We shall next, in somewhat natural sequence, as we have step by step been working up to it, consider the teaching of that important branch of Arithmetic comprehended under the generic term Percentages-the commercial or higher-business man's practical Arithmetic. The word has, in the singular number, for some years obtained such a terrible ascendancy in the scholastic vocabulary that it has almost come to have the same effect on the profession that a red flag has on the lord of the bovine creation. Here, however, we trust it will be found perfectly innocuous.

In commencing the teaching of Percentages, I first try to instil into the minds of the scholars a clear idea of the meaning of the term per cent.-so much or so many per 100. When money is referred to, it means on each 100 pounds, and when any other concrete number is considered, it means so many out of each 100 of that number. For instance, when we speak of money making 5 per cent. interest, or of goods being sold at a profit of 15 per cent., we simply mean in the former case that Lioo makes £5 interest, or that some principal greater or less than 100 makes at the rate or proportion of £5 for £100; and in the latter case that goods bought for £100 were sold at a profit

of £15, thus selling for £115; or at the rate of £15 profit on £100, whatever the money originally spent on the goods might be. When concrete numbers other than money are referred to, as 20 per cent. of a number of boys, it denotes 20 boys out of 100 boys, or that proportion of the boys, whatever the number on which the percentage to be taken might be.

I now give a number of easy examples to be worked mentally-Find the interest of £200 at 5 per cent. ; £250 at 4 per cent. ; £650 at 3 per cent.; £825 at 6 per cent. In the last example explain-each of the others being, however, explained if required-that £8258 hundreds, consequently the interest will be 8 times 6 £49 10s. Give a score or two of such questions till even the slower ones are able to answer, having heard question after question worked out.

Now work out mentally some other concrete exercises-not money. In a school of 200 children, 5 per cent.—that is, per 100-are late; how many are late? Out of 300 four per cent. were late? Out of 250 two per cent. were late? Here, as 250 = 2 hundreds, there must have been 24 times 2 = 5 scholars late. I now present a different phase of these exercises :-If 10 boys are late out of 200, what is the rate per cent. ? If 12 are late out of 300? If 15 are late out of 250? Here 2502 hundreds, and as we want to know how many were late out of every 100, then 15÷22, or 30÷ 5-6 per cent. Ans.

Having given some general notion of the application of the term per cent., we will now consider seriatim and in detail the various specific operations included in the general term Percentages.

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(a) The sub-division generally taken first is Simple Interest, with which we at once proceed. I first define interest-money paid for the loan of a sum of money, called the principal; and rate per cent. as previously defined, remarking that this rate is so much for one year, unless otherwise specially mentioned. We now at once commence working exercises freely on the board, the first being in finding (a1) the simple interest of a sum of money for a given time, and first for a year. Find the interest of £420 at 5 per cent. We will work this exercise by different methods, in order that the rationale may be thoroughly understood, so that in the future working of such questions the method that commends itself as the easiest in any particular case can be easily resorted to. (1) I explain that 5 per cent. 2, hence that the interest is of the principal, so that £420+20= £21. Ans. In practice I adopt this plan more frequently than any other. (2) As £420=4} hundreds, £5×4 = £21. Ans. This method is convenient when the principal can be easily expressed in hundreds and the decimal or fraction of a hundred. (3) As 5 per cent. is, the interest is just a shilling for every £ in the principal, hence the interest must be 420 shillings = £21. Ans. This method is handy when the rate is also 2 per cent., that is, sixpence in the £, or ; also when it is 3 per cent., that is, ninepence in the . The application of this method is limited, being only available when the rate is a multiple or sub-multiple of those just given-thus 7 per cent. just given-thus 7 per cent. Is. 6d. in the £, 14 per cent. = 3d. in the, or, etc. (4) One per cent. of £420 = £42, hence £42×5=£21. Ans. This method is useful when there are £'s only-no shillings and pence-or when these will easily come out in the terminate decimal of a £. (5) Multiply by the rate per cent. and divide by 100, then (£420×5)

The scholars who work percentages being fairly up in fractions and decimals, each of the above methods is easily intelligible to them. In No. I we have taken 4 per cent., which is of 100, and then, which is of the 4. A boy at this stage ought to be so well up in mechanical working that he can easily divide by 25 by short division. In No. 2 we have brought the 12s. 6d. to the decimal of a £, which most boys can easily do mentally, being = 625. We then proceed as in No. 1. In No. 4 we have mentally divided the £625625 by 100 = £625625, which is the interest at one per cent., and then multiplied by the rate required-4. In No. 3 we have adopted method 5 as referred to above. In giving a class an early lesson, I exhibit all these methods simultaneously on the board, and remark well on each of them, leaving the scholars to adopt which measure they prefer in their own working.

Another example,-What is the simple interest of £476 13s. 9d. for 4 years at 6 per cent. per annum ?

(1) £ s. d.

5 p. c. is 476 13 9

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42 years

£12870 11s 3d, 1d. Ans.

