Dividing by One-Digit Numbers

Same as the division, here i use the below notation to represent the rod:

Example :    8/ 2 = 4 

Step 1: Set the dividend 8 on rod F and the divisor 2 on rod A, with four vacant rods between the two numbers. Make sure that F is a unit rod marked with a unit point. 

Step 2: Mentally divide 8 by 2 ( 8 / 2 = 4 ); set the quotient 4 on D, the second rod to the left of the dividend; and clear F of its 8. The row of figures designated Result in the diagram show the result of this step.

Example:     837+ 3 = 279 

Step 1: Set 837 on the rods FGH, with H as the unit rod, and set 3 on A.

Step 2: Compare the 3 with the 8 in 837. 3 goes into 8 twice with 2 left over. Set the quotient figure 2 on D, the second rod to the left of 8 in 837. Next multiply the divisor 3 by this quotient figure 2, and subtract the product 6 from the 8 on F. This leaves 2 on F. 

Step 3: Compare the 3 with 23 on FG. The 2 on F is the remainder left over as a result of the previous step. 3 goes into 23 seven times with 2 left over. Set 7 as the quotient figure on E. Next multiply the divisor 3 by this 7, and subtract the product 21 from the 23 on FG. This leaves 2 on G. 

Step 4: Compare the 3 with 27 on GH. The 2 on G is the remainder left over as a result of the second step. 3 goes into 27 nine times. Set the quotient figure on F. Next multiply the 3 by this 9, and subtract the product 27 from the 27 on GH. This clears GH and leaves the answer 279 on DEF. 

Note: Answers to problems in division can be easily checked by multiplication. Thus, to check the foregoing answer, simply multiply the quotient 279 on DEF by the divisor 3, that is, the number you originally divided by, and you will get the product 837 on FGH, i. e., the same rods on which you had 837 as the dividend. By this checking the student will see that the position of the quotient in division is that of the multiplicand in multiplication, and that the position of the dividend in division is that of the product in multiplication.

Example:     6 013 / 7 = 859 

Step 1: Set 6,013 on FGHL with 1 as the unit rod, and set 7 on A. 

Step 2: Compare the divisor 7 with the 6 in 6,013. 7 will not go into 6. So compare the 7 with the 60 in 6,013. 7 goes into 60 eight times. In this case set the quotient figure on E. the first rod to the left of the first digit of the dividend. Next multiply the divisor 7 by this 8, and subtract the product 56 from the 60 on FG. This leaves 4 on G. 

Step 3: Compare the 7 with 41 on GH 7 goes into 41 five times. Set the quotient figure 5 on F. Next multiply the 7 by this 5, and subtract the product 35 from the 41 on GH. This leaves 6 on H. 

Step 4: Compare the 7 with 63 remaining on HI. 7 goes into 63 nine times. Set the quotient figure 9 on G. Next multiply the 7 by this 9, and subtract the product 63 from the 63 remaining on HI. This clears HI, and leaves the answer 859 on EFG. 

Note: When the divisor is larger than the first digit of the dividend, compare it with the first two digits of the dividend. In this case set the quotient figure on the first rod to the left of the first digit of the dividend. The chief merit of this procedure is that, in checking, the quotient multiplied by the divisor gives the product on the very rods on which the dividend was located previous to its division.  This procedure is the same as the principle of graphic division. In dividing 36 by 2, you write the quotient figure 1 above the 3 in 36. But in dividing 36 by 4, you write the quotient figure 9 above the 6 in 36. On the abacus board the quotient figure cannot be put above the dividend. So in dividing 36 by 2, the first quotient figure 1 is set on the second rod to the left of 36, while in dividing 36 by 4, the quotient figure 9 is set on the first rod to the left of 36.