234 



THE POPULAR EDUCATOR 



natural divisions of the earth's surface, we have given two maps, 

 one of the Atlantic Ocean and the other of the Pacific Ocean, 

 which will be found useful in showing on a larger scale than that 

 on which our Map of the World was drawn, the shores of the 

 countries that are washed by these great bodies of water, and 

 their mutual position with regard to each other. These maps are 

 arranged in such a manner that the student may ascertain the 

 longitude East or West from Greenwich or Washington of any 

 place that is marked in them. In the map of the Atlantic Ocean 

 the meridians are marked according to their position east and 

 west from Greenwich along the top, and from Washington along 

 the bottom. In the map of the Pacific Ocean the distance of 

 each meridian from Greenwich is marked along the bottom of 

 the map, and from Washington along the top. A representation 

 of by far the greater part of the surface of the globe between 

 the Arctic and Antarctic Circles is given in these maps. The 

 part which does not appear in them is that portion of the Indian 

 Ocean which washes the southern coast of Asia and the eastern 

 shore of Africa, with the whole of Central and Western Asia 

 and the east part of Africa. 



LESSONS IN ARITHMETIC. XXX. 



DECIMALS IN CONNECTION WITH COMPOUND QUANTITIES 

 REDUCTION OP DECIMALS. 



1. To reduce any given Compound Quantity to tfie Decimal of 

 another given Compound Quantity of the same kind. 



This is the name given to the process of finding, in the form 

 of a decimal, what fraction the one quantity is of the other ; or, 

 in other words, of expressing the ratio of the two quantities as 

 a decimal fraction. 



Hence, clearly, all we have to do is to find the ratio of the 

 two quantities expressed as a vulgar fraction, and then to reduce 

 that fraction to a decimal. 



Thus 4s. is <i, and i = -2. 



Hence 4s., reduced to the decimal of a pound, is '2. 



Again, 14s. 6d. = ,fg. 



And we find, as in the margin, that |g, expressed 40) 29-000 

 aa a decimal, is '725. 



Hence, 14s. 6d., reduced to the decimal of a pound, '725 



is .-725. 



2. But instead of reducing the whole of the one compound 

 quantity to. the fraction of the other, and then reducing this 

 fraction to a decimal, we can, in many cases, obtain the result 

 more conveniently by reducing the separate portions successively 

 to decimals of the next higher denomination. 



Thus, if it be required to reduce 3 4s. 4Jd. to the decimal 

 of a pound, we may proceed as follows : 



$d. = '5 of a penny. 

 Therefore 4'd. = 4'5 



And 4id. = Vj of a shilling = '375 of a shilling. 

 Therefore 4s. 4Jd. = 4-375 of a shilling. 

 And 4s. 4-Jd. = *-p of a pound. 



= '21875 



Hence 3 4s. 4-Jd., reduced to the decimal of a pound, is 3'21875. 

 In practice we should arrange the process thus : 

 4) 2-0 farthings. 



12 ) 4*5 pence. 



20 ) 4-375 shillings. 



3-21875 pounds. 



Writing down the two farthings and dividing 6y 4, we get 

 5 pence, before which we place the 4d. of the given sum. 

 Dividing this again by 12, we get '375 shillings, before which 

 we place the 4s. of the given sum ; and, similarly, dividing this 

 by 20, we get '21875 pounds, before which we place the 3 pounds 

 of the given sum. 



3. EXAMPLE. Eeduce 5 days 3 hours 36 minutes to the 

 decimal of 3 weeks. 



60 ) 36-00 minutes. 



24 ) 3-60 hours. 



7) 5-150000 days. 



3 ) -73571428 weeks. 



24523809 decimal of 3 weeks. Answer. 



4. To reduce a Decimal of any Compound Quantity to swo 

 cessive lower Denominations. 



For instance, suppose it be required to reduce ,3'21875 to 

 pounds, shillings, and pence. 



This is the reverse process to that already explained in Art. 2, 



01 07*: v 90 



Now -21875 = 



sM1 - = 



= 4 ' 375 s " Kil - 



375 x 12 

 375 shil. = /o s o shillings = 1AAA pence = fg = 4'5 pence. 



IvUV 



And "5 pence = TO pence = -JT farthings = ?o = 2 farthings. 



Hence, 3-21875 = 3 4s. 4-Jd. 



An examination of the above will sufficiently explain the fol- 

 lowing method of arranging the work : 



3-21875 

 20 



4-37500 

 12 



4-500 (leaving out unnecessary 

 4 ciphers.) 



2-0 3 4s. 41d. Answer. 

 Notice that the decimal part only of each line is multiplied. 



5. Hence we get the following 



Rule for finding the Value of a Decimal of any one Denomina- 

 tion in successive lower Denominations. 



Multiply the decimal part by the number of units of the next 

 lower denomination which makes one of the denomination in 

 which the decimal is expressed, and cut off from the result a 

 number of decimal places equal to the number in the multi- 

 plicand. The integral part in this result will express the number 

 of units of the lower denomination. Proceed to reduce the 

 remaining decimal part to the next lowest denomination exactly 

 in the same way, and continue the process until the lowest 

 required denomination is arrived at. 



6. EXAMPLE. Eeduce -4258 days to hours, minutes, etc. 



4258 

 24 



17032 

 8516 



10-2192 hours. 

 60 



13-1520 minutes. 

 60 



9'1200 seconds. 

 Hence '4258 days = 10 h. 13 m. 9-12 a. 



7. It is evident that, since each of the two processes we have 

 explained is the converse of the other, we can prove the correct- 

 ness of our operations in any case by reducing the result to the 

 original form. 



Thus we showed that .3 4s. 4^d. was <3'21875, and then, 

 by the converse process, we proved that .3-21875 = =3 4s. 4^d. 



8. EXAMPLE. Eeduce -21 73 of a pound to shillings, pence, eta 

 This may be performed in two ways. 



2173 

 20 . 



4-347 

 12 



4-169 

 4 



678 farthings. 

 Hence -2173 = 4a. 4d. and -678 farthings. 



Here, in multiplying we are obliged to take in additional figures 

 of the recurring period, in order to obtain the recurring period 

 after the multiplication correctly ; and this might give rise to! 

 considerable trouble if the number of figures in the recurring 

 period were large. It will be often better, therefore, in such 

 cases, to reduce the recurring decimal to a vulgar fraction, and 

 proceed to perform the operation as follows : 