In method 2 we

(3) £476 13s. 9d. x.6 x 41 which divided by 100 = £128 14s. Method I above is self-evident. have £476 13s. 9d. = £476·6875= £4*766875 hundreds of £'s, the remainder being easy to follow. In this case method 3 is the shortest, though the least rational. I here explain that instead of dividing the principal by 100 at first, and then multiplying by 6 and 4, we simply multiply first, and then divide afterwards.

I should now explain the term amount-the sum obtained when the interest has been added to the principal-and work a question requiring it. What is the amount of £756 15s. for 3 years 10 months at 3 per cent. per annum? Here, first elicit that 33 is of 100, hence that a year's interest is of the principal-£756 15s.÷30= £25 4s. 6d., a year's interest. Then, 3 years 10 months = 33 years, hence £25 4s. 6d. × 3 = £96 13s. 11d. the total interest, which added to the principal £756 15s. = £853 8s. 11d., the amount. Ans.

(a2) We will now find the rate per cent. when the interest of some specified principal for a given time is known.

Example:-At what rate per cent. will £450 amount to £508 10s. in 34 years? Here reiterate what rate per cent. really means-the interest of one hundred for one year; and we have in the question the interest of a number of hundred £'s (4) for a number of years (34).

£508 10s. - £450= £58 10s., int. of
£450 for 34 yrs.

£58 105.4 (hundreds) = £13, int. of
£100 for 3 yrs.

£13÷34 (years) = £4, int. of £100 for 1 yr. Ans.

Another example :-At what rate would £162 10s. amount to 184.8s. 9d. in 4 years, simple interest? We will work this question decimally. £162 10s. = £1625 1625 hundreds, and £184 8s. 9d = £184 4375, then

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£184 4375-£1625 £21'9375, int. of 1625 hrds. for 4 yrs.

£219375÷1625-135, int. of 1 hrd. for 4 yrs. (£135÷42)=(£27÷9)= £3. Ans. Int. of i hrd.

for 1 year

We will also work this question by compound proportion

case.

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(a3) We will next find the time in which a sum of money will make a given amount of interest in a specified time. Example:-In what time would £1,625 amount to £1,824 6s. 8d. at 33 per cent. per annum? Here we simply work on first principles, finding the interest produced in one year, and then seeing how often this one year's interest is contained in the given interest. £1,824 6s. 8d. - £1,625 £199 6s. 8d. the given interest. 1,625 16 hundreds, and (£3} × 164) = (4×65)= £52, the interest for one year; then £199 6s. 8d. ÷ £52 = 33, the number of years 3 years 10 months. Ans. In dividing, the £199 6s. 8d. = £199 by £52, bring both to thirds of £'s 598 15631383 years. Ans. In teaching a class I should here remark that although one sum of money cannot be multiplied by another, both being concrete numbers, we may by dividing see how often one sum of money is contained in another, just on the same principle as we find (say) how many times 9 apples is contained in 72 apples. A caution here is needed that the quotient obtained is not money (£ s. d.), but an abstract number, representing, however, it may be, something concrete, as 3 years above.

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Before commencing to divide, the two amounts should be brought to the same denomination.

Another example:-In how many years would £600 amount to £750 at 3 per cent. per annum? We will work this question by compound proportion, first reading it in a more explicit form, thus:-If £100 makes £33 (=) interest in one year, in how many years will £600 make (£750 - £600 =) £150 interest at the same rate? First bringing the two amounts of interest, £3 and £150, to fourths of £'s, to get rid of the fraction (), we have 15 and 600; then stating,

we have :

In the above statement we have represented the two principals in hundreds of £'s, their simplest ratio, instead of £100.

(a4) We now proceed to find the principal which will amount to a certain sum, or, in other words, produce a specified interest in a given time at a given rate per cent. Example:-What principal at simple interest would amount to £487 18s. at 4 per cent. for 4 years? Here we might find what any principal would amount to at 4 per cent. for 4 years, and then whatever fraction or proportion this supposed principal is of its amount, so the principal required will be of the given amount-£487 18s. The most convenient supposed principal is £100, then £4×4}= £19, interest of £100 for the given time; hence 100 would amount to £119, consequently the principal is of the amount, and 9 of £487 18s. 410. Ans. The simplest method of obtaining 19

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of 487 18s. is by representing it as £4879, then mentally multiplying it by 100 = £48,790, which divided by 119 = £410.

= 24

Another example:-What principal will amount to £369 os. 2 d. in 4 years at 34 per cent. per annum? Here 3x413× 25-325-13, interest of £100 for the 4 years, hence £100 would amount to 113 in the given time; the principal is therefore 100 2400_480 96 of the amount, then 113 2725 545 109

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£369 os. 24d. x 96)÷109 = £325. Ans.

