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latter case with the units' figures. The reduction to a common denominator recommended in oral division of fractions, is seldom employed in slate work.

The average scholar is able to handle "mental” problems containing small numbers in a way that he cannot always explain, although he may endeavor to stultify himself by repeating a prescribed form of analysis. It is next to impossible, with the average teaching, to get the same pupil to work some varieties of "written " problems containing the same conditions.

In order to furnish a general method of treating some classes of examples, it has been thought best to commence with written work, leaving the mental exercises with their various devices until the former task is accomplished. ..

The accomparving exercises are so simple as not to need explanation by the teacher; but sufficient time should be given the pupil to work them out in his own way. They differ in this respect from the oral examples of a single operation containing larger figures, but which do not require any effort on the part of the scholar to determine which process is required.

1. Yearly interest is $6; a year and a half will be needed to make the interest $9.

2. The yearly interest is $ 8, making the rate 4%..

3. Yearly interest is $4, requiring a principal of $ 100, at the given rate.

5. The pupils may remember (Art. 878, No. 15) that 5% for a year is 1% for 72 da.

6. 4% per year is 1% for 90 da. 11. 2 mo. 12 da. = 72 da. See 5. 12. 1% for 80 da. is (360 - 80) % for a year. 17. 2% for 6 mo. 18. $ 3.60 per year is 1 cent per day. 20. See 18.

935. First payment = , leaving no se remainder ; second payment = f of maintenant, leaving remainder ; third payment = f of na ; last payment, = $ 2000. The total cost of the house, r= the sum of the payments, of ++5 + 2000.

936. The books contain many methods of calculating interest, but it is questionable whether it is not time wasted in giving so much attention to this topic. The average person is required to do comparatively little work in this line; while those called upon to compute interest often, learn short methods of their own or use interest tables.

If a second method is to be taught at all, the one by aliquot parts is the most useful, as modifications of this method may be applied to other operations.

6. See Arithmetic, Art. 384.
937. A modification of the so-called “ 60-day method.”
16. See Art. 901 as to days of grace.

938. 21. 10% gives 2 years' interest; then 1 yr. (4 of the foregoing); 6 mo.; 1 mo.; 18 da. (1 of 6 mo.).

942. 46. Term, 57 da. (54 da.). 47. Term, 92 da. (89 da.). 49. Term, 34 da. (31 da.). 48. Term, 16 da. (13 da.). 50. Term, 187 da. (184 da.).

943. 9. See Table, Arithmetic, Art. 1303.

944. 11. The net price of goods catalogued at x dollars, and sold at a discount of 20 and 10%, will be (x – 200 or 302) - 2 of 80x) – 80x _ 8x _ 72:

TO" 100 100 100 100

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13. If the selling price of the above is $360, 724 = 360 ; 72x= 36000; x = 500. Catalogue price = $ 500. Ans. 14. 750 – (1 of 750) = 500; 500 – 60 of 500) = 500 – 5x = net price.

500 – 5x = 450. Transposing,

– 5x = -50. Changing signs of both terms, 5 x = 50,

x = 10. 945. 7. Let x = selling price of muslin.

(84 x 40) + 105x = (84 x 55) + (105 x 20). Another way: He loses 15$ per yard on 84 yd., which is a loss of 154 x 84. This he must make up on 105 yd., which is (15$ x 84) -- 105 on each yard, or 124. Selling price of muslin, 20¢ +12%, or 324. Ans.

8. I of them brought $120; } of remainder, or of them, brought $96; 1 of remainder, or of them, brought $40; remainder, or of them, brought $30. Total amount received, $286.

9. Proceeds of gas stock, $25 x 165 = $4125. Cost of lots, $4125 – $27=$4098. Number of square feet in lots, (32 x 115) + (30 x 105) = 3680+ 3150 = 6830. Value per square foot, $ 4098 -- 6830 =$0.60. Ans.

10. Two walls, each 16 x 14, and two others, each 12 x 14, contain (32 +24) x 14, or (56 x 14) sq. ft. = 784 sq. ft. The ceiling contains (16 x 12) sq. ft. = 192 sq. ft. Adding this to the walls, makes a total of 976 sq. ft.

The deductions are (8 x 4) sq. ft. X 2, and (7 X 3) sq. ft. X 3, or 64 sq. ft. +63 sq. ft. = 127 sq. ft. Number of square feet to be plastered = 976 – 127 = 849. Cost at 1,8¢ per square foot = 2¢ x 849 = $ 16.98. Ans.

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946. 1. A can do f of the work in 1 hr., and B can do 4 of it in 1 hr. ; together they can do in 1 hr. (} + ) of the work, or of it; and to do the whole work it will take as many hours as it is contained times in 1.

1:1=1xjh=ii=211. Ans. 211 hours. 2. Commission of 21% = L of amount collected = $1.60. Amount collected=$1.60 X 40=$64. Amount remitted=$64 – $1.60 = $62.40. Ans.

3. 1% of all of $12000) = 1% of $ 9000 = £ of $90= $ 22.50. Ans.

NOTE. — It may be advisable to explain to the pupils that property is seldom insured for its full value, because it is not likely that a fire will completely destroy a building, and insurance companies reimburse the person insured, only to the extent of his loss.

4. 32 x x = 6 x 4; 32x = 24; x = 44=. Ans. & yd. or 27 in.

5. 5% for 360 days=1% for 72 days=2% for 144 days. 2% of $87 = Ans.

6. 2% of $176.

7. Let x=commission; 40x = amount invested ; x +40 x = 41x = 8200; x = 200. Ans. $200.

8. $500 is of cost, $ 4000.

9. Let x = loss, or 20% of cost; 5x=cost; 5 x - x = 4x = selling price.

x, the loss, is of selling price, 4x. 10. Let x=gain, which is 20%, or }, of the cost of the goods ; 5 x = cost ; 5 x + x, or 6x, = selling price.

X, the gain, is į of selling price, 6x. NOTE. — The amount of money given in these two examples, $ 1200, does not affect either result. It may be used or not, as the pupil prefers.

11. 3 men earn $72 : 8 in one day, or $3 per day each. 5 men earn $15 a day, or $165 in 11 days.

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947. The following is the solution without days of grace : Let

x= face of the note.

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Proof. Face of note,

$ 1005.03 – Deduct 30 days' discount, 1%, 5.03 –

Proceeds, $ 1000.00

949. 1. When days of grace are omitted, the term of discount is 90 da.

10. Find the term, and add the number of days to March 15.

950. 2. (a) 1 trillion, 500 billions, etc.

5. The first quarter of 1888 contained (31 + 29 +31) da., or 91 da. The man was employed 60 da., and unemployed 31 da. His $3 additional paid the expenses of the working days. Deducting $2 x 31, or $ 62, for the expenses of the other days, his net income = $350 – $62= $288. Ans.

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