ARITHMETIC 



is 654 shillings, or 32 pounds 14 shillings ; then 

 53 times 25 pounds gives 1325 



pounds, and 32 pounds added 23 12 4^X3 

 gives in all 1357 pounds ; the 10 



complete product will there- 



fore be ^1357, Us. o^d. We 2 5 6 3 i 



might have obtained the 5 



same result by taking first 1280 17 8i 



10 times the given quantity, 76 17 of 



then 5 times the result ; this - 



will then be 50 times the A 1 357 H 9i 



given quantity ; and to this, 



if we add 3 times the given quantity, we will have 



the required result. 



Again, let it be required to find the price of 

 5432 articles at 2, 33. 7^d. 



each ; that is, let it be re- 2 3 7^ X 2 



quired to multiply 2, 33. 10 



7d. by 5432. Here we first 7 



find the price of 10 articles 3X3 



by multiplying the price of 

 i by 10 ; then the price of 218 26X4 



10 multiplied by 10 gives 10 



the price of 100, and the 



price of loo multiplied by 2181 5 o 



10 gives the price of 1000. 5 



Having found the price of 10906 5 o 

 looo, we multiply this by 5, 

 and have the price of 5000 ; 65 8 9 



then the price of 400 is $j 2 10 o 



found by multiplying the 

 price of i oo by 4; the price Il &4% n 

 of 30 is found by multiply- 

 ing the price of 10 by 3 ; and the price of 2 

 is found by multiplying the price of i by 2. 

 Those four results being added give the result 

 required. 



When the number of articles is large, it is usual 

 to find their value at a given price by the following 

 method : for example, let it be required to find the 

 price of 5432 articles at 2, 33. 7^d. each. We 

 first write down the price at one pound each ; this 

 will be 5432 pounds. Then, at 2 pounds each, it will 

 be the double of this, or 10864 pounds. We next find 



5432 = price at ^i. 



2 



2S. 

 IS. 



6d. 

 lid. 



^11848 II 



(,200 

 O 2 O 

 O I O 

 OO6 

 O O l 



2 3 7i 



the price at 2s., which is the tenth of the price at 

 one pound ; then the price at is. is half the price 

 at 2s. ; the price at od. is half the price at is. ; 

 and the price at i^d. is one-fourth the price at 

 6d. : these several prices added together give 

 the price at 2, 33. 7^d. This method is called 

 Practice. 



Example 2. What is the rent of 20 acres 3 

 roods, at $ per acre ? 



Here rent of i acre = 

 20 acres = 

 2 roods = 

 i rood = 



5 X 20 = ^100 o o 



5 = 2 IO O 



r } 2, IOS. = 150 



20 acres 3 roods = ^103 15 o 



Division of Compound Quantities. 



Let it be required to divide I7 <>4, * * 



273, I3s. lojd by 17. Here 7 ' 2 ? 3 I3 



we first divide the 273 pounds ' 



by 17 ; this gives 16 pounds, 103 



with a remainder of i pound 102 



or 20 shillings ; these shillings, 



taken along with the 133. we 



have in the given sum, give 



33 shillings, to be divided by 17)33(1 



17 ; this gives a quotient of 17 



i shilling, with a remainder of ;. 



16 shillings or 192 pence ; and 

 these, taken along with the 10 



pence, give 202 pence to be 17)202(11 

 divided by 17 ; this gives a 17 



quotient of ii pence with a 

 remainder of 14 pence or 56 

 farthings ; then these farthings, 

 with 2 added, when divided by 



17 give a quotient of 3 and a 

 remainder of 7: so that the 

 complete quotient is .16, 



is. nfd., with a remainder of 5 1 



7 farthings. _ 



RATIO. 



The relation which one number has to another, 

 as measured by the quotient of the one divided by 

 the other, is called the ratio of these numbers. 

 Thus the ratio of 1 8 to 9 is measured by 2, because 

 it is the quotient which arises from dividing 18 by 

 9. That of 5 to 2 is measured by the mixed num- 



r 2\, and that of 4 to 13 by the fraction . 



The former number is called the antecedent, and 

 the latter the consequent. A ratio is expressed as 

 follows : 



The ratio of 36 to 6 is expressed 36 : 6. 

 it it 40 to 16 ii it 40 : 1 6. 



If the ratio of two numbers be equal to the ratio 

 of two others, the fractions whose numerators and 

 denominators are the antecedents and consequents 

 of these ratios respectively are equal. For ex- 

 ample, the ratio of 7 : 3 and 21:9 are equal, and 

 their equality is thus expressed : 



7 21 



There is another way of expressing the equality of 

 two ratios, namely : 



7 : 3 : : 21 : 9. 



These four numbers are said to be proportionals ; 

 the first and last are called the extremes, the second 

 and third are called the means. 



If four numbers are proportionals, the product 

 of the extremes is equal to the product of the means. 

 Thus, since 



7 _ 21 



the fractions are equal after both have been mul- 

 tiplied first by 3 and then by 9, or we have 



7X9 = 21X3. 



Hence it follows, that of four numbers which are 

 proportionals, if any three be given, the fourth 

 may be found ; thus, the first, second, and third 



603 



