10. 9 × 9, -6, +5, ÷20, −1, × 8, +9, ×2, ÷11, ×5.

11. 87, 3, 5, +6, x 9, +2, ÷ 8, × 6, +3, ÷ 9. ÷ − ×

12. 64÷8, + 7, × 4, ÷ 3, + 8, ÷ 7, × 9, − 20, × 3, ÷ 4, × 0.

13. 68, 10, × 9, ÷ 3, + 12, − 6, + 2, ÷ 5, — 4, + 17.

14. 9 × 5, ÷ 3, + 6, ÷ 7, + 9, × 8, ÷ 3, + 3, ÷ 7, × 0. 15. 6 × 3, +7, ÷ 5, × 3, + 5, ÷ 4, + 13, ÷ 6, + 7, × 12.

Similar examples should be dictated daily.

MISCELLANEOUS PROBLEMS

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1. Lake Ontario is 234 feet above sea level, Lake Erie is 330 feet higher than Lake Ontario, Lake Huron 10 feet higher than Lake Erie, Lake Michigan 4 feet higher than Lake Huron, and Lake Superior 22 feet higher than Lake Michigan. How many feet above sea level is Lake Superior?

A leaf from the time book of Mr. Blake, a contractor, showing the number of hours each employe worked each day for a week, and the pay of each per hour.

2. Find (1) the total number of hours each man worked; (2) the wages of each for the week; and (3) total pay of all.

3. A plumber receives 80 cents an hour for a day's work of 8 hours, and double pay for overtime. How much does he earn in a week when he works 6 hours overtime?

4. What are the average daily earnings of a boy who receives $0.88, $0.25, $1.15, $0.75, $0.50, and $0.61 in one week?

5. One suit of clothes cost $12.75 and another twice as much. What did both cost?

6. Two farm wagons cost $50.

One cost $27.50.

How much did the other cost?

7. What is the cost of 8 plows at $6.75 each?

8. How many churns at $6.75 each can be bought for $108?

9. 18 watches cost $135; what is the price of one? 10. What is the gain on a dozen cans of tomatoes bought at $1.32 per dozen and sold at 15 cents apiece?

11. When 75 pounds of ham are bought for 24 cents a pound, for how much must the lot be sold to gain 4 cents a pound?

12. With money received by a will a man bought a house for $3950 and 16 acres of land at $125 an acre. What was the amount of the legacy?

13. Mr. Holt raised 288 bushels of rye on 24 acres of land. At the same yield per acre, how many bushels did he get the next year from a field of 8 acres?

14. A dealer in farm supplies paid $675 for hay rakes at $18.75 each. How many did he buy?

15. At another time he bought 28 potato diggers for $567. How much apiece?

16. If he paid $18.75 apiece for cotton planters and sold them at $25 each, what was his gain on 36?

17. In one month he made $180 by buying disc cultivators at $22.50 and selling at $30. How many did he sell? 18. At $1.28 each, how much will 24 umbrellas cost? 19. What is the cost of 15 tons of coal at $6.75 a ton, and 6 cords of wood at $7.50 a cord?

(1) Total cash sales in each department for the week. (2) Total charge sales in each department for the week. (3) Total cash sales on each day of the week. (4) Total charge sales on each day of the week. (5) Total sales in each department for the week. (6) Total sales in all departments for the week. (7) Total sales in all departments for each day. (8) Total daily sales in all departments for the week.

21. A telephone rental is $3.50 a month. What is the yearly rental?

22. A business man pays yearly $36 for his office telephone, and $27 for his house telephone. How much does he pay every month?

23. Boxwood 2-foot rules are bought at 8 cents and sold for 10 cents. How many must be sold to gain one dollar? 24. At $1.62 a yard, a piece of silk cost $84.24. How many yards in the piece?

1. Name five numbers between 10 and 100 and tell their factors.

2. What are the factors of a number?

A number that can be separated into factors is a composite number.

3. Name the composite numbers below 26; between 26 and 47; 47 and 73; 73 and 100.

A number that cannot be separated into factors is a prime number.

4. Name the prime numbers below 25; between 25 and 50; between 50 and 75; between 75 and 100.

A prime number used as a factor is a prime factor.

5. Name the prime factors of 60.

We may think of 60 as 6 × 10. The prime factors of 6 are 2 and 3; the prime factors of 10 are 2 and 5; therefore, the prime factors of 60 are 2, 2, 3, and 5.

6. Name the prime factors of:

28 36 40 48 54 56 72 80 81 144

NOTE. Going rapidly around the class, let the pupils recite as follows: 1 is a prime number; 2 is a prime number; 3 is a prime number;

4 is a composite number, its prime factors are 2 and 2; 5 is a prime number; 6 is a composite number, its prime factors are 2 and 3; and so on to 100.

When several equal factors occur in the answer, a small figure, called an exponent, is written at the right and a little above the factor to show how many times the factor is used. Thus, 23 means that 2 is used as a factor three

times. 2 x 2 x 2 = 8.

The factors of 72-2, 2, 2, 3, 3 are written 23 x 32. 7. What number does 52 represent? 25?

8. 2 x 52 are the prime factors of what number? 9. Of what number are 22 x 32 the prime factors?

10. The prime factors of a number are 22 × 3 × 52. What is the number?

11. What are the prime factors of 168?

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To find the prime factors of a number not readily factored by inspection, we divide the number by one of its prime factors; then divide the resulting quotient by one of its prime factors, and continue the division until the resulting quo tient is prime. The several divisors and the last quotient are the prime factors.

1. What is the greatest number that will exactly divide 12 and 18?

The greatest number that will exactly divide two or more numbers is their greatest common divisor (g. c. d.).