As every question in simple interest must come under one or other of the above four headings, before proceeding further I devote a week or two to working out a number of dissimilar questions, so as to bring into requisition the knowledge acquired. Here the first 80 questions from 'Percentages' in my own series come in opportunely, and are supplemented by a number of questions from Colenso's, Mansford's, or any other manual of arithmetic, and by setting ad libitum original questions on the board. In working original questions I very rarely have any difficulty in ascertaining the true answer, as when three or four of the best workers have honestly the same answer, they are nearly certain to be right. Should there be the least doubt, I work out the question on the board in presence of the whole class. I would here remark that in finding the interest of a sum of money from one date to another, the worker is often in doubt whether he should reckon both the given days, or only one of them, or neither of them. Of course the true interest, as a banker would reckon it, and as common sense would suggest, would be obtained by reckoning one of the days only, if no directions were given; as a sum of money invested one day and taken out the next would only be one day at interest, or if invested one (say) Monday and taken out the next Monday would only bear 7 days' interest, not 8 days. For instance, from April 14th to July 26th, I should reckon 16 days for April, omitting the 14th, but reckoning July 26th, making in all 103 days; hence the interest for these days would be 10 of a year's interest. In every question in arithmetic, the data should be explicit, nothing doubtful or ambiguous about them; yet nearly in every batch of questions issued by the Education Department, whether to pupil teachers, Queen's scholars, or students, I can find one or more questions puzzlingly lacking in definiteness. If the full working of a question is carefully looked over by the examiner, which takes considerable time, so that the view the worker had of what was required is appreciated, and the work be valued accordingly, not much harm will be done; but if only a specific result (answer) must be obtained, then the worker may be seriously wronged. I remember many years ago a pupil teacher being shown his worked paper by an Inspector, who told him gravely that one of the questions was seriously wrong and seriously wrong it certainly was in the answer and casting my eye cursorily over his paper I saw and pointed out that this 'serious wrong' was, that in a question in interest, in which the answer was to be the amount, he had only given the interest-his omission to add the interest to the principal.

(To be continued.)

Recent Enspection Questions.

[The Editor respectfully solicits contributions—all of which will be regarded as STRICTLY PRIVATE-to this column. For obvious reasons, it cannot be stated in which district the questions have been set.]

Arithmetic.

STANDARD I.

(1) Add together, three hundred and eighty-nine, fifty-seven, eight hundred and sixty, seven hundred and twenty-six, and five hundred and sixty-four. Ans. 2596.

(2) From six hundred and forty-three, take four hundred and seventy-nine. Ans. 164.

(3) Find the difference between seven hundred and five, and two hundred and sixty-nine. Ans. 436.

STANDARD II.

(1) Multiply sixty-eight thousand four hundred and seventy-five, by seventy-eight. Ans. 5,341,050.

(2) Divide sixty-nine thousand nine hundred and thirty-one, by nine. Ans. 7770,-1.

(3) From sixty-four thousand two hundred and fifty, take thirty-nine thousand eight hundred and twelve. Ans. 24,438.

STANDARD III.

(1) Divide four millions and seventy-six thousand three hundred and sixty-one, by six hundred and seventy-three. Ans. 6057.

(2) Add together,-eight hundred and forty three pounds thirteen shillings and eightpence three farthings, one hundred and sixteen pounds six shillings and elevenpence, eight thousand and ninety-four pounds eight shillings and fourpence halfpenny, three hundred and nineteen pounds three shillings and threepence three farthings, and six thousand and twenty-eight pounds eighteen shillings and sixpence halfpenny. Ans. 15,402 10S. Iold.

(3) Find the difference between twenty-three thousand eight hundred and forty-three pounds thirteen shillings and sixpence halfpenny, and nine thousand nine hundred and eighty-nine pounds seventeen shillings and sixpence three farthings.

Ans. 13,853 15s. 11 d. (4) I pay £13 175. 11d. to one person, and £29 2s. 6d. to another. I then receive £8, and find £9 5s. od. in my purse when I get home. How much had I at first? Ans. £44 5s. 5d.

STANDARD IV.

(1) Multiply seven hundred and fifty-five pounds twelve shillings and ninepence farthing, by ninety-two. Ans. £69,518 14s. 11d.

(2) Divide five hundred and twenty-eight thousand and seventy-six pounds ten shillings and a penny three farthings, by five hundred and twenty-seven. Ans. 1002 os. 10 d.

(3) Bring eleven million and twenty-two ounces to tons. Prove the answer. Ans. 306 tons 18 cwt. I qr. 17 lbs. 6 oz.

(4) How many persons can receive 15s. 1d. each out of £40 14s. 6d? Ans. 54 persons.

STANDARD V.

(1) 3191 @ £1 9s. 53d. Ans. £4703 8s. old.

(2) 79 cwts. 3 qrs. 16 lbs. @ £2 12s. 6d. a cwt. Ans. 209 14s. 4 d.

(1) Divide the sum of and (1) by the sum of ,,; and find the value of (3 × 2}) ÷ (1's × 2) of 3. Ans. 14; £81 13s. 4d.

(2) If 20 men do a piece of work in 12 days, working 9 hours a day, how many men will do twice as much work in 8 days, working 6 hours a day? Ans. 90 men.

(3) A man owns 375 of a ship. He sells of his share for £1260. What will be the value of the whole ship? Ans. £5040.

(4) 37'004 x '00473. 1892. Ans. 9251.

Divide the product by