foot

49. It requires 144 square inches to make one square how many square feet are there in 46800 square inches? Ans. 325 square feet.

;

50. A cistern which holds 7250 gallons, is to be filled with water by a pipe which pours into it 290 gallons per hour; in what time will the cistern be filled?

Ans. 25 hours.

51. Allowing a steamship to run 375 miles per day, in what time would she make a voyage of 7875 miles?

Ans. 21 days.

52. It requires 1728 cubic inches to make one cubic foot; how many cubic feet then will 525312 cubic inches make? Ans. 304 cubic feet.

53. A reservoir containing 106950 gallons of water, is to be emptied by a pump which discharges 465 gallons per hour. In what time will the reservoir be emptied?

Ans. 230 hours.

54. If the population of a State containing 135 counties, be 3180600, what is the average number of inhabitants to each county? Ans. 23560.

55. If the velocity of light is 192500 miles per second, and the distance from the Sun to the Earth is 95000000 miles; how long is light in coming from the Sun to the earth? Ans. 493-97500 seconds.

Successive Divisions.

192500

(44.) Dividing by one number, and the quotient thus obtained by another, and so on, is equivalent to dividing by the product of the several divisors.

The last remainder x the last divisor but one, + the preceding remainder, the next preceding divisor, + the next preceding remainder, and so on, will be the true remainder of the dividend.

For example, dividing 836 by 3, and the quotient thus obtained by 7, is equivalent to dividing the first number by the product 21 of 3 and 7.

3)836

7)278 2 over.

39 5 over.

The quotient is 39, and the remainder is 5 x 3 + 2 = 17.

The remainder 2 is units of the given dividend; the remainder 5 is units of the dividend 278, or first quotient; and since the first quotient × the first divisor 3, produces units of the given dividend, it follows that the remainder 5 × the divisor 3, produces units of the given dividend, to which the 2 must be added, for the entire remainder of the dividend.

56. Allowing a railroad car to run at the rate of 45 miles an hour, in what time would it run 6120 miles?

Ans. 136 hours.

57. A hogshead of ale or beer contains 54 gallons; how many hogsheads then might be filled with 11070 gallons? Ans. 205 hogsheads.

58. If 81 men take equal shares of 13846 dollars, how inany dollars will fall to the share of each inan?

Ans. 170 dollars.

59. A field which contains 96 acres, produced 14304 bushels of corn; what was the average yield per acre?

Ans. 149 bushels.

60. A canal which is 132 miles in length, cost 268637 dollars; what did this canal cost per mile, one mile with another? Ans. 2035 dollars.

Cancellation in Division.

(45.) Equal factors may be canceled from a dividend and its divisor, without altering the quotient; for this is equivalent to dividing the dividend and divisor by the same number (38.) For example, suppose that 132 is to be divided by 48.

The division and cancellation may be denoted by placing the dividend over the divisor, and drawing lines across the canceled equal factors; thus

Cancellation may always be thus employed to simplify Division, when the dividend and divisor contain equal factors. To give other examples ;

Use of the Parenthesis, or Vinculum.

(46.) A parenthesis ( ) enclosing a numerical expression, or a vinculum drawn over it, connects the value of that expression with the sign which immediately precedes or follows it.

Thus 20 (4+5), the sum of 4 and 5 subtracted from 20. (4×5) × 6, or 20 – 6 × (4+5), denotes 6 times the sum of 4 and 5 subtracted from 20.

20

20(4+5) x 6, or 6 x 20 (4+5) denotes

6 times the difference between 20 and the sum of 4 and 5. The use of these, as well as of the other signs which have been explained, will be fully seen hereafter.

MISCELLANEOUS EXERCISES.

ON NOTATION, ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION. ABBREVIATED MULTIPLICATION AND DIVISION.

1. Find the sum of 19 thousand and thirteen, 105 thousand and ten, 3 millions 94 thousand and sixty, and 45 millions two hundred and five. Ans. 48218288.

2. Find the sum of 135 thousand four hundred, 500 millions thirty-five thousand, 350 millions and forty, and 4 billions 24 millions 30 thousand. Ans. 4874200440.

3. Find the difference between 9 trillions 31 millions 360 thousand five hundred and three, and 10 billions 5 millions 273 thousand 8 hundred and four.

Ans. 8990026086699.

360 billions 204 millions.

4. Find the difference between 34 thousand three hundred, and 2 trillions 375 millions 183 thousand 7 hundred and sixteen. Ans. 1640171149116. 5. Find the product of 350 thousand and nineteen, multiplied by the sum of 5 thousand four hundred, and 100 thousand two hundred and ten. Ans. 36965506590.

6. Find the product of 4 millions 3 thousand seven hundred, multiplied by the sum of 300 and 21 thousand nine hundred and seventy-seven. Ans. 80190424900.

.

7. Find the quotient of 95 trillions 200 millions, divided by the difference between 275 millions 100 thousand and 75 millions 100 thousand. Ans. 475001.

8. Find the Quotient of 495 millions 66 thousand 570, divided by the Difference between 150 thousand 100 and 55 thousand four hundred and 41. Ans. 5230.

9. The Sum of two numbers being 346790, and the less number 39035, what is the greater number? (22)

Ans. 307755

10. The Sum of two numbers being 5706803, and the greater number 3983000, what is the less number?

Ans. 1723803.

11. The Product of two numbers being 64605700, and the multiplier 740, what is the multiplicand? (37)

Ans. 87305.

12. The Product of two numbers being 72958584, and the multiplicand 8076, what is the multiplier?

Ans. 9034.

13. The Sum of two numbers is 12384, and the Difference is 3600; what are the two numbers?

Since the Difference of two numbers, added to the less number, makes the greater number, it is plain that

The Sum of two numbers + their difference, is twice the greater number; and the Sum of two numbers their difference, is twice the less number.

The two numbers required above are therefore

of (12384+3600), and of (12384-3600).

Ans. 7992 and 4392.

14. The Sum of two numbers is 3864, and their Difference is 2400; what are the two numbers? Ans. 3132 and 732. 15. The Sum of two numbers is 45831, and their Difference is 3500; what are the two numbers?

Ans. 24665 and 211651. 16. A and B together have 9875 dollars, and A has 1260 dollars more than B; what sum has each of them?

Ans. A has 55671, and B 43071⁄2 dollars. 17. A borrowed of B 8794 dollars, of which he paid to B at one time 2340 dollars, and at another time 1375 dollars; what balance remains to be paid?

The balance that remains is the Difference between the sum borrowed and the aggregate of the two payments made, which may be denoted thus:

8794 — (2340 + 1375).

Ans. 5079 dollars.