Algebraic Problems

1. A man paid $468 for an equal number of cows and sheep. He paid $35 each for the cows and $4 each for the sheep. How many of each did he buy?

2. I have $1.68 in dimes, 5-cent pieces, and cents. I have twice as many dimes as 5-cent pieces, and three times as many cents as 5-cent pieces. How many of each have I?

3. A gentleman, after giving away and of his money, had $10,000 left. How much had he at first?

4. A piece of land containing 180 acres is divided into 3 fields. The first field contains 10 acres more than the second, and the second 20 acres less than the third. How many acres are there in each of the 3 fields?

5. I bought 70 stamps. I bought 10 more 2-cent stamps than 1-cent stamps and 25 more 2-cent stamps than 5-cent stamps. How many of each did I buy?

6. The length of a rectangular field is 12 rods more than its width. The distance around it is 84 rods. Find its length and width.

7. The length of a rectangular field is five times its width. It contains 1125 square rods. Find its length

and width.

8. In 6 days 16 men and 16 boys earn $259.20. The boys earn as much as the men. Find the daily wages of

each.

9. The time past 12 o'clock equals of the time before 1 o'clock. What time is it?

10. If the time past 4 o'clock is of the time remaining before 6 o'clock, what time is it?

The additions will consist of two rectangles, each 60 units long, and a small square whose length will be equal to the width of the rectangles. These additions may be imagined to form a continuous rectangle whose approximate length is 2 × 60, or 120. Since 1161 is the number of square units in the additions, and 120 is their approximate length, their width will be approximately the number of times that 120 is contained in 1161, or 9. If 9 is the true width of the additions, it must also be the length of the small square, and 120 + 9, or 129, must be the true length of all the additions. Multiplying this number by 9, which is the width of the additions, we have 1161 square units for the entire surface of the additions. In making the additions we have thus used all the square units that remained.

Find the square root and explain the process geometrically :

2. 1225

3. 2209

4. 2809

5. 3364

Square Root Formal Explanation

1. Find the square root of 54,756.

5'47'56(234

4

43)147

129 464)1856

1856

Beginning at the right and separating the fig ures into sections of two figures each, we have three sections, hence there will be three figures in the root. The greatest perfect square in the first section is 4, and its square root is 2. Placing the root figure at the right and subtracting its square, we have 1 remaining. Annexing the next section of figures, we have 147. We double the root figure, 2, making 4, which we place at the left of 147. This trial divisor, 4, which is to be regarded as 40, is contained in 147 three times. We write the 3 in the root and also annex it to the 4, making 43 for the complete divisor. Multiplying the complete divisor by the root figure, we have 129. Subtracting this and annexing the next section of figures, we have 1856. We next double the root figures, 23, for a new trial divisor, writing it at the left of 1856 and regarding it as 460. This is contained in 1856 four times. We write the 4 in the root and annex it to 46, making 464 for the complete divisor. Multiplying the complete divisor by the last root figure, we have no remainder, hence the square root of 47,560 is 234.

2. Find the square root of 162,409.

In this example the trial divisor 8, which is to be considered as 80, will not be contained in 24. We write zero both in the root and at the right of the 8, making the new trial divisor 80, which is to be regarded as 800. We now bring down the next section of figures.

800 is contained in 2409

16/24/09 (403

16

803)2409

2409

three times. We write the 3 in the root and annex it to the trial divisor to complete the divisor. Multiplying the complete divisor by 3, we have no remainder.

Find the square root of the following:

Miscellaneous Problems

1. To make 18% profit, at what price must goods be marked that cost $7.50 ?

2. An agent collected bills for me on a commission of 10%. He paid me $630. How much did he collect?

3. A board 16 ft. long and 9 in. wide contains the same number of square feet as another board that is 8 ft. long. How wide is the latter board?

4. Find the number of square yards of plastering in a room 16' x 14' x 9', making no allowance for doors, windows, etc.

How many

5. A field 80 rods long contains 72 acres. feet of wire will it take to build a fence 4 wires high around it?

6. A room 36 ft. x 22 ft. x 14 ft. contains 40 persons. How many cubic feet of air space are there for each person?

7. When a gold dollar was worth $1.12 in paper money, what was the value in gold of a $20 bill?

8. Find the value of half a dozen silver spoons, each weighing 3 oz. 2 pwt., at $1.15 per ounce.

9. If silver coin is pure silver, how many ounces of silver are there in 500 silver dollars, each weighing 4121 grains?

10. If gold coin contains 90% gold, 9% silver, and 1% copper, find the amount of pure gold in a $10 gold piece weighing 258 grains.

11. If the specific gravity of gold is 19.258, how many pounds Avoirdupois would a gold brick weigh of the size of a common brick?

12. If the specific gravity of silver is 10.474, what would be the weight of a silver brick ?

1. A tank 12 filled with water.

Measurements

ft. long, 21 ft. wide, and 16 in. deep is How many gallons does it contain?

2. How many bushels of oats can be put into a bin that is 6 ft. long, 21 ft. wide, and 3 ft. deep?

3. In a rainfall of 1.2 inches, how many pounds of water fall upon a lot 100 ft. long and 60 ft. wide?

4. How much will it cost to pave a street 18 rd. long and 32 ft. wide, at 55 a square yard?

5. How much will it cost to build a road 3 miles, 90 rods long, at $2000 a mile?

6. How many square rods are there in a lot that is 60 ft. wide at one end, 40 ft. wide at the other end, and 10 rd. long?

7. What will be the cost of a piece of land 4200 ft. long, 3600 ft. wide, at $ 200 an acre?

8. A lot of land is 24 rd. wide. How long a piece of it will make an acre?

9. Find the cost of 15 boards, each 17 ft. long, 8 in. wide, and 1 in. thick, at 43 a foot, board measure.

10. Find the cost of a pile of wood 42 ft. long, 4 ft. wide, and 7 ft. high, at $ 5.60 a cord.

11. A pile of wood 60 ft. long, 4 ft. wide, contains 12 cords. How high is it?

12. How much will 5 lb. 10 oz. of butter cost, at 23¢ a pound?

13. Find the cost of 5250 lb. of coal, at $5.25 a ton.

14. If 7 lb. 4 oz. of cheese costs $1.16, how much will 3 lb. 2 oz. cost?