Ninety-six on the frame, divided by three. Nothing here is hard to move. What is missing is somewhere to put the answer. On paper the quotient is written above the dividend, and a soroban has no above — it has left. So before any arithmetic happens, division needs a rule for which rod, and that rule is what this page is. 96 ÷ 3 = ?
Interactive walkthrough · Higher B into Grand Level
Abacus division: which rod the answer goes on
Set the dividend on the right of the frame with its units digit on the rightmost rod, and park the divisor far to the left. Divide the digit you are working on, write the result two rods to its left, then subtract what that digit cost out of the dividend and move one rod right. The units digit of the answer always lands two rods left of the units digit of the dividend — that offset is the technique. Nine chapters, twenty-six steps, and the last third hands you the beads.
What this page asks of you
This is not a drill sheet and not an article. Every chapter is a frame of a real abacus with an explanation beside it, and the last third hands the beads over. One warning about the reading under the frame: it announces the whole row of beads as a single number, so a set-up frame reads 300096, with no separators. That number is true of the beads and meaningless as arithmetic — the frame is three regions sharing one row, and the text beside each step keeps naming which is which.
Division is the last technique on the ladder. A learner meets it at the end of Higher B, once multi-digit multiplication is steady, and works it through Grand Level, the ninth rung. Multiplication is the literal prerequisite: every step of this method subtracts a product, so a learner who is still working those out belongs on that walkthrough first.
How do you divide on a soroban?
Set the dividend on the right with its units digit on the rightmost rod, and the divisor far to the left with empty rods between them. Look at the leading part of the dividend, work out how many times the divisor goes into it, and write that digit two rods to the left of the digit you divided. Multiply it back by the divisor and subtract the result from the dividend, digit by digit. Then move one rod to the right and do the same again. When the dividend region is empty, the middle of the frame is the answer.
Two rods, not one, is the part worth remembering. A one-digit divisor can produce an answer with as many digits as the dividend has, so with a single rod of gap the answer would want a rod the dividend is still sitting on. Two is the smallest offset that never collides. Measure it from the digit you are dividing rather than from the edge of the number, and it keeps holding for three-digit dividends and longer.
Move the beads yourself
The frame is showing 96.
The whole lesson, written out
Act I — a soroban has no above
The dividend sits on the right. Set 96 with its units digit on the rightmost rod: 9 on the tens rod, 6 on the ones rod. This region is the part of the problem that gets eaten. By the end of the method it reads zero, or it holds the remainder. 96 ÷ 3
The divisor sits far to the left. Put the 3 on the leftmost rod, with rods to spare between it and the dividend. The frame now reads 300096, and that number means nothing as arithmetic — it is three separate things sharing one row of beads. Read the regions, never the total. 96 ÷ 3
The rods in between are the answer, waiting. The empty rods between divisor and dividend are not slack. They are where the answer gets written, one digit at a time. The rod glowing here is where its units digit will land — two rods to the left of the units rod of the dividend, and it is always two. 96 ÷ 3
The finished frame, shown before it is earned. This is 96 ÷ 3 with the method complete: the divisor 3 still on the left, the dividend region emptied to 00, and 32 standing in the middle. The 2 — the units digit of the answer — is on the glowing rod, exactly two rods left of the rod the 6 was sitting on. 96 ÷ 3 = 32
Why two rods and not one. The rod glowing now is the tens rod of the dividend, and it was in use for most of the method. Dividing by a single digit can produce an answer with as many digits as the dividend has, so with only one rod of gap the answer would need this rod while the dividend was still on it. Two is the smallest gap that never collides. No beads moved between this frame and the last one — only which rod is being pointed at. 96 ÷ 3 = 32
Act II — worked twice: exact, then corrected
Set the problem. Divisor 3 on the far left, dividend 96 on the right with its units digit on the unit rod, and empty rods between them. The line under the frame will call this 300096, in one unpunctuated run; that is the honest total of the beads and it is not a number this method ever uses. 96 ÷ 3 = 32
Divide the leading digit, write the result two rods left. The 9 is standing on the tens rod of the dividend. Three goes into nine three times, so the digit is 3 and it goes two rods to the left of that 9. Three earth beads rise there. Nothing in the dividend has changed yet — the digit has been claimed, not paid for. 9 ÷ 3 = 3
Now pay for it. The digit you just wrote costs 3 × 3 = 9, and that comes off the rod you divided. Heaven bead up and four earth beads down, and the tens rod of the dividend reads 0. Every bead that move needs is already on the side it needs to be, so no complement is involved. This is the step that makes the next one legitimate. 3 × 3 = 9, 9 − 9 = 0
Move one rod right and do it again. What is left of the dividend is 6, standing on the units rod. Three goes into six twice, so a 2 is written two rods to its left — immediately right of the 3 from before. The answer is being built left to right, one rod at a time, in the order it is read. 6 ÷ 3 = 2
Pay for that one too. 2 × 3 = 6 comes off the units rod: heaven bead up, one earth bead down, and the rod reads 0. The dividend region is empty, which is what divides exactly looks like on a soroban. 2 × 3 = 6, 6 − 6 = 0
Read the middle. The dividend region reads 00, so nothing is owed. The quotient region reads 3 then 2, so the answer is 32, and 32 × 3 = 96. Its units digit is on the rod two to the left of where the units digit of the dividend was, exactly as Act I promised. 96 ÷ 3 = 32
Same setup, harder question. Divisor 6 on the far left, dividend 42 on the right. The arithmetic is easy — most people can say 42 ÷ 6 = 7 without a frame in front of them. The whole difficulty is which rod that 7 goes on, so this is the one chapter on the page where the tempting move is shown being wrong. 42 ÷ 6 = ?
