 Arranging these fractions as in the 

 margin, and bringing them to others 

 having a common denominator, we find 

 the fractions transformed into and 

 ||, and their difference will therefore be 



5i = SA 



3A 



hav 

 will 



I 



Again, let it be required to subtract 2| from 7. 

 We cannot take | from zero, but to both frac- 

 tions we will add i ; adding this to the upper 7 

 line as fifths, and to the under as i whole, 2f 

 the numbers will stand as in the margin ; 

 we are now enabled to take from , and 3 

 from 7. 



Lastly, let it be required to subtract 3$ 7 

 from 5|. Bringing the fractions in each 3! 

 to others having a common denominator, the 4! 

 numbers will be 3^ and 5^. Arranging 

 them as in the margin, we cannot take -fa from 

 ^r, but to both we add i whole ; add 

 this to the upper number as twenty- $\ 5^ 

 fourths, adding 24 of them, and to 3^ = 3^ 

 the under number as I whole, the 

 numbers will now stand as in the margin. We 

 can take A from J> and 4 from 5. The 

 result is therefore i^f. H 



Multiplication of Fractions. 



The definition which has been given to multi- 

 plication of whole numbers applies to the multipli- 

 cation of fractions. 



First, the multiplicand being a whole number, 

 and the multiplier a fraction, let it be required to 

 multiply 12 by \, Now, to produce \, we have 

 taken the unit, and divided it by 2. Therefore, 

 to multiply 12 by \, we must perform the same 

 operation on 12 as has been performed on the 

 unit that is, we must divide it by 2 ; therefore, 

 12 X \ equal to 6. 



Again, multiply 12 by |. To produce |, the 

 double of the unit has been divided by 3 ; hence, 

 to multiply 12 by |, we must divide the double of 



12 by 3, or 



12 x 2 



equal to 8. 



Again, to multiply a fraction by a fraction. Thus, 

 \ by |. Now, \ has been formed by dividing 7 

 times the unit by 8 ; we must, therefore, divide 

 7 times \ by 8 ; but 7 times \ is V, and this 

 divided by 8 gives J. Hence, \ X $ equal to H- 



It follows from these examples, that one fraction 



is multiplied by another by multiplying together 

 the two numerators for the numerator of the re- 

 sult, and the two denominators for the denominator 

 of the result. 



Division of Fractions. 



For example, let it be required to divide by 

 f. We begin by dividing \ first by 3 ; this evi- 

 dently gives for a result ^ ; but on dividing by 3, 

 we have been dividing by 5 times too much ; 

 hence this result must be made 5 times larger ; 



it will therefore be 2-2U or &|. To divide one 

 24 



fraction by another, therefore, 'we invert the 

 divisor, and proceed as in multiplication! 



Conversion of Vulgar Fractions into Decimal Fractions. 



Thus, let it. be required to find the decimal 

 fraction which is equivalent to . This fraction, 

 written as it is here, shews that I is to 

 be divided by 2 : the operation is 2)1 '00000 

 merely indicated, and of course in whole 

 numbers it cannot be continued, but I '5 

 is the same as 10 tenths ; if 10 can be 

 divided by 2, the first figure in the result will be 

 tenths. Hence \ is the same as -5. Similarly, 

 \ is the same as -75, and f is the same as -625. 



4)3 ooo 

 75 



8)5-0000 

 625 



Again, let it be required to convert -&* into a 

 decimal. Performing this division, we find ^ 

 equivalent to -015625. 



64) i -oooooo( -o 1 5 62 5 

 64 



360 

 320 



400 

 384 



160 

 128 



320 

 320 



Again, if we take , and convert it 3) i .0000... 

 into a decimal, the result is -3333. .. 

 and this is continued, so that | is '3333--- 

 equivalent to -333. .. or, as it is 

 usually written, -3, and read, decimal 3 repeating. 

 Similarly, $ is -1. i is -16666... or -16. \ is equal 

 to -142857142857... or | is equal to -142857. 

 Such results are called repeating decimals. 



Conversion of Decimal Fractions into Vulgar Fractions. 



First, if the decimal is finite, the conversion 

 may be performed at once : for example, -2 is 

 equal to -ft, -02 is yihr, -H 2 7 is rVVW? & c - Again, 

 if the decimal is infinite that is, if it repeats : 

 for example, let it be required to find the vulgar 

 fraction equivalent to -3. This is the same as 

 finding the sum of the terms ^, -H^, TT & T , &c. on 

 to infinity, and this has been shewn to be or i. 

 Again, -7 is L -14 is H> ""fl is v$ and so on. 



601 



