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answered, by taking the aliquot parts, the same as in pratice; which the following will show.

Gave £1 1s. 8d. for 3lb. of coffee, what must be given for 291 lb. ?

Here then, if £1 1s. 8d. be 3lb., 30lb. will be 10 times £1 1s. 8d., or £10 16s. 8d., too much by the price of lb. And lb. of 3lb., therefore deduct of £1 1s. 8d.

value of lb.

as £10 16s. 8d.

5 5

£10 11s. 3d. Answer.

If 14 yards of broad cloth cost £13 10s., what is the purchase of 75 yards?

Here (14×5)=77, too much by 2 yards,
And 2 yards of 14 yards.

Now £13 10s. X5=

13_1

£74 5s. Od.

Also £131os. value of 2 yards which deduct £1 18s. 64d.

Answer, £72 6s. 54d.

If 27 yards of cloth cost £7 12s. 6d., what will 90 yards
Answer, £25 8s. 4d.

cost?
If 17 lasts of corn cost £19 15s., what will 72 cost?

Answer, £83 18s. 9d.

In the rule of three, in whole numbers, the operation is made shorter when it is discoverable that the first term will divide the 2d and 3d terms; or contrariwise, when either the 2d or 3d terms will divide the 1st; so likewise in the rule of three in vulgar fractions, the operation may be made shorter when the numerator or denominator will divide the numerator or denominator of either the 2d or 3d terms, or the product of them in the same manner as in whole numbers.

EXAMPLES.

If of a lb. cost 8s. 4d., what will 54 lb. cost?*

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181

of a £. £.

53: Here the divisor is reduced to an unit.

2

Hence X11-£, or 905 at 10d.=£37 14s. 2d.

answer.

NOTE.

* When the denominator of the divisor, with any one of the other terms can easily be brought alike, then the numerator of such only may be brought down, thus, }=1%; now,

If of a yard cost of a £, what will of an ell English Answer, 9s. 4d.

cost?

If of of an ell English of Cambric cost 1s. 7 d., what will 16 yards cost? Answer, £2 5s. 6d. As this depends on the ingenuity of the reader, no rule can be given. It may, however, be proper to show what is the best method by the following operations.

EXAMPLES.

If of a yard cost of a £, what will of a yard cost? Here a trifling attention shews that &=.

And

yard cost of a £., or 13s, 4d.

Also, the difference between and is, or to of

the 19.

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If of a cwt. cost £14 4s., what will 7 cwt. cost?

Answer, £118 6s. 8d

Here, by multiplying the denominator of the divisor by the sum £14 4s. by 10 we have £142, which is the value of 9 cwt. (see the numerator of the divisor); and, as 9 cwt. is more than 7 by 1, or the of 9 cwt., we have the operation as follows:

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If 3 ells of Holland cost £2, what will 15 ells cost?

NOTE CONTINUED.

Answer, £9 9s.

as the denominator is 10, the above example may be thus stated:

6543 : £37 14s. 2d

In endeavouring to answer this question mentally, I should multiply the 1st term into the denominator of the third, and place it under the middle term, (it being the greatest number, and capable of being reduced.) I have then the fraction thus, =W=£7, 4 of a £., or 13 tenpences, or 10s. 10d., thus. £7 10s. 10d. multiplied by 5, (the numerator of the third term produces £37 14s 2d. as before.)

If 4 tuns of wine cost £138 15s., what will be the price of 18 tuns? Answer, £520 6s. 3d. When the quantity whose value is to be found is any measure or multiple of the numerator of the divisor.

It has been shewn by the operations of the above examples, that if the denominator of the divisor be multiplied by the value given, the numerator of such divisor denotes the number in integers that may be purchased for that money.

Hence, if we want the value of any number of times, or a third, fourth, &c., of the quantity the numerator of the divisor represents, it may easily be obtained mentally, as the operations of the following two examples will show.

EXAMPLES.

If of a yard cost 12s. 6d., what will 27 yards cost?
Answer, £18 15s.

Here 12s. 6d. x 10 £6 5s. the value of 9 yards, (see numerator of the divisor,) and, consequently, £6 5s. X3= £18 158., the value of 3 times 9 or 27 yards, which is the

answer.

