DECIMAL FRACTIONS 



1733 



DECIMAL FRACTIONS 



9 9.00 



50 50 

 17^17.0000 

 80 80 

 2_2.0000 

 3~ 3 



.18 



= .2125 



.6666 + 



Then follows the common fraction that can- 

 not reduce to a decimal fraction without hav- 

 ing a fraction in the numerator of the decimal : 

 1 1.0 l 



5_5.00 

 ____ 



5_5.0000 



1 

 . 83 



There are two points of interest here : 



FIG. 5 

 In .ji, i is - of a tenth (Fig. 5) 



33 3 



In .83 5 , 5 is i of a hundredth 



O O O 



In .33335, 5 > | 



333 



a ten-thousandth 



(2) When shall one drop the fraction? This 

 depends upon the real problem that gives rise 

 to the fraction, and only the conditions of that 

 problem can determine when the fraction may 

 be discarded. This depends upon the value 

 of the material and the degree of accuracy de- 

 sired. Is it a fraction of a pound of coal or 

 of a gram of radium? Is the answer required 

 a rough estimate or a refined exactness? But 

 for practice in abstract numbers the class may 

 carry to two places at one time, three places 

 another time, and so on. These points deter- 

 mine when the fraction may be dropped. 



Multiplication of Decimals. Moving the deci- 

 mal point changes the place of the digits that 

 express a number, and therefore changes the 

 value of the number; for example, 67.3. Here 

 6 represents 60, 7 is 7 ones, 3 is 3-tenths. Move 

 the point one place to the right, 673.; 6 be- 

 comes 600, 7 becomes 70 and 3 becomes 3 ones, 

 making the number 10 times what it was be- 

 fore. Move the point one place to the left, 

 and have 6.73; 6 becomes 6 ones, 7 becomes 

 7-tenths and 3 becomes 3-hundredths, which 

 makes the number Ho what it was at first. So 

 we see that moving the decimal point one 



place to the right multiplies the number by 

 10, and moving it one place to the left divides 

 the number by 10. 



Multiply the following by moving the point 

 one place to the right, and we have: 

 7.6 76. 



4.16 41.6 



17.09 170.9 



6.004 60.04 



Divide the following by 10 by moving the 

 point one place to the left, and we have: 



8.6 .86 



12.07 1.207 



.18 .018 



It is easily seen from the above discussion 

 that moving the point two places to the right 

 multiplies, and two places to the left divides, 

 the number by 100. This can be seen for any 

 number of places when the effect of moving 

 one place is clear. 



Multiplier changes: 



(1) 424X142 = 60208 



(2) 424xl4.2 = n 



How does the second multiplier compare 

 with the first? It is Ho as great; therefore the 

 second product is Ho as great as the first. We 

 find a number Ho as great as 60208 by mov- 

 ing the decimal point one place to the left. 

 So (2) above becomes 424X14.2=6020.8. 



(3) 424X1.42 = n 



How does the third multiplier compare with 

 the second? It is Ho as great; therefore the 

 third product will be Ho as great as the second 

 product. This number we get by moving the 

 point in the second product one place to the 

 left, and we have 424X1.42=602.08. Now let 

 us look at the three together: 



424X142 = 60208 

 424X14.2 = 6020.8 

 424X1.42 = 602.08 



We see that when the multiplicands are the 

 same and the multiplier changes, the product 

 changes in the same proportion as the multi- 

 plier. 



The same thing can be shown where the 

 multipliers are the same and the multiplicand 

 changes, as: 



424X142 = 60208 

 42.4X142 = 6020.8 

 4.24X142 = 602.08 



The second product is Ho of the first, and the 

 third is Ho of the second, because in each case 

 the multiplicand is Ho as great as the preced- 

 ing one and the multiplier is the same as the 

 preceding multiplier. Here the product 

 changes as the multiplicand changes. 



