CHAPTER
4

Division

In This Chapter

• The flag method of division
• Using altered remainders to deal with negative numbers
• Finding the decimal value of fractions with auxiliary fractions

Division can be a trying process without a calculator, especially when you don’t have single-digit divisors. But with a couple new processes in your arsenal, you will soon be doing division faster than ever. And who knows? You may even learn to like it!

In this chapter, I show you how the flag method, altered remainders, and auxiliary fractions can speed up division for you.

Division with the Flag Method

To help you with division beyond two one-digit numbers, I’d like to introduce you to the flag method. To begin the flag method, you write the divisor down in a “flag” formation—for example, if it’s a two-digit number, the first digit is the “flag pole,” and the second digit is the “flag.” The following shows how it looks using 31 as a divisor.

You then follow two basic rules to get the answer:

1. Divide by the flag pole.
2. Subtract by the flag times the previous quotient digit; think of it like Digit − (Flag × Previous Quotient Digit).

So there are only two basic rules for applying this method: divide and subtract. It’s a cyclical process, so if you divide in a step, you subtract in the next step, and so on.

Just keep this cycle in mind, and you’ll have it down in no time!

Two-Digit Divisors

If you have a two-digit divisor, you set up the problem with a line between the divisor and dividend. You then put another line in to indicate where the decimal point goes in the answer. The decimal point is based on how many digits you have in the flag; for a two-digit number, you only have one digit in the flag, so you count over one from the left in the divisor and place the line. You work the problem by following the two simple steps of the flag method: divide and subtract.

Example 1

Solve the problem 848 ÷ 31 to one decimal point.

Step 1: Lay out the problem. Because 848 is the dividend, put a line before it. For the divisor 31, put it to the left of the line, with the 3 serving as the flag pole and the 1 serving as the flag. You only have one digit in the flag, so put a line one to the left of the dividend to indicate where the decimal point goes.

Step 2: Divide 8 by the flag pole, 3. This gives you a quotient digit of 2 with a remainder of 2 (because 3 × 2 = 6, which is the highest multiple of 3 that can go into 8). Put down 2 as and prefix the remainder to 4 so it becomes 24.

Step 3: