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Three hints on Interest. The three following hints concerning Interest, the most important of those rules not discussed, will be found of advantage :

1. Accustom the child to place the multipliers and divisors in the form of a fraction before he begins to work, as by this means he will more readily understand many of the abbreviated processes given in the usual elementary treatises, and, which is even better, he will be enabled very frequently to make out others for himself applicable to the peculiar question before him.

For instance, to find the interest of £12 16s. 8d. for one year at 5 per cent., the child, before he begins the work, writes the question down thus:

and he can see at a glance that that is the same as dividing by 20. Or, to find the interest for four years at 5 per cent., he writes the multipliers thus:

and sees that dividing by 5 saves a vast amount of trouble.

2. Make him clearly perceive that to divide by 20 is merely calling the pounds shillings; and by 12, calling the shillings pence.

3. Do not let the children divide by 100, according to the long process of cutting off two figures, multiplying by 20, by 12, &c., but teach them to divide by it in one line, in accordance with the following rule. Consider all but the first two figures as pounds; (2) take the one-fifth of these two figures for shillings; (3) place the

Mr. Fletcher says (Min. of Council, 1846-7, vol. ii. p. 96): Nor is there in most schools that occasional revertal to the early rules, and to those even of Notation, which would awaken a child's perception to the value of the whole.'

It is a bad plan, however, to put children back from one rule to another to begin the study of it anew. Nor is it necessary: all that is required is to take the opportunity, as it offers, to give test questions upon what is already passed over, and to frame the questions on the rule at which they are actually engaged in terms of the others.

When children are learning Compound Rules, it is easy to give in

remainder to the left hand of half the shillings in the dividend, and consider the amount as farthings: this will give the pence &c. of the answer.

Rules in interest. The following rules may be found of advantage:

(a) To find the interest at any rate for any time.

Multiply the principal by the rate, and divide by 5: then count the quotient as shillings for a year, and as pence for a month.

(b) To find the interest at 6 per cent. for a month.

Cut off the units figure of the pounds; consider all to the left of it as shillings, and add the one-fifth of the units figure to itself, and call the sum pence.

NOTE.-The interest for any other time or rate is easily found from this.

(c) To find the interest at any rate for 5 months.

Multiply the principal by the rate, and count the product as pence.

NOTE. This is but a particular application of Rule (a).

In some books of arithmetic, distinct rules are given for the solution of the following questions :

(1) What principal in a given time would produce a given interest at a given rate per cent. per annum?

(2) In what time would a given principal produce a given interest at a given rate?

(3) At what rate would a given principal produce a given interest in a given time?

These three ought not to be separated, as they can easily be reduced to one rule. In each case there are given the interest, and some two of the three following terms, viz. principal, rate, time. And the rule for the solution of the questions may be thus stated: Multiply the interest by 100, and divide the result by the product of the two given terms. This is a simple rule, and one easily remembered.

Questions on averages, stocks, and per cent. In the majority of Civil Service examinations, questions on stocks, and those in which the terms 'average' and 'per cent.' occur, are now frequently given. It is necessary, therefore, that the children of our national schools, many of whom will possibly be competitors at these examinations, should know how to solve such questions. The following are a few examples :—

1. Find the cost of £3,456, 3 per cent. consols at 903, brokerage being per cent.

2. What income will £5,000 of 3 per cent. stock at par, and £3,420 in the same stock at 1061, produce?

3. A person invested a sum of money in the 3 per cent. consols

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when they were at 90, and some when at 75; what interest did he get in each case?

4. A person invests £720 in the 3 per cents. at 86; the funds rise 2 per cent.; he then sells out and invests in the 4 per cents. at 94; what is the change in his income?

5. If sugar be bought at 4s. 6d. and sold at 5s. 2d. what is the gain per cent. ?

6. By selling a cow for £12, the seller loses 3 per cent. on his total outlay; what did he lose, and what would have been his loss or gain if he had sold her for £13 15s. ?

7. What is the premium of a policy of insurance for £356, at £2 78. 11d. per cent. ?

8. The per-centage of children learning to write is 52 in one school of 80 children, and 47 in another of 96; what is the percentage in the two together?

