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Example 1.-If 9 lbs. of tea cost 27s., how many lbs. may be purchased for 18s. ? State and work the question as follows :
8. $. lbs. Here the answer wanted is lbs.: the 9 lbs., 27 : 18 :: 9 therefore, are made the third term; and since a
less quantity will be bought for 18s. than for 27.,
the answer will be less than the third term; 27) 162
therefore 18s., the lesser of the other two terms, Ans. 6 lbs. is made the second, and 27 s., the greater, the first:
then multiplying the second and third terms
together, we have 162, which, divided by 27, give 6 lbs. for the fourth or previously unknown term.
Example 2.-If 12 men can execute a piece of work in 18 days, how long will 3 men take to do the same ? Men. Men. Days. In this example the answer wanted is 3 : 12 :: 18 days, the 18 days are therefore made
the third term; and as 3 men will take 3)216
longer than 12 men to execute the work,
and the answer should consequently be Ans. 72 days. greater than the third term, the greater of
the other two is placed second, and the
lesser first. Example 3.-If 1 cwt. 1 qr. 6 lbs. of sugar cost £3, 6s. 11d., what will 2 cwts. 3 qrs. 5 lbs. cost ? culs. qrs. lbs. cuts, qrs. Ibs. £ $. d. 1 1 6 : 2 3 5 :: 3 6 11
Here, in order to make
the first term a simple 28
number, it is reduced to 803
its lowest denomination,
namely, lbs.: the second 313
term is therefore re
duced to lbs. also, that 803
both may be alike. The 939
third term is reduced to 2504 12
its lowest denomination, 251339 ( 1721;
pence. After multiplying
and dividing, according 146 2,0 ) 14,3 5} to rule, the quotient is 1053 £7 3 57 Ans.
1721 pence, and a remain1022
der of 73d.: these being
further reduced to far313
things, and divided by the 292
first term, give 2 farthings, 219
which are placed in the 146
quotient: then convert
£7, 3s. 5d.
CONTRACTION OF THIS RULE. The working of the questions may often be much shortened as follows:
Divide the two first terms, or the first and last terms (but never the second and third) by any number that will divide both without a remainder; then proceed according to the rule, with the quotients, instead of the original terms which are now said to be cancelled--a stroke being drawn across them to indicate this.
The cancelling does not alter the relative proportion between the two terms, because both have been divided by the same number, and the answer to the question will be the same as if the original terms had been employed; whilst, from the terms having been lessened, the calculations are more easily performed.
Example 1.- If 15 yards cost £9, 12s., what will 55 yards cost ? yes. yds. £ $. d.
Here the first two terms are 15 : 55 :: 9 12 0
divided by 5, and the quotients 3 11
employed instead of the original 3)105 120
terms, which are cancelled.
See Note as to the multiplica£35 4 0 Ans. tion of the third term, p. 56. Example 2.- If 5 lbs. cost £1, 7s. 6d., what will 45 lbs. cost ? lbs. lbs. £ $. d.
Here the first two terms are 5 : 45 :: 1 7 6
divided by 5; and the first being 9
thus entirely cancelled, all that
is necessary is to multiply the £12 7 6 Ans.
third term by 9, the cancel of
the second. Example 3.—If 34 yds.cost £4, 8s., how much will 118. purchase ?
Here the first two terms are divided by ll; and the second
being thus entirely cancelled, pohy 8) 34
it is only necessary to divide
the third term by 8, the cancel 8
41 yds. Ans. of the first. Example 4. If 7 yards cost 28s., how much will £5, 3s. purchase ?
Here the first and last terms
are divided by 7; and the last 4 20
being entirely cancelled, the 4) 103
second is divided by 4, the Ans. 25% yds.
cancel of the first. Example 5.—If 7 yards cost 28s., what will 26 yards cost ? yds,
Here the first and last terms
are divided by 7; and the first 4 . 4
being entirely cancelled, 26 is 104 = £5, 4s. Ans.
multiplied by 4, the cancel of the last.
ANOTHER METHOD OF STATING THE TERMS IN PROPORTION.
Instead of stating the terms according to the rule, as given at page 56, it will often be found simpler to state them in the following order-the working of the question being the same as before : I. When the greater of the two similar terms requires the greater answer, or
the lesser requires the less answer. RULE.-Write down the terms in the order in which the sense can be plainly expressed in words, making that the middle term which is of the same kind as the answer required: then, as before, multiply the two last terms together, and divide the product by the first. The answer will be of the same kind as the middle term.
The terms may be cancelled, when practicable, as already explained.
Here the terms are stated in the same 10 : 20 :: 30
order as they are expressed in words : 20
they might have been cancelled by strik10) 600
ing out the nothings in the first and last 60s. = £3 Ans.
terms. The reason for multiplying and dividing is as follows:- The price of 10 yards being 208., if we divide 20 by 10, we will get the price of 1 yard-namely, 28. : if we then multiply the price of one yard by 30, we will get the price of 30 yardsnamely, 60s. In practice, we multiply before dividing, as it is usually more convenient.
II. WHEN the greater of the two similar terms requires a less answer, or the
less requires a greater answer. RULE. --State the two similar terms in the inverse order in which they are stated by Rule I. Thus-first suppose the question to be stated according to the meaning, as in Rule I., then reverse the terms, placing the first as the third, and the third as the first.
Example.- If 12 men take 18 days to execute a piece of work, how many days will 3 men take? Men. Days. Men. 3 : 18 :: 12 . • By Rule I., this would be stated-12 : 18 :: 3; but as 12
the smaller number of men, 3, will require the greater 3)216
answer, the 3 is placed first, and 12 last. Ans. 72 days.