10. What time elapsed from the battle of Lexington, 1775, to the firing upon Fort Sumter, 1861?

11. If I buy real estate for $854657, agreeing to pay for it in yearly payments of $37159 each, how many payments will I make?

12. The improved land of the United States is estimated at 207198720 acres; how many townships of 23040 acres each could be made from this land?

13. In a pile of 4701625 bricks how many loads are there, if each load contains 1000 bricks?

14. Charlie saw a box car marked "Weight 28000 pounds." A ton is 2000 pounds. What is the weight of the car in tons?

15. What will be the cost of whitening the ceiling and walls of a schoolroom 44 feet long, 28 feet wide, and 13 feet high, at 5 cents a square yard, if 20 square yards are deducted for doors, windows, and baseboards?

16. Find the sum of sixteen million one thousand twenty, twelve million one hundred twentyeight, nine million thirteen thousand two, seven million sixteen thousand seven, and three hundred million nine.

17. How many years from the Centennial exhibit at Philadelphia in 1876 to the Columbian exhibit at Chicago in 1893?

18. In a field of turnips there are 296 rows, and each row yields 18 bushels. How many loads of 30 bushels each in the field?

19. $557283 added to a certain number of dollars will produce $1157003; what is the number?

20. If 68 pounds of coal are consumed in carrying a train one mile, how many pounds will be consumed at that rate in going 1894 miles?

21. What is the weight of a train of six cars if the baggage car weighs 6000 pounds, the express car 58000 pounds, the sleeping car 80000 pounds, and each of three passenger cars 54000 pounds? How many tons is that?

30. DIVISORS AND MULTIPLES.

What integer greater than 1 will exactly divide 12, 20, and 24? 4 will divide each of these numbers, and is called a common divisor; it is a divi'sor common to all of them. What number greater than 1 will exactly divide 25, 30, and 35?

is

31. A Common Divisor of two or more numbers any whole number greater than 1, which will exactly divide each of them.

Find the greatest number which will exactly divide 18, 27, and 36. 9 is the greatest common divisor of these numbers.

What is the greatest common divisor of 30, 45, 60, and 90?

32. The Greatest Common Divisor of two or more numbers is the greatest exact divisor of each of them.

The product of all the prime factors common to two or more numbers is their Greatest Common Divisor (G. C. D.).

33. Find the greatest common divisor of 70 and 180.

34. MULTIPLES.

Name a number that is 3 times 9.

tiple of 9; it is also a multiple of 3.

27 is a Mul

A multiple of 3 is the product of 3 multiplied by any whole number. A multiple of 9 is any number of times 9. 24, 21, and 28 are multiples of what numbers?

35. A Multiple of a Number is any integral (or whole) number of times that number.

Find the multiples which 3 and 5 have in comOf the common multiples 15, 30, 45, and 60, which is the least?

mon.

36. A multiple that is common to two or more numbers is called a Common Multiple, and the least of the common multiples is called the Least Common Multiple (L. C. M.).

37. Find the L. C. M. of 27, 24, and 30.

2 × 2 × 2 × 3 × 3 × 3 × 51080, L. C. M.

Resolve 27, 24, and 30 into their prime factors. Select the different factors and repeat each as many times as it is found in any one of the numbers. The product of 2, 2, 2, 3, 3, 3, and 5 is the L. C. M. of 27, 24, and 30.

A multiple of a number must contain at least all the prime factors of that number.

The least common multiple of two or more numbers must contain all the prime factors that enter into each of them, and no others.

Find the quotient of 4 x 6 divided by 2 x 6.

Of 4 x 8 divided by 2 × 8.

(4 × 7) ÷ (2 × 7) = ?

(6 x 8) ÷ (3 x 8) = ?

2 times any number is contained in 4 times that number, how many times?

(6 × 7) (3 × 7) = ? (9 × 5) ÷ (3 × 5) = ? 3 × 9 is contained in 6 × 9 as many times as 3 is contained in what? 3 times any number is contained how many times in 6 times that number?

39. To find quotients by the shortest method, strike out equal factors from the dividend and the divisor.

divisor by the same

Dividing both dividend and

number does not affect the quotient.

The process of striking out equal factors from

the dividend and the divisor is called Cancellation.