This frame is a mistake, on purpose. Here is the 7 dropped on the first quotient rod, the way the last example started. But that rod is the tens rod of the answer, so the frame is now claiming 70 — and 70 × 6 = 420, not 42. Every rod in this picture is a legal soroban digit, the arithmetic behind it was right, and the answer is out by a factor of ten. That is the only real hazard this method has. 70 × 6 = 420, not 42
Take it back, and ask the question properly. Before writing anything, look at the digit you are dividing. Six does not go into four, so this answer has no tens digit at all and that rod stays empty. Drop the 7 back off and the frame is as it was. 6 > 4
The chunk is 42, and its low digit is on the units rod. Because 6 did not go into 4, the part being divided is the whole of 42, whose low digit stands on the units rod. Two rods to the left of that is where the 7 belongs: heaven bead down, two earth beads up. The rule did not change — the digit it was applied to did. 42 ÷ 6 = 7
Pay for it, tens first. 7 × 6 = 42, and that 42 comes off the dividend digit by digit. Four earth beads drop on the tens rod of the dividend and it reads 0. 7 × 6 = 42, 4 − 4 = 0
Then the ones. Two earth beads drop on the units rod and it reads 0 as well. The dividend region is clear, so this one came out exact too. 7 × 6 = 42, 2 − 2 = 0
Read the middle again. The quotient region reads 0 then 7, so the answer is 7 — and it is standing on the same rod the 2 of 32 landed on in the previous chapter. A one-digit answer and a two-digit answer put their units digit on exactly the same rod, because that rod is fixed by the dividend rather than by how big the answer turns out to be. 42 ÷ 6 = 7
Act III — your turn
Write the first digit of the answer. The frame is set — 3 on the left, 96 on the right — and the beads are live. Nine divided by three is three; put that 3 two rods to the left of the 9. The step is graded on the whole frame, so the same digit one rod out is a different number and will not pass. The rod is the test here, not the arithmetic. 9 ÷ 3 = 3, two rods left of the 9
Now clear what it cost. The frame reads 303096. Take 3 × 3 = 9 off the tens rod of the dividend so the frame reads 303006 — heaven bead up, four earth beads down. If it lands somewhere unexpected on the way, leave it and carry on; only where the beads stop is judged. 3 × 3 = 9, 9 − 9 = 0
One rod right, and write again. What remains of the dividend is 6 on the units rod. Six divided by three is two — write it two rods to the left of that 6, which puts it immediately right of the 3 you wrote a moment ago. The answer is built left to right, one rod at a time. 6 ÷ 3 = 2, two rods left of the 6
Clear the last of the dividend. Take 2 × 3 = 6 off the units rod so the frame reads 303200. The dividend region is empty, the middle of the frame reads 32, and the division is finished. 2 × 3 = 6, 96 ÷ 3 = 32
The digit is easy. The rod is the test. Divisor 6, dividend 42, and you already know the answer is 7. Six does not go into 4, so there is no tens digit: the 7 belongs two rods left of the units rod of the dividend. Put it one rod further left instead and the frame claims seventy — the step will not take it, and that is the entire lesson of this page. 42 ÷ 6 = 7, two rods left of the units rod
Subtract the product and read the answer. 7 × 6 = 42. Take four off the tens rod of the dividend and two off the units rod so the frame reads 600700. The dividend region is empty, the middle reads 7, and 7 × 6 = 42. 7 × 6 = 42, 42 ÷ 6 = 7
Seven rods, no target. One rod wider than every taught frame, so a three-digit dividend fits. Set one — 837 with a 3 parked on the far left, for instance — and work it through: the answer marches right into rods the dividend has already emptied, and the two-rod offset still holds, because it is measured from the digit being divided rather than from a gap. Nothing here is graded.
Where to go from here
Same technique, four different jobs. This page was the walkthrough; these are the others, and none of them replaces it.
The other walkthroughs
Five techniques, one ladder. Each is its own walkthrough with its own beads to move, and they are listed here in the order a learner meets them.