660. 1. $289.52 79.68 81.73 Add: NOTE.Problems 2-6 are taken from an arithmetic published over one hundred years ago. 2. There are two numbers; the less number is 786.39 8761, the difference between the numbers is 597. 496.38 What is the sum of the numbers? 809.99 78.63 61.92 5.48 689.73 738.93 604.52 900.68 99.83 86.49 3. What is the length of the road, which, being 33 ft. wide, contains an acre? 4. A bankrupt whose effects are $3948 can pay his creditors but 28 cents 5 mills on the dollar. What does he owe? 5. The river Po is 1000 feet broad and 10 feet deep, and it runs at the rate of 4 miles an hour. In what time will it discharge a cubic mile of water (reckoning 5000 feet to the mile) into the sea? 808.70 6. At the late census, taken A.D. 1800, the number of inhabitants in the New England states was as follows, viz.: New Hampshire, 183,858; Massachusetts, 422,845; Maine, 151,719; Rhode Island, 69,122; Connecticut, 151,002; Vermont, 154,465. What was the entire number? 7. Draw two straight lines having the ratio of 3 to 2. 8. What is the selling price of 48 yd. of cloth bought at 38. 6d. per yard and sold at a gain of 21% ? 9. Estimating a bushel of coal to weigh 80 lb., find to the nearest tenth the number of cubic feet of space needed for the storage of one ton of coal. 10. Find the product of the common prime factors of 1395 and 1736. 661. 1. 4937 × 398 = ? 2. A note drawn for 90 da. without interest was discounted 24 da. after date, at 6% per annum, yielding $553.84 proceeds. What was the face of the note? 3. a. How many kiloliters of water can be contained in a rectangular cistern 2.5 m. by 3.6 m. and 75 cm. deep? b. What is the weight of this water in kilograms? 4. a. How many shares of preferred stock, paying 51 % dividends, must I buy to secure an annual income of $500.50 ? b. What will the stock cost, at 1248, brokerage %? 5. A barn roof is 58 ft. long and the slant height is 24 ft. on each side. Find the cost of the shingles for this roof at $5.00 per M, allowing 1000 shingles for 120 square feet. 6. When it is noon at Boston, 71° 4' west longitude, what is the time at Rochester, 77° 51' west longitude? 7. a. A six months' note for $900 without interest, dated Oct. 26, 1906, is discounted Feb. 21, 1907, at 6%. What are the proceeds? b. If the note were interest-bearing, what would be the proceeds? 8. A tract of land is 424 rods long and 324 rods wide. It cost $36919.80. What was the cost per acre? 9. Three loads of coal weighing respectively 3805 lb., 3965 lb., and 4730 lb., cost $38.75. What was the price per ton? 10. Find the square root of 160 correct to four decimal places. APPENDIX CUBE ROOT THE cube of a number composed of tens and units may be found as follows: 3 times the square of tens multiplied by units 3 × (202 × 4) The cube of a number composed of tens and units is equal to the cube of the tens plus 3 times the square of the tens multiplied by the units, plus 3 times the tens multiplied by the square of the units, plus the cube of the units. By reversing the process, we may find the cube root. 1. What is the cube root of 13,824? SOLUTION. Separating into periods of three figures each, beginning at units, we have 13'824. Since there are two periods in the power, there must be two figures in the root, tens and units. 392 The greatest cube of tens contained in 13824 is 8000, and its cube root is 20 (3 x tens2 + 3 x tens × units + units2) × units = 5824 Subtracting the cube of the tens, 8000, the remainder, 5824, consists of 3 × (tens2 x units) + 3 × (tens × units2) + units3. 5824, therefore, is composed of two factors, units being one of them, and 3 × tens2 + 3 × tens x units + units2, being the other. But the greater part of this factor is 3 × tens2. By trial we divide 5824 by 3 x tens2 (1200) to find the other factor (units), which is 4 if correct. Completing the divisor, we have 12002 + 3 × (20 + 4) + 42 = 1456, which, multiplied by the units, 4, gives the product, 5824, proving the correctness of the work. Therefore, the cube root is 20+ 4, or 24. To find the cube root by the aid of blocks. Finding the cube root of a number is equivalent to finding the thickness of a cube, its volume being given. The following formulas illustrate the principles that underlie operations in cube root. NOTE. - For convenience, 7, b, t, and v will represent length, breadth, thickness, and volume, respectively. (1) lxbxt = v. (2) v ÷ (1 × b)=t. (4) v÷(bxt)=l. (3) v÷ (lxt) = b. 2. What is the thickness of a cube whose volume is 13824 cubic feet? mainder of 5824 cu. ft., which are added in solids of equal thickness to three sides of A, as seen in Fig. 2. It now remains to find the thickness of the additions (b, c, d), (e, f, g), and h, which have a uniform thickness. As the solids, b, c, d, form the greater part of the volume of the additions (5824 cu. ft.), and the length and breadth of each is 20 ft. (the length and breadth of A), by trial, using Formula 2, we find 5824 ÷ (3 × 202) = 4 ft., thickness of the additions, if correct. Knowing the thickness, which is also the breadth of e, f, g, h, we find the product of the length and breadth of e, f, g = 3 × 20 × 4 = 240 sq. ft.; and that of h = 42 = 16 sq. ft.; both of which added to 1200 sq. ft. = the product of the length and = breadth of all the additions. This product, by Formula 1, multiplied by the thickness, 4 ft. = 5824 cu. ft., proving the correctness. Therefore, The thickness of a cube whose volume is 13824 cu. ft. is 20 + 4 ft., or 24 ft. The numbers in the middle column (Ex. 2) all indicate volume: 5824 volume of the additions (b, c, d), (e, f, g), and h. = The numbers in the left-hand column indicate product of length and breadth: The numbers in the right-hand column indicate thickness: |