why can you estimate a quotient without completing the division?

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If you are doing the long division, you get closer to the correct answer the more decimal places you carry it out. You do not have to "complete" the division.

Another way to get an answer "in your head" is by memorizing a few relationships like 1/2 = 0.500, 3/8 = 0.375 and 0.714 = 5/7 etc.

Take a look at the quotient and try to approximate or "bracket" it with a fractions with smaller integers. Then use the relationships you have memorized.

Here's an example: 23/87

That is more than 1/3 (23/69 = 0.33) and less than 1/4 (23/92 = 0.25) but closer to 1/4. I would guess the answer to be 0.27

The correct answer is 0.264

You can estimate a quotient without completing the division by using a technique called "compatible numbers." Compatible numbers are numbers that are easy to work with and are close to the numbers being divided.

To estimate a quotient using compatible numbers, follow these steps:

1. Identify the numbers being divided. For example, if you have 648 ÷ 6, the numbers being divided are 648 and 6.

2. Look for compatible numbers that are close to the numbers being divided. In this case, you might choose 600 and 6 as compatible numbers.

3. Divide the compatible numbers. In our example, 600 ÷ 6 equals 100.

4. Use the quotient from step 3 to estimate the actual quotient. In our example, since 600 ÷ 6 equals 100, we can estimate that 648 ÷ 6 is close to 100.

Estimating the quotient can be useful when you want to quickly check if an answer is reasonable, or when you don't need an exact answer and only need an approximate value.