10. A house rents for $55.17 per month; what is the rent for 12 months or 1 year?

11. What is the sum of the following items?—

5 lb. sugar at 17 ct. a pound;

7 lb. coffee at 43 ct. a pound;

5 boxes raisins at $2.25 a box.

12. What is the amount of the following bill of goods?— 3 yds. calico at 17 ct. a yard;

13 yds. tape at 1 ct. a yard;
11 skeins of silk at 5 ct, a skein;
6 pieces of braid at 91 ct. a piece.

50. Division.-The method of finding how many times one number is contained in another, or of separating one number into as many equal parts as there are units in another, is called division.

51. The Dividend is the number to be divided.

52. The Divisor is the number by which the dividend is divided.

53. The Quotient is the number of times the divisor is contained in the dividend.

54. In Short Division the quotient is set down at once. It is the preferable method, when the division can be as readily performed.

55. In Long Division each step of the process is written. It is the method generally used, when the divisor is greater than twelve.

56. A Remainder in division is the part of the divi-. dend which remains undivided.

57. The Sign of Division.-Division is indicated by a short horizontal line with a dot above and below it, thus (÷), and it shows that the number before it is to be divided by the number after it. Thus, 20÷4 means that 20 is to be divided by 4.

58. The Fractional Sign of Division.-Division is also indicated by writing the dividend above a line and the divisor below it. Thus, 12 means that 12 is to be divided by 3. 59. The Axioms used in division are,

1. If equal numbers be divided by the same or equal numbers, the quotients will be equal.

2. If a number be both multiplied and divided by the same number, its value will not be changed.

60. ANALYSIS OF SHORT DIVISION.

(a.) Problem.-Divide 70301 by 4.

MODEL OPERATION.

(a.) Anal. Form.*-1. For convenience write the divisor at the left of the dividend, and commence by dividing the left hand digit.

2. One-fourth of 7 tens of thousands is 1 ten of thousands with a remainder of 3 tens of thousands, equal to 30 thousand; write the 1 in the place of tens of thousands, and add the 30 thousands to the thousands.

3. One-fourth of 30 thousands is 7 thousands with a remainder of two thousands, which is equal to 20 hundreds; write the 7 in the place of thousands, and add the 20 hundreds to the hundreds; 20 hundreds and 3 hundreds are 23 hundreds.

4. One-fourth of 23 hundreds is 5 hundreds with a remainder of 3 hundreds, equal to 30 tens; write the 5 in hundreds' place, and add the 30 tens to the tens.

In this analysis we prefer the fractional form, as it is simpler and more readily understood, although liable to some objection. The teacher will find another form in the analysis of Long Division (62), which he can use here if he prefers it.

·5. One-fourth of 30 tens is 7 tens with a remainder of 2 tens, equal to 20 units; write the 7 in the place of tens, and add the 20 units to the units; 20 units and 1 unit are 21 units.

6. One-fourth of 21 units is 5 units with a remainder of 1 unit, which divided by 4 gives 1 fourth of a unit, written; write the 5 in the place of units, and annex the of a unit.

(c.) Conclusion.-Therefore one-fourth of 70301 is 175754.

! (d.) Rule for Short Division.—(Write a Rule from the Analysis.) (e.) Proof.-Division may be proved by multiplying the QUOTIENT by the integral part of the DIVISOR, and adding to the product the remainder, if there is any. The result will be equal to the dividend, if the work is correct.

TEST PROBLEMS FOR REVIEW.

Problem,-Divide $3786 by 7.

$3785.9997 equal to the dividend lacking 3 tenths of a mill. (a.) Anal. Steps.-1. Reduce $3786 to tenths of a mill.

2. Find 1 seventh of 37860000 tenths of a mill.

FORMULA.-One seventh of 37860000 tenths of a mill is 5408571 tenths, equal $540.8571+ (with a remainder); therefore $3786 divided by 7 equals $540.857+. NOTE.-Division of United States currency may be proved as simple division.

1. Divide $384. by 4.

2. Divide $417. by 5.
3. Divide $613. by 7.
4. Divide $4.13 by 2.
5. Divide $5.20 by 5.
6. Divide $3.42 by 4.
7. Divide $5.783 by 6.
8. Divide $4.173 by 4.
9. Divide $5.132 by 5.

10. Divide $5 by 8.

11. Divide 38 ct. by 8.
12. Divide $9 by 5.
13. Divide 3 dollars by 9.
14. Divide 1 dollar by 7.
15. Divide 7 mills by 8.
16. Divide 1 cent by 7.
17. Divide $1 by 3.
18. Divide $1 by 5.
19. Divide $ by 6.
20. Divide $2 by 8.

62. ANALYSIS OF LONG DIVISION.

(a.) Problem.-Divide 26306241 by 483.

(b.) FORM.-1. For convenience,* write the divisor at the left, and the quotient when obtained at the right of the dividend.

2. 483 is contained in 2630 tens of thousands 5 ten-thousand times, with a remainder of 215 tens of thousands, equal to 2150 thousands; write the 5 in the place of tens of thousands in the quotient, and to the 2150 thousands add the 6 thousands of the dividend, which makes 2156 thousands.

3. 483 is contained in 2156 thousand 4 thousand times, with a remainder of 224 thousand, equal to 2240 hundreds; write the 4 in the place of thousands in the quotient, and to the 2240 hundreds add the 2 hundreds of the dividend, which makes 2242 hundreds.

4. 483 is contained in 2242 hundreds 4 hundred times with a remainder of 310 hundreds, equal to 3100 tens; write the 4 in hundreds' place in the quotient, and to the 3100 tens add the 4 tens of the dividend, which makes 3104 tens.

5. 483 tens is contained in 3104 sixty times with a remainder of 206 tens equal to 2060 units; write the 6 in the tens' place in the quotient, and to the 2060 units add the 1 unit of the dividend, which makes 2061 units.

NOTE.-It is perhaps more convenient to place the divisor at the right of the dividend and the quotient under it.

6. 483 is contained in 2061 units 4 times with a remainder of 129 units; write the 4 in the units' place in the quotient, and divide the remainder of 129 units by 483, giving 19 of a unit to be annexed to the quotient.

12

129

(c.) Conclusion. Therefore 26,306,241 divided by 483 equals 54,464 33 (d.) Rule for Long Division.-Write a rule from the Analysis.)

(e.) Proof-The proof of Long Division is the same as that of Short

Division.

NOTES.-1. The first step in Long Division is to find the quotient figure of the highest denomination.

2. Multiply the divisor by that figure, and write the product under the first partial dividend, then observe that if the product is greater than the partial dividend the quotient figure is TOO LARGE.

3. Subtract that product from the first partial dividend, and observe that if the REMAINDER is equal to or GREATER than the DIVISOR, the quotient figure is

TOO SMALL.

4. Change the remainder to the next lower denomination, and add to it the corresponding denomination in the dividend.

If 27 pounds of tea cost $30%, what will 1 pound cost?

FORM.-$30 equals 302000 tenths of a mill.

cost 1 lb.

Since 27 lb. cost 802000 tenths, 1 lb. will cost one twenty-seventh of 302000 tenths, which is 11185 tenths (with a remainder), equal to $1.1185+

NOTE.-If there is any remainder in dividing tenths, put the sign plus (+) after the answer; for an example, see above problem. Such problems will not prove exactly unless the remainders are kept and added, as in simple division.