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The ladder drawn rung by rung, with the exact decimal and the remainder.
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The Coddy math course teaches the method itself - you work each step on an interactive board and get told exactly where a move went wrong.
Every rung of the ladder is the same four moves: bring down the next digit, ask how many times the divisor fits, multiply, subtract. What is left after the subtraction carries into the next rung, and the digits you write on top assemble the quotient. Nothing else ever happens - long division is a short algorithm applied as many times as the number has digits.
The first rung is the only special one: you take just enough digits for the divisor to fit at all. Dividing 1234 by 56, neither 1 nor 12 is big enough, so the first window is 123 - and 56 fits into it twice. After that it is strictly one digit per rung, zeros included: when 56 fits zero times, the 0 still goes on top, and skipping it is the classic way to lose a place value.
When the digits run out and something is still left over, the same loop keeps going past the decimal point by bringing down zeros - which is where decimals actually come from. And when a remainder shows up that you have seen before, the digits between the two sightings must repeat forever: that is why 7 ÷ 12 = 0.58333… repeats, spotted by the ladder itself rather than asserted.
Read digits of the dividend from the left until the number is at least the divisor. For 731 ÷ 4 that is just the 7; for 1234 ÷ 56 it is 123.
How many whole times does the divisor fit? Write that digit on top, write the product underneath, subtract. 4 goes into 7 once; 7 - 4 leaves 3.
Append the next digit of the dividend to the leftover and run the same three moves on it. 3 becomes 33, and 4 fits 8 times.
What is left at the end is the remainder: 731 ÷ 4 = 182 R 3. To get a decimal instead, put a point in the quotient and keep bringing down zeros: 182.75.
The same division, in the three forms a question might want - and when each one is the right answer.
| Division | Quotient and remainder | Exact decimal | As a fraction |
|---|---|---|---|
| 731 ÷ 4 | 182 R 3 | 182.75 | 731/4 = 182¾ |
| 1234 ÷ 56 | 22 R 2 | 22.03(571428) | 617/28 |
| 7 ÷ 12 | 0 R 7 | 0.58(3) | 7/12 |
| 100 ÷ 8 | 12 R 4 | 12.5 | 25/2 |
| 45 ÷ 9 | 5 R 0 | 5 | 5 |
| 10 ÷ 3 | 3 R 1 | 3.(3) | 10/3 |
| 1 ÷ 7 | 0 R 1 | 0.(142857) | 1/7 |
731 ÷ 44 into 7 goes once, leaving 3. Bring down the 3: 4 into 33 goes 8 times, leaving 1. Bring down the 1: 4 into 11 goes twice, leaving 3. That is 182 remainder 3 - and pushing past the point, 30 gives a 7 and 20 gives a 5, so the exact decimal is 182.75.
1234 ÷ 56Neither 1 nor 12 fits 56, so the first window is 123: 56 goes in twice (112), leaving 11. Bring down the 4: 56 into 114 goes twice again (112), leaving 2. So 1234 ÷ 56 = 22 R 2. With a bigger divisor the only extra work is estimating each quotient digit - the loop is unchanged.
7 ÷ 127 is smaller than 12, so the integer part is 0 and the ladder goes straight past the point: 70 gives 5 (leaving 10), 100 gives 8 (leaving 4), 40 gives 3 (leaving 4 again). The remainder 4 has appeared before - so from there the ladder can only repeat itself, and the answer is exactly 0.583333… = 0.58 with the 3 repeating.
12.5 ÷ 0.5Multiply both numbers by 10 to clear the points: 125 ÷ 5. Scaling both together leaves the quotient untouched - the answer is 25 either way - which is why the school method says "move both decimal points the same amount". The ladder then runs on whole numbers.