9. Suppose a person expended £420 one year, £560 the next, £321 the next, and £472 in the next; what did he spend on an average each year?

10. In a school there are three children of 6 years, four of 7, eight of 9, and nine of 10; what is their average age?

Monthly exercises. In all schools the children of the advanced classes should be required to solve, once a month, on paper, without the use of slates, miscellaneous test questions on the rules already learned by them. This they may occasionally do at home, but in general it is better to have it done in the school, under the master's own supervision. Special books should be provided for the purpose. I give a few examples of what each month's exercise should consist of:

I.

1. Subtract 20,506,701 from 42,079,804, and the first again from the remainder.

2. Find by reduction how many statute perches there are in 3 Irish acres.

3. In three Irish miles of road, how many English perches ? and calculate the cost of keeping the road in repair, at 3s. 4d. per statute perch.

4. What is the price of 16 cwt. 3 qrs. 12 lbs. at £5 10s. per cwt.? 5. Define the terms Prime Numbers, Composite Numbers, Aliquot Parts and Aliquant Parts; Greatest Common Measure and Least Common Multiple ?

6. Find the least common multiple of all the even numbers up to 20 inclusive.

7. (1) Add together of of, of of 5, and of of

7

(2) Express 14s. 11⁄2d. as the fraction of 5 of of 30s. (3) Value of 115 of a cwt.

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8. (1) Define Practice, Interest, Discount, and state how the last differs from the second. (2) Why is the first called Practice?

9. Calculate the interest on £5,600 for 23 years at 4 per cent. ; and of the same sum for 120 days at 45 per cent.

10. A property was bought for £3,200, and sold for £5,681; find the gain per cent.

11. Extract the square root of 3; of 5.

12. Extract the cube root of 40,306,407.

II.

1. Multiply 421 separately by 300, 20, and 6, adding the results together. Point out the agreement between this step by step, and the ordinary multiplication of 421 by 326. Show also the difference.

2. Resolve 43678 into its prime factors.

3. Divide £450 into parts proportional to 4, 5, and 7.

4. A person leaves Sligo for Dublin, (distance 135 miles), at 6 A.M., travelling for the first four hours at the rate of 223 miles per hour, and the remainder of the journey at 15 miles per hour. At what o'clock did he arrive?

5. A can give B 20 points in a game of billiards, and C 30; how many points can B give C ?

6. Two men fire at a target, having 60 cartridges each. The first fires three times in 4 minutes, and the other twice in 6 minutes; how many times will the last have to fire when the first has finished?

7. A person invested £400 in the 3 per cents. at 951, and sold them when they rose to par; what did he gain by the transaction? 8. My income is £620; what is the amount of income-tax at 7d. in the pound?

9. Which is greater, the 45 of a guinea or the 36 of £1, and by how much?

III.

1. How many times must 152 be added to itself to produce 1,368 ?

2. Divide 456789 by the factors, 4, 5, 7, expressing the remainders first as decimals, and secondly as vulgar fractions.

3. The 3 per cents. are offered at 901, the 4 per cents. at 1063; which is the most profitable investment?

4. A person's income is reduced by £40 4s. 6d. when the taxes were raised from 4d. to 7d. in the £1; what is his income?

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5. Two clocks strike together at 8 on Monday morning. On Tuesday one wants 7 minutes to 12 when the other strikes 12; how much must the slower be put on so as to strike 9 together in the evening?

6 A merchant buys 3 pipes of wine for £100, £115, and £132 respectively, and mixes all together, selling the mixture at 428. per gallon; does he gain or lose, and how much ?

7. A person pays £620 for a bill of £676 due three years hence; what is the rate of interest?

lb.; per

8. On what sum is the daily interest at 5 per cent. sixpence? 9. A person bought 120 lbs. of tea at 3s. 4d. immediately afterwards there was a rise of 44d. per lb.; how much money did he save?

10. A wishes to settle on B an annuity of £150 a year free of income-tax, which is 5d. in the pound; what sum must he invest in the 4 per cent. consols at 903, paying the usual brokerage?