CHAMBERS'S INFORMATION FOR THE PEOPLE. 



being given, the fourth is evidently found by 

 dividing the product of the second and third by 

 the first 



Direct Proportion. 



When two magnitudes vary simultaneously, 

 and in such a manner that any two values 

 whatever of the first have the same ratio as 

 the corresponding values of the second, we say 

 that the magnitudes are directly proportional 

 the one to the other ; or when two magnitudes 

 are such, that if one of them becomes a certain 

 number of times greater or less, the other becomes 

 the same number of times greater or less, the two 

 magnitudes are said to be directly proportional. 



Inverse Proportion. 



When two magnitudes are such, that if one of 

 them becomes a certain number of times greater 

 or less, the other becomes the same number of 

 times less or greater ; these two magnitudes are 

 said to be inversely proportional. Thus the time 

 necessary to finish a certain work is inversely 

 proportional to the number of men employed. 



RULE OF THREE. 



In all such questions the following problem is 

 proposed, namely : Having given the simultaneous 

 values of two magnitudes which are directly or 

 inversely proportional, to find the value of one of 

 the magnitudes which corresponds to a new given 

 value of the other. Thus, if 25 yards cost ,40, 

 what will 200 yards cost ? Here the magnitudes 

 under consideration are, yards bought and money 

 paid. These magnitudes are supposed to be 

 directly proportional. The simultaneous values 

 of the two magnitudes are 25 and 40, and the 

 question proposed is to find what the second 

 becomes when the first is changed into 200 ; the 

 proportion will then be stated : 



25 : 200 : : 40; 



and we have seen that when the first three terms 

 are given, we find the fourth by dividing the pro- 

 duct of the second and third by the first ; 



therefore -- 320 = fourth term ; 

 2 5 



that is, if the first magnitude change from 25 to 

 200, the second will change from 400 to 320. 



Example 2. If 3 Ibs. of tea cost 9 shillings, 

 how many Ibs. can I purchase for 21 shillings? 

 This will evidently be stated thus : 



9 : 21 : : 3 ; 



and the fourth term will be found to be. 

 2iX 3 



Example 3. If 10 men do a work in 8 days, 

 how long will 4 men take to do the same ? The 

 magnitudes here are workers and days, and it is 

 evident that they are inversely proportional, for 

 the greater number of men will require the less 

 number of days to complete the same work. The 

 simultaneous values of the two magnitudes are 10 

 and 8, and the question proposed is to find what 

 the second magnitude becomes when the first 

 becomes 4 ; the proportion will then be stated 

 thus: 



4 : 10 : : 8; 



604 



and the fourth term will evidently be 20. In this 

 way all such questions may be stated and wrought 

 out. 



The following simple and elegant method of 

 working questions in proportion will be easily 

 understood from the annexed examples ; it is called 

 the unitary method, for a reason which will be at 

 once apparent (See Munn's Theory of Arith- 

 metic?) 



Example I. If 25 yards cost ^40, what will 

 200 yards cost ? Here we say 



.40 = 



25 



.'. 40 X 200 

 25 



price of 25 yards. 



i yd. ; since i yd. is the 25th 

 part of the price of 25 yds. 



= 2Ooyds. ; since price of 200 yds. 

 is 200 times price of i yd. 



Simplifying this fraction, we have 



8 



40 X ^ 

 = ^320 = price of 200 yards. 



Example 2. If 3 Ibs. of tea cost 9 shillings, how 

 many Ibs. can I purchase for 21 shillings? 



3 Ibs. = number of Ibs. I get for 9 shillings. 

 3 



9 

 .'. 3X 21 



9 



Or simplifying 



7 

 $X 5ft 



_ *y _, 



7 



i shilling. 

 21 shillings. 



21 shillings. 



Example 3. If 10 men do a work in 8 days, 

 how long will 4 men take to do it ? 



8 days = time which 10 men take to do the work. 



/. 8 X 10 = ii i man takes (since i man 

 takes 10 times longer). 

 .'. 8 X 10 

 = 4 men take. 



4 

 Simplifying 



2 



& X 10 



= 20 days = 4 men take. 

 i 



Example 4. If 5 men build 40 yards in 8 days, 

 how many men wiU be required to build 120 yards 

 in 12 days? 



5 men = number required to build 

 40 yards in 8 days. 



=i yard in 8 n 



40 



5 X 8 



= i i, i day. 



40 



5 X 8 X 120 



40 



5 X 8 X 120 



- 

 40 X 12 



1 20 yards in i 

 120 ii 12 days. 



