11. Four men together paid $20,000 for some land. The first puts in $2350, the second $5820.35, the third $7640.75. How much must the fourth man pay? 12. What will be the cost of uniforms for a base-ball nine at $2.87 for each uniform?

13. At $15.87 a ton, what will be the value of 637 tons of hay?

14. If peaches are worth $1.25 a basket, and it takes 3 dozen for a basket, what is the value of 2892 dozen peaches?

15. If 964 baskets of peaches are sold for $1301.40, what is the price per basket?

16. If 324 men contribute together $2647.08, what is the contribution of each?

17. A boy picks blueberries in a pasture, giving to the owner of the pasture for the privilege 1 quart out of every 8 quarts. In 2 days he picks 48 quarts, and sells his share of the berries for $3.78. What did he get a quart?

18. If 150 men work on a railroad at the same price per day, and if, at the end of the week, they all together

receive $1575, what price per day does each man receive?

19. If a kite-string is 213.86 feet long, and the kite breaks away and carries off 94.38 feet of the string, how much will be left? How much more must be bought to make up 1000 feet?

20. At $9.17 a barrel, how many barrels of flour can be bought for $876.35, and how much money will be

left over?

Ex. 60.

If the length of the diameter of a circle is multiplied by 3.1416, the product is the length of the circumference.

1. Find the length in inches of the circumference of a circle if the diameter is 6 inches.

Circumference

Diameter

2. Find the length in inches of the circumference of a circle if the diameter is 17 inches.

3. If a carriage wheel is 4 feet in diameter, what is its circumference in feet?

4. If the carriage wheel in Example 3 rolls on the ground without slipping, how many feet will it go in turning 27 times?

5. How many times will the carriage wheel in Example 3 turn in going 1160 feet?

If the length of the circumference of a circle is divided by 3.1416, the quotient is the length of the diameter.

6. Find the diameter in feet of a circle if its circumference is 1000 feet.

7. Find the diameter in feet of a wheel which revolves 19.5 times in going 253.5 feet.

8. If the circumference of a circle is 198 yards, what is its diameter in yards?

9. How deep is a well if the wheel whose diameter is 2 feet makes 10 revolutions in raising the bucket?

10. If a carriage wheel makes 440 revolutions in travelling a mile (5280 feet), what is its diameter in feet?

11. If 1964.52 bushels of corn are to be put into bags holding 2.14 bushels each, how many bags will it take? 12. A boy has 3 pieces of twine: one is 58.74 feet long, another is 97.86 feet, and a third 57.26 feet. How

long a kite-string can he make, making no allowance for knots?

13. Boys in playing hare and hound run 3.876 miles. The hares drop a piece of paper every 4.75 feet on

the average. How many pieces do they drop? A mile is 5280 feet.

14. If a man earns $23.25 a day, how many days will it take him to earn $1964.87?

15. A grain merchant bought corn at 60 cents and rye at 75 a bushel. He bought the same number of bushels of both kinds of grain and paid for both together $607.50. How many bushels of each kind did he buy?

16. When potatoes are worth $0.77 per bushel, and corn $1.10, how many bushels of corn should a farmer receive in exchange for 50 bushels of potatoes?

17. How many gallons of 231 cubic inches are contained in a cubic foot (1728 cubic inches)?

18. How many cubic feet are contained in a bushel, a bushel containing 2150.42 cubic inches?

19. For $7624.13 how many tons of hay can be bought at $18.75 a ton?

20. The large wheel of a bicycle is 14.37 feet around. How many times will it turn in going a mile (5280 feet)?

CHAPTER VII.

MULTIPLES AND MEASURES.

85. When the multiplier is an integral number, the product is called a multiple of the multiplicand; and, in division, when the quotient is an integral number, the divisor is called a measure of the dividend. Thus, 8 x 7=56; the number 56 is a multiple of 7. Again, 56÷ 7 = 8; the number 7 is a measure of 56.

86. A number which cannot be divided by any other number except unity without remainder is called a prime number.

Thus, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, etc., are prime numbers.

87. Other numbers are each the product of a fixed set of prime numbers, and are called composite numbers.

88. Numbers which can be divided by 2 without remainder are called even numbers; and all other numbers are called odd numbers. Even numbers end in 2, 4, 6, 8, or 0; odd numbers end in 1, 3, 5, 7, or 9.

89. By way of distinction, when a number is used without reference to any designated unit, it is called an abstract number; and, when used with reference to a specified unit, it is called a concrete number.

Thus, 5, 7, 8 are abstract numbers, and 5 horses, 7 chairs, 8 dollars are called, by way of distinction, concrete numbers.

90. To factor a composite number is to separate the number into its factors.

Find the prime factors of 144.

That is, 144 = 2×2× 2 × 2 × 3 × 3.

91. To avoid the necessity of writing long rows of equal factors, a small figure called the exponent is written at the right of a number to show how many times the number is taken as a factor.

Thus, 2 x 2 x 2 x 2 x 3 x 3 is written 2' x 32.

The expression 2a is called the fourth power of 2, and 32 is called the second power of 3.

92. It is evident from § 90 that the method of separating a composite number into its prime factors is,

Divide the given number by any prime number that is contained in it without remainder; then the quotient by any prime number that is contained in it without remainder; and so on until the quotient is itself a prime number. The several divisors and the last quotient are the prime factors. If no prime factor is found before the quotient becomes equal to or less than the divisor, the number is prime.

93. The following tests are useful for determining without actual division if a number contains certain factors: 1. A number is divisible by 2 if its last, or right hand, digit is even.

2. A number is divisible by 4 (22) if the number denoted by the last two digits is divisible by 4.

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