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27. 672 yd. @ $24 = $1512. Discount without grace = $1512 Xito xd=$17.64. Profit is $1 per yard less discount = $ 672 – $17.64 = $654.36. Ans.

The discount for 3 days' additional (grace) = zło of $ 17.64 = $.88, making the profit 88¢ less than the above, or $653.48. Ans.

28. 40¢ x [(55 x 600 x 54) -- 27]. Cancel.

1023. 5. The circumference of the wheel = distance traveled in 1 revolution=1 mi. 94 rd. 2 yd. 1 ft. -- 526 = 6838 ft. = 526 = 13 ft. = 4 yd. 1 ft.

6. The weight in pounds = 1969 X 9 X 9} x 64.

8. Rate of income received on 6% bonds = 6; 1.18; rate on 44% bonds = 44 - . The income being the same, and the same amount being invested, the rates must be equal; there

600 450 a €, 118a

°; 600 x = 118 x 450 = 53100 ; x =88. Price per $ 100 = $88.50.

9. The shrinkage being 1 lb. in 10, he must sell 9 lb. for the cost of 10 lb. to suffer no loss. 10 lb. cost $1.80; by charging 20¢ per pound, he receives the cost. To gain 20%, he must sell for of 20€, or 24¢ per pound. Since he loses 4% of the amount of sales, or zs, he receives only of the price charged per pound. Therefore to receive 24¢, he must charge 24¢ = = 25¢ per pound.

[(184+%)*] 1024. 1. 12 xx=20 x 8.

2. This may be solved by analysis, or the following method may be employed :

The solid contents of first beam in cubic feet = 16 x 24 x }; of the second = x x 34 x 24. The second weighs 2.38 times the first; its contents, therefore, = .38 times the contents of the first.

X X 31 x 2 = 16 x 24 x š x 1278;
x=(16 x 24 x Î x 1938) = (34 x 29)

= 16 x x x X A X 24 Cancel.

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3. The carpet costs 50¢ per foot. $1 x 224 x 152 = 24 = Ans. Or, changing all dimensions to yards: $ 11 x 74 x 54 - = Ans.

4. As a sight example, some pupils may see that the width of the large box is double that of each small one, and its depth is three times that of each small one, so that with the same length as the small one, it would contain 2 x 3, or 6, small ones. A length twice as great —81 ft. — is required to enable it to hold 12 boxes.

6. [1 of (18} x 115)] sq. ft. . 22 14 x 12 x 1728

2150.4 Drop the decimal point in the denominator, and annex a cipher to one of the numbers in the numerator. Cancel.

9. 49 x 44 x 27 = 231= number of gallons. Cancel. 10. Solve at sight. 7 yd., 6 yd., 4 yd.

11. Number of gallons = 51 X 6 X 7 x 1728 = 231. Canceling, we obtain 1728 gal. One empties it in (1728 -- 9) min. = 192 min.; the other in (1728 = 7) min. = 2464 min.; both in (1728 - 16) min. = 108 min.

12. The dimensions of the room are 6 yd. and 5 yd., and the carpet is yd. wide. 6 contains å an exact number of times (8), so that if the carpet runs across the room it will take 8 strips each 5 yd. long. As 5- = 63, to lay the carpet lengthwise would require 6 strips, and f of a seventh strip, which would have to be cut.

Carpet 30 in. wide, byd., could be laid lengthwise without splitting the breadths, 5:6, or 6, strips being needed, each 6 yd. long.

13. 36 yd. are needed to cover the floor; including 41 yd. cut off in matching the pattern, 401 yd. must be bought at 95%. At 10$ per yard, the sewing and laying should cost 10¢ x 36 = $3.60, but the custom is to charge for the number of yards purchased, 401, making $ 4.05; (5 x 6 sq. yd. or 30 sq. yd. @ 5%

= $1.50 for lining. Total cost, $38.471 + $ 4.05 + $ 1.50 = $44.022, or $ 44.03.

14. A strip į of (18 – 15) or ļof (21 – 18) is left uncovered on each of the four sides, or 11 ft. The area of the uncovered space in square feet =(21 x 18) – (18 x 15).

15. The number of square feet in the walls = (18+ 24 + 18 + 24) X 9. The ceiling contains (18 x 24) sq. ft. Deduct 60 sq. ft. for two doors, 48 sq. ft. for two windows, 25 sq. ft. for the fireplace. The total number of feet around the four walls = 18+ 24 + 18+ 24 = 84 ft. Baseboard will not be required at the doors, 8 ft.; nor at the fireplace, 5 ft. — a deduction of 13 ft., making 71 running feet of baseboard, 1 ft. wide, containing, therefore, 71 sq. ft. The total deduction from the area of walls and ceiling, 1188 sq. ft., are 60 sq. ft. +48 sq. ft. + 25 sq. ft. + 71. sq. ft. = 204 sq. ft., leaving 1188 sq. ft. — 204 sq. ft., or 984 sq. ft., to be plastered.

16. The first pile contains (25 x 20 x 10) cu. ft. and costs $1400. 1 cu. ft. cost $ 1400 = (25 x 20 x 10). Multiplying by (50 x 40 x 20), the number of cubic feet in the second pile, gives the cost :

$1400 x 50 x 40 x 20

25 x 20 x 10 17. Pupils that endeavor to solve a problem without examining the conditions, will be likely to assume that this example resembles 16. In the latter, the cost of the second pile is 8 times the cost of the first; in this one, the surface to be painted in the second room is 4 times that of the first room, making the cost $56. As they may not be familiar enough with similar surfaces to know the ratio, they should find the surface of each.

1025. In dividing decimals, change the divisor to a whole number. See Arithmetic, Art. 663. .

1026. 2. XXV == 25000.

2.

5. The furlong is seldom used. 3.7082 mi. - 4=.92705 mi., the length of one side. Multiplying by 8 to reduce to furlongs, we obtain 7.4164 fur. Change the decimal part, .4164 fur., to rods by multiplying by 40, obtaining 16.656 rd.; .656 rd. = 3.608 yd.; etc.; etc.

6. See Art. 986, 7. That selling price, $3.54, must be increased }, or $.59, to gain 163%. $3.54 +$.59 = $ 4.13. Ans.

7. $$ x 24

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Cancel.

27

10. The inventors of the expression “true discount” assume that interest is not payable in advance. They claim that a borrower that promises to pay $380 at the end of 2 yr. 5 mo. should receive as a loan the principal that will amount to $380 in that time.

Let x= principal.
Interest=ex 16 x 29 – 2012
Amount=x+202 = 380 ;
200x + 29x = 229x = 76000;

x = 331.88 –
The principal (" present worth ") = $331.88.
The “true discount" = $380 - $331.88 =$48.12.

This “true discount" is the interest at 6% on $331.88 for 2 yr. 5 mo.

The interest on $380 at 6% for 2 yr. 5 mo. is $ 55.10; the difference between the interest and the “true discount" being $55.10 – $48.12 = $6.98. Ans.

1027. 1. A gain of 25%, or ļof cost, makes the selling price $9, equal of cost; & of cost, the present gain, is $9 = 5, or $1.80. A gain of 50% would be $1.80 x 2 = $3.60.

N. B. -- It is not necessary to find the cost.

2. $30 in one case represents of cost, making the gain $6; in the other case, $30 represents of cost, making the loss $10. Net loss $4.

NOTE. — The thoughtful teacher will recollect that every member of the class does not "see through” an example in the same way, nor with equal rapidity. While quick work should be exacted where the question involves but a single arithmetical operation, time should be given, in problem work, to pupils that do not quickly grasp the conditions. Such as can dispense with unnecessary figures should be encouraged to do so as much as possible; but care should be taken not to injure others by requiring them to adopt a short method whose underlying principles they do not thoroughly understand. Each should, to a certain extent, be allowed to use his own mode of “analyzing” oral problems and of setting down his written ones; shorter ways, however, being presented from time to time in the oral and blackboard work of his brighter classmates. The scholar that reaches his results by a circuitous course will, by these models, be led to see the time saved by shorter methods, and he will probably try to master some of them.

3. Together they have $300 = x + 2x.

6. $ 40 in 31 yr. = $12 per year. This is produced at 6%. by $200. Ans.

7. The interest of x dollars for 5 yr. at 6%=32; x + 32, or

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8. Yearly interest = $3. To obtain $18 interest will require 6 yr.

9. The interest is $2, į of principal in 33 yr.; in 1 yr. it is x of principal, or zo = 5%.

Or, the interest on $12 @ 1% for a year is 12€, or 40¢ for 3} yr.; to obtain $2.00 interest, which is 5 times as much, the rate must be 5%.

10. I lose $25, or 1 of cost = 331%.
1028. 4. 23} +(23} + 3) +(23} + 3+31).

6. Let 3x represent the amount received by one; and 4x the amount received by the other. Then, 7x=21.63; x=3.09; 4x = 12.36, and 3 x = 9.27. Ans. $9.27 and $12.36.

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