This result is more easily obtained when-regarding £2 178. 6d. as the difference between £3 and 2s. 6d., which is an aliquot part of £1 I-we subtract the rent at 2s. 6d. an acre from the rent at £3 an acre. Multiplying £35 188. 9d., the rent at 1 an acre, by 3, we find the rent at 3 an acre to be 107 16s. 3d.; dividing £35 188. 9d. by 8, we find the rent at 2s. 6d. ( of £1) an acre to be £4 98. 10дd.; and, subtracting £4 99. 10gd. from £107 168. 3d., we find the rent at (£3 —2s. 6d.=) £2 178. 6d. an acre to be (£107 16s. 3d.—£4 98. 10ģd.=) £103 68. 47d.:

103 6 43=

=

The following is a third solution, which, however, involves the employment of a larger divisor than 12. Regarding the given quantity of land as the difference between 36A. and 10P. (an aliquot part of an acre), we find the answer by subtracting the rent of 10 perches from the rent of 36 acres. Rent of 36 acres= £2 178. 6d. × 36=£2 178. 6d. × 12 X 3=103 10s. ; rent of 10 perches (of an acre) of £2 178. 6d.= 38. 7 d., rent of 35A. 3R. 30P.=£103 68. 43d. :·

£ s. d.

6 rent of 1 acre.

EXAMPLE XVII.-Find the price of 29 cwts. 2 qrs. 18 lbs. of sugar, at £278. 10d. a cwt.

If the price of 1 cwt. were 1, the price of 1 qr. (4 of I cwt.) would be of 1=58., and the price of 1 lb.

I

(2 of 1 qr.) of 58.=24d. So that when we set down I for each hundred-weight, 5s. for each quarter, and 24d. for each pound, we have, in the sum of the three amounts, the price of the given quantity of sugar at a cwt.: £29+2×58.+18x24d.=£29+108.+ 38.57d.=£29 138. 2.57d. At £2 a cwt., the price would be 29 138. 2.57d. x2=£59 68. 5°14d.; at 58. (of 1) a cwt., of £29 13s. 2.57d.=£7 88. 3.64d.; at 28. (1 of 1) a cwt., o of £29 138. 2.57d.= £2 198. 3·86d.;* at 8d. (3 of 2s.) a cwt., of £2 19s. 3.86d. =198. 9.28d.; and at 2d. (4 of 8d.) a cwt., 4 of 198. 9.28d.

48. 1132d. Therefore, at (2+58.+28.+8d.+2d.=) £2 78. 10d. a cwt., the price must be (£59 6s. 5°14d.+ £7 88. 3.64d.+£2 198. 3·86d.+198. 9.28d. +48. 11·32d. =) £70 188. 9'24d., or (very nearly) £70 188. 94d.: cwts. qrs. lbs.

58.=

29 2 18 @ £2 78. 10d. a cwt. £1 58. 24d.

[£29+108. +38·57d.=]
£ 8. d.

58: of £1 29 13 2'57 =price @ £1 a cwt.

28.=

2

NOTE 1.-When extreme accuracy is required, smaller amounts than id. should each, as in the last example, be expressed as a decimal (carried not farther than two places) of Id.; and the resulting decimal in the answer, when too large to be rejected, should afterwards be converted into d., d., d., or id.t—according to the value of the decimal: ·25d.=4d.; •5d.=1d.; ·75d.=2d.

*The figure 6, although too high, is nearer to the truth than 5. When the figure in the first decimal place is 9, the amount is nearer to id. than to id.

NOTE 2. Notwithstanding what has just been said, the pupil, in working examples like the last, should occasionally be required-if only as an exercise in Fractions to express smaller sums than id. as fractions of Id. Thus, returning to the last example, and writing 18x24d. as (384d.=) 38. 24d., we have

The first two fractions (and 4) explain themselves. The others require very little explanation:

(a) After setting down £7 8s. 3d. as of £29 138. 2 d., we have 2 d. remaining. The division of this remainder by 4 gives d. for quotient: 24d.÷4=1&d.÷4=18d. =14d.

(b) After setting down £2 198. 3d. asof £29 138. 2 d., we have 8 d. remaining. The division of this remainder by 10 gives d. for quotient: 84d.÷10=d.÷10= $8d.=&d.

(c) After writing 19s. 9d. as of £2 198. 3 d., we have d. remaining. The division of this remainder by 3 gives d. for quotient: d.÷3=2o1d.=4d.

(d) After writing 48. 11d. as of 198. 94d., we have 14d. remaining. The division of this remainder by 4 gives d. for quotient: 14d.÷4=&d.÷4=2°%d.

The addition of the fractions, 14, 4, 4, and 28 presents no difficulty: +14+&+&+28=28+18+ +2%+2%==2, or 24.

8

INTEREST.

176. The charge made for a loan of money is called INTEREST.

177. The money lent is termed the PRINCIPAL; the sum of the principal and its interest for a given time is called the AMOUNT for that time; and the interest of £100 for a year is known as the RATE PER CENT. per annum-the words " per annum," however, being usually omitted.

178. Interest is either SIMPLE or COMPOUND. When charged upon the principal only, and paid in yearly, half-yearly,* or quarterly instalments (according to agreement), interest is called simple; but when such instalments, on becoming due, are added to the principal, and made themselves to bear interest, money is said to be at compound interest.

SIMPLE INTEREST.

EXAMPLE I.-What interest would £78 98. 10d. produce in a year, at 3 per cent. (per annum)?

At the given rate, £100 would produce £3 in a year at the same rate, therefore, and in the same time, £78 98. 10d. would produce a sum as many times less than £3 as £78 9s. 10d. is less than £100. The answer being represented by a, we thus have the proportion

£100: £78 98. 10d. :: £31: a ;

or, by Alternation (see p. 214),

£100 £3:: £78 9s. 10d.: a.

From this last proportion † we find a=£78 98. 10d. X 3÷100=2148. 11d.

* As a general rule, (Simple) Interest-like rent—is paid halfyearly.

+ The first proportion would involve the reduction of the first two terms (100 and £78 9s. 10d.) to pence.

Or thus: £78 98. 10d.=£78.492 (§ 88); £33=£3'5 ; a=£78'492 x 3'5÷100=2747=£2 148.114d. (§ 89).

Here the Principal is £78 9s. 10d.; the Interest, £2 148. 114d.; the Amount (£78 9s. 10d.+£2 148. 114d.=) £81 48, 91d.; and the Rate per cent., £32, or £3 10s.

179. To find the Interest of any Principal for a Year, at any Rate per cent.: Multiply the Principal by the Rate per cent., and divide the product by 100.

NOTE 1.-To find the interest for two or more years, we multiply the interest for one year by the number of years. Thus, at 3 per cent., £78 98. 1od. would produce-in 2 years, ₤2 14s. 114d.×2=£5 9s. 10d.; in 3 years, £2148. 11d.x3=£8 48. 93d.; &c.

1

NOTE 2.-When the rate is 5 per cent., a year's interest can be found with great facility. Instead of multiplying the principal by 5, and dividing the product by 100, we simply take 20 of the principal; and this we do by setting down Is. for every pound, 3d. for every crown, and d. for every fivepence in the principal. Thus, in £345 128. 7d. there are 345 pounds, 2 crowns, and (28. 7d. =31d.) 6 fivepences. As the interest of this sum for a year, therefore, at 5 per cent., we write 345 shillings +2 threepences+6 farthings = £17 58.+6d.+13d. = £1758. 7 d.

From 5 per cent. we can easily pass to either 4 or 6 per cent., by subtracting or adding the interest at i per cent.-obtained from the division of the interest at 5 per cent. by 5. Thus, interest (for a year) of £345 128. 7d. at 5 per cent.₤17 58. 7d.; at 1 per cent. 17 58. 73d.÷ 5=£3 9s. 1d., at 4 per cent.=17 58. 73d.-£3 9s. 11⁄2d. =13 168. 6d.; and at 6 per cent. 17 58. 7 d. + £399. 1d.=£20 148. 9d.

NOTE 3.-The time, when not an exact number of years, is always expressed in DAYS, or in years and DAYS. Notwithstanding what several treatises on Arithmetic imply to the contrary, there is no such thing in the commercial world as a month's-or a number of months'—