If of a yard cost 17s. 6d., what will 24 yards cost? Here the value of 11 yards, by proceeding as above, is £10 10s.; and as 24 yards is of 11.

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If of a yard cost 4s. 94d., what will 14 yards cost?

Answer, 10s. 94d.

If of a yard cost 7s. 8d., what will 44 yards cost?
Answer, £46 7s. 8d.

PRACTICE.

Practice is of such extensive use to persons engaged in mercantile affairs, that, to give it a place in this little work, cannot but be desirable.

As to the rule for performing it, no particular one can be laid down; it depends much on the ingenuity of the reader. If he get well acquainted with the aliquot and fractional parts of a £, shilling, &c., he will find it both easy and delightful.

When the price is an aliquot part less than a shilling, substract it from the quantity, viz., as many shillings, and the answer in shillings is obtained.

If the quantity has fractional parts, let the quantity with the fractional parts be converted into shillings.

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The following modes of working are all quite new, or what few are acquainted with.

When the price is an aliquot part less than a pound, substract that part from the quantity considered as so many pounds, and it will give the answer in pounds at once.

But if the quantity has a fractional part annexed to it, convert that fractional part into its proper value of a pound.

98 yards at 13s. 4d. 6s. 8d. £98 10s. 32 16 8

Answer, £65 13s. 4d.

8474 lb. at 168.

Answer, £677 16s.

7683 yards at 15s. Answer, £576 13s. 1 d. 468 yards at 16s. 8d. Answer, £390 4s. 2d.

377 lb. at 17s. 6d. Answer, £330 12s. 9 d.

The following being the aliquot parts of a £., it may be proper to advise the reader to give answers to those examples without using pen or pencil.

63 yards at 1s. 8d.

56 yards at 3s. 4d.
781% yards at 48.

1s. 8d. £63 5s. Od.

Answer, £5 5s. 5d.

Answer, £9 9s. 2d.

Answer, £15 15s. 7 d.

The following examples may also be answered mentally; thus, if it is asked how much is 96 times 19s. 9d.? Here 19s. 9 d.=3% of a £, therefore 96 times (see the denominator) must be £95, (see the numerator,) therefore, according to the nature of fractions, whether you multiply or divide

95_23
96 24

the terms by a whole number, or a fraction, the value of it cannot be altered; for that is 24 at 19s. 9 d., is £23 15s. ; in this manner a person may proceed as far as he pleases. And again, by multiplication, if a cipher or ciphers be put to the right of the denominator, the same number of ciphers must also be put to the right of the numerator, numerous questions in rapid succession may be asked and given with great rapidity; thus, for example, 960 at 19s. 9 d. is £950; 9,600 at 19s. 9 d. is £9,500; 96,000 at 19s. 9 d. is £95,000; 9,600,000 at 19s. 9 d. is £9,500,000, &c

EXAMPLES.

What will 1,600 yards at 18s. 9d. come to?

Here 18s. 9d.=£5, now, if this denominator be multiplied by 20, we shall have the quantity, namely, 1,600. Hence the numerator multiplied by 20 will be 1,500 for £., which is the answer.

600 yards at 19s. 2d. Answer, £575.

7 yards at 19s. 4d.

Answer, £7 5s.

4000 yards at 19s. 6d.

Answer, £3,900.

24 yards at 19s. 7d.

Answer, £23 10s.

64,000 yards at 19s. 84d. Answer, £63,000.

Having given some idea of practice, I shall now treat upon finding the value of any quantity when the shillings are even; and which may be found in almost all the Tutor's Assistants, though it is not explained in them why it is so done.

To give a just idea of this, we will suppose the price to be 8s.; now 8s. of a £, therefore, if any quantity at this price be multiplied by, we shall have the answer in manner following. For example.

What is the value of 647 yards at 8s. ?

647
2

5) 1294

£258 16s. answer

Here, because the denominator is 5, we are obliged to divide; but, if instead of reducing the fraction to its lowest terms, we had only divided the terms by 2, we should have had, and having the denominator 10 by this process, no divisor, like the above, would have been necessary; for it being so well known that when the right hand figure is

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