Dividing by Two-Digit Numbers or More Numbers

Example :   552 / 23 = 24  

Step 1: Set 552 on GHI, with 1 as the unit rod, and set 23 on AB. 

Step 2: Compare the 2 in 23 with the 5 in 552. 2 goes into 5 two times. Set the quotient figure 2 on E, the second rod to the left of the 5 in 552. Next multiply the 2 in 23 by this quotient figure 2, and subtract the product 4 from the 5 on G. This leaves 1 on G. 

Step 3: Now multiply the 3 in 23 by the same quotient figure 2, and subtract the product 6 from 15 on GH. This leaves 9 on H. 

Step 4: Compare the 2 in 23 with the 9 on H. 2 goes into 9 four times. Set the quotient figure 4 on F. Next multiply the 2 in 23 by this quotient figure 4, and subtract the product 8 from the 9 on H. This leaves 1 on H. 

Step 5: Multiply the 3 in 23 by the same 4, and subtract the product 12 from the 12 remaining on HI. This clears HI and leaves the answer 24 on EF.

Example :    6 308 / 83 = 76 

Step 1: Set 6 308 on GHIJ, with J as the unit rod, and set 83 on AB. 

Step 2: Compare the 8 in 83 with the 6 in 6,308. 8 will not go into 6. So compare the 8 with the 63 in 6,308. 8 goes into 63 seven times. Set the quotient figure 7 on F, the first rod to the left of the 6 in 6,308. Next multiply the 8 in 83 by this 7 in the quotient, and subtract the product 56 from the 63 on GH. This leaves 7 on H.  

Step 3: Multiply the 3 in 83 by the same quotient figure 7, and subtract the product 21 from 70 on HI. This leaves 49 on HI. 

Step 4: Compare the 8 in 83 with the 49 on HI. 8 goes into 49 six times. Set the quotient figure 6 on G. Next multiply the 8 in 83 by this 6, and subtract the product 48 from the 49 on HI. This leaves 1 on I. 

Step 5: Multiply the 3 in 83 by the same quotient figure 6, and subtract the product 18 from 18 on IJ. This clears IJ. and leaves the answer 76 on FG. 

Note: In case the divisor is a two-digit number, do not take the trouble of comparing its two digits with the first two or three digits of the dividend to work out the correct quotient figure mentally. Simply compare the first digit of the divisor with that of the dividend. When the first digit of the divisor is larger than that of the dividend, compare it with the first two digits of the dividend.

Example : 4,698 / 54 = 87 

This example shows how the process of division must be revised when too large a quotient figure has been used. 

Step 1: Set 4 698 on GHIJ, with J as the unit rod, and set 54 on AB. 

Step 2: The 5 in 54 will not go into the 4 in 4 698. So compare the 5 with the 46 in 4 698. 5 goes into 46 nine times. Now suppose you have tried 9 as the quotient figure instead of the correct 8 and have set it on F. Then you will multiply the 5 in 54 by 9, and subtract the product 45 from the 46 on GH. This leaves 1 on H. Next multiplying the 4 in 54 by the same 9, you will find that the product 36 is larger than the 19 remaining on HI and that you ought to have tried a quotient figure one less than 9. 

Step 3: To revise the incorrect quotient figure 9 to 8, subtract 1 from the 9 on F, and you get the new quotient figure 8 on F. Next multiply the 5 in 54 by 1, i. e., the difference between the quotient figures 9 and 8, and add the product 5 to the 1 on H. Now you have 6 on H. 

Step 4: Multiply the 4 in 54 by the new quotient figure 8, and subtract the product 32 from 69 on HI. This leaves 37 on HI. 

Step 5: Compare the 5 in 54 with the 37 on HI. 5 goes into 37 seven times. So set the quotient figure 7 on G. Next multiply the 5 in 54 by this 7, and subtract the product 35 from the 37 on HI. This leaves 2 on I. 

Step 6: Multiply the 4 in 54 by the same quotient figure 7, and subtract the product 28 from 28 remaining on IJ. This clears IJ and leaves the answer 87 on FG. 

Example:    1 666 / 17 = 98 

This example shows how a problem of division is worked when the first digit of both divisor and dividend are the same. 

Step 1: Set 1 666 on GHIJ, with J as the unit rod, and set 17 on AB. 

Step 2: When the first digit of the divisor and the dividend are the same, as in this example, compare the second digits of the two numbers. In such a situation, if the second digit of the dividend is smaller than that of the divisor, try 9 as the quotient figure. If 9 is too large, try 8 as in Step 5 of this example. If 8 is still too large, go on trying a quotient figure one less till the correct one is found. In such a case 9 is the figure likeliest to be correct.  Now try 9 as the quotient figure and set it on F, the first rod to the left of the first digit of the dividend. Next multiply the 1 in 17 by this 9 and subtract the product 9 from 16 on GH. This leaves 7 on H. 

Step 3: Multiply the 7 in 17 by this same 9, and subtract the product 63 from 76 on HI. This leaves 13 on HI.

Step 4: The 1 in 17 and the 1 remaining on H are the same. So compare the 7 in 17 and the 3 remaining on I. 3 is smaller than 7. So try 9 as the quotient figure and set it on G. Now multiply the 1 in 17 by this 9 and subtract the product 9 from the 13 on HI. This leaves 4 on 1. Next, multiplying the 7 in17 by this same 9, you will see that the product 63 is larger than 46 remaining on IJ. So you will find that you ought to have tried 8 as the quotient figure. 

Step 5: To revise the incorrect quotient figure 9 to 8, subtract 1 from the 9 on G. Next you must revise the division in Step 4. So multiply the 1 in 17 by 1, the difference between the 9 and 8, and add the product 1 to the 4 remaining on I. Then you get 5 on I. 

Step 6: Multiply the 7 in 17 by the new quotient figure 8 and subtract the product 56 from 56 on IJ. This clears IJ and leaves the answer 98 on FG. 

Note: In cases where the first digits of both the divisor and the dividend are the same, if the second digit of the dividend is larger than that of the divisor, set 1 as the quotient figure on the second rod to the left of the first digit of the dividend.

Example:    7 644 / 84 = 91

This example is to show how division is to be revised when the quotient figure tried is too small. 

Step 1: Set 7 644 on GHIJ, with J as the unit rod, and set 84 on AB. 

Step 2: The 8 in 84 will not go into the 7 in 7 644. So compare the 8 with the 76 in 7 644. 8 goes into 76 nine times. So you ought to try 9 as the quotient figure. But suppose by mistake you have tried 8 as the quotient figure instead of the correct 9 and have set it on F. Then you will multiply the 8 in 84 by 8 and subtract the product 64 from the 76 on GH. This will leave 12 on GH. 

Step 3: Multiplying the 4 in 84 by the same quotient figure 8, you will subtract the product 32 from 124 on GHI. Then you will find that the remainder 92 is larger than 84 and that you ought to have tried 9, i.e., a quotient figure one more than 8. 

Step 4: To revise the incorrect quotient figure 8 to 9, add 1 to the quotient figure 8 on F. Next multiply the divisor 84 by 1, i. e., the difference between the two quotient figures, 8 and 9, and subtract the product 84 from the 92 on HI. This leaves 8 on I.

Step 5: The 8 in 84 and the 8 remaining on I are the same. So compare the 4 in 84 with the 4 remaining on and you can see that they are also the same. Therefore, set the quotient figure 1 on G. Now, multiplying the 8 in 84 by 1, subtract the product 8 from the 8 on I. Next multiplying the 4 in 84 by the same 1, subtract the product 4 from the 4 on J. This clears IJ and leaves the quotient 91 on FG.

Example :    3 978 / 234 = 17 

Step 1: Set 3,978 on HIJK, with K as the unit rod, and set 234 on ABC. 

Step 2: Compare the 2 in 234 with the 3 in 3 978. 2 goes into 3 one time. Set the quotient figure 1 on F, the second rod to the left of the 3 in 3978. Now multiply the 2 in 234 by this quotient figure 1, and subtract the product 2 from the 3 on H. This leaves 1 on H.  

Step 3: Multiply the 3 in 234 by the same quotient figure 1, and subtract the product 3 from 9 on 1. This leaves 6 on 1 and 167 on HIJ. 

Step 4: Multiply the 4 in 234 by the same quotient figure 1, and subtract the product 4 from 7 on J. This leaves 3 on J and 1638 on HIJK. 

Step 5: Compare the 2 in 234 with the 16 remaining on HI. 2 goes into 16 eight times. Suppose you have tried 8 as the quotient figure instead of the correct 7 and have set it on G. Then you will multiply the 2 in 234 by 8, and subtract the product 16 from the 16 on HI. This clears HI. Next, multiplying the 3 in 234 by the same 8, you will find that the product 24 is larger than 3 remaining on J, and that you ought to have tried a quotient figure one less than 8. 

Step 6: To revise the incorrect quotient figure 8 to 7, subtract 1 from the 8 on G, and you get the new quotient figure 7 on G. Next multiply the 2 in 234 by 1, i. e., the difference between the quotient figures 8 and 7, and set the product 2 on I. Now you have 2 on I and 234 on IJK. 

Step 7: Multiply the 3 in 234 by the new quotient figure 7, and subtract the product 21 from 23 on IJ. This leaves 2 on 3 and 28 on JK. 

Step 8: Next multiply the 4 in 234 by the same new quotient figure 7, and subtract the product 28 from 28 on JK. This clears JK and leaves the answer 17 on FG.

Example :    7,061 / 307 = 23 

Step 1: Set 7 061 on HIJK, with K as the unit rod, and set 307 on ABC, leaving as always four vacant rods between the two numbers. 

Step 2: Comparing the 3 in 307 and the 7 in 7,061 you can see that 3 goes into 7 two times. Set the quotient figure 2 on F. Next multiply the 3 in 307 by this 2, and subtract the product 6 from the 7 on H. This leaves 1 on H. 

Step 3: Multiply the 7 in 307 by the same quotient figure 2, and setting the product 14 on IJ, subtract it from 106 on HIJ. This leaves 92 on IJ. Since the second digit in 307 is zero, see that you set the product 14 on IJ instead of HI. 

Step 4: The 3 in 307 goes into the 9 on I three times. So set the quotient figure 3 on G. Next multiply the 3 in 307 by this quotient figure 3 and subtract the product 9 from the 9 on 1. This leaves 21 on JK. 

Step 5: Multiply the 7 in 307 by the same quotient figure 3 and subtract the product 21 from the 21 on JK. This clears JK and leaves the answer 23 on FG.

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.