How to Do Division Using Partial Quotients

A Step-by-step Guide to Doing Division Using Partial Quotients

Here is a step-by-step guide to using the Partial Quotients method in the division for grade 4. Let’s use an example: 657 divided by 3.

Step 1: Write the problem out

Start by writing the problem like you would in long division. The number being divided (657) is inside the division bracket, and the number you are dividing by (3) is outside.

3 | 657

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Step 2: Make an initial estimate

Estimate a multiple of 3 that can go into 657. It’s okay if it’s not exact. Since we are working with a 4th grader, let’s start with an easy multiple – 100 (because 100 × 3 = 300, and 300 is less than 657).

Step 3: Subtract and write the partial quotient

Multiply your estimate by the divisor (3), subtract that from 657, and write the result below. Also, write the estimate above the division bracket. This estimate is your partial quotient.

 100 _______
3 | 657 - 300 ----- 357

Step 4: Repeat the process

Now, repeat the process with the new number (357). Let’s estimate 100 again. Subtract 300 (100 × 3) from 357.

 100 100 _______
3 | 657 - 300 ----- 357 - 300 ----- 57

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Step 5: Continue until the remainder is less than the divisor

Now, we have 57 left. We can’t subtract 300 anymore, so let’s choose a smaller estimate. Let’s choose 10 (because 10 × 3 = 30, and 30 is less than 57). Subtract 30 from 57.

 100 100 10 _______
3 | 657 - 300 ----- 357 - 300 ----- 57 - 30 ----- 27 

We can estimate 10 again and subtract 30 from 27.

 100 100 10 10 _______
3 | 657 - 300 ----- 357 - 300 ----- 57 - 30 ----- 27 - 30 ----- -3

Oops! We’ve gone too far. We can’t subtract 30 from 27. So, let’s remove the last estimate of 10. That leaves us with 27.

 100 100 10 _______
3 | 657 - 300 ----- 357 - 300 ----- 57 - 30 ----- 27

We can still subtract a multiple of 3 from 27. Let’s choose 9 (because 9 × 3 = 27, and 27 is equal to 27, so we can subtract it). Subtract 27 from 27.

 100 100 10 9 _______
3 | 657 - 300 ----- 357 - 300 ----- 57 - 30 ----- 27 - 27 ----- 0

Step 6: Add the partial quotients

Now add up all the partial quotients (100, 100, 10, 9) and write that number: 100 + 100 + 10 + 9 = 219. So 657 ÷ 3 = 219.

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Worked example: 684 ÷ 57 with partial quotients

A two-digit divisor looks scary, but the method stays the same. Use easy multiples you can find in your head.

1. Pick an easy multiple. 10 × 57 = 570. That is less than 684, so it works. Write 10 on the side.
2. Subtract. 684 − 570 = 114.
3. Pick another multiple. 114 is about twice 57. Check: 2 × 57 = 114. Write 2 on the side.
4. Subtract. 114 − 114 = 0. Nothing is left, so there is no remainder.
5. Add the partial quotients. 10 + 2 = 12.

So 684 ÷ 57 = 12. Check by multiplying: 57 × 12 = 570 + 114 = 684.

Partial quotients with a remainder

Sometimes the division does not come out even. Try 389 ÷ 6.

1. 60 × 6 = 360. Write 60. Then 389 − 360 = 29.
2. 4 × 6 = 24. Write 4. Then 29 − 24 = 5.
3. 5 is less than 6, so stop. The remainder is 5.
4. Add the partial quotients: 60 + 4 = 64.

So 389 ÷ 6 = 64 R 5. Check: 6 × 64 = 384, and 384 + 5 = 389.

There is no single right set of chunks. One student might take 50, then 10, then 4. Another might take 60 and 4. Both reach 64. Bigger chunks just mean fewer steps.

Frequently Asked Questions

How can you use partial quotients to find 684 divided by 57?

Start with an easy multiple of 57: 10 × 57 = 570. Subtract to get 684 − 570 = 114. Next, 2 × 57 = 114, and 114 − 114 = 0. Add the partial quotients, 10 + 2, to get 12. So 684 ÷ 57 = 12, and you can check it because 57 × 12 = 684.

How do you do partial quotients step by step?

Write the problem. Choose an easy multiple of the divisor, such as 10, 20, or 100 times it, that is not more than the number you are dividing. Subtract it and write the multiplier on the side. Repeat with what is left until it is smaller than the divisor. Add the multipliers to get the quotient. Whatever is left is the remainder.

Is partial quotients taught in 4th grade?

Yes. Many 4th grade classes use partial quotients to divide numbers up to four digits by one-digit divisors, because it builds on place value and multiplication facts. In 5th grade, students often use it with two-digit divisors before or alongside the standard long division method.

What is the difference between partial quotients and partial products?

Partial products is a multiplication method: you split numbers by place value, multiply each part, and add the results. For example, 23 × 4 = 80 + 12 = 92. Partial quotients is the division version: you subtract easy chunks of the divisor and add up how many chunks you used.

What happens if I pick a chunk that is too big?

If the multiple is bigger than what is left, the subtraction would go below zero. Just cross it out and choose a smaller multiple. For example, if 30 is left and the divisor is 8, 4 × 8 = 32 is too big, so use 3 × 8 = 24 instead. This flexibility is one reason students like the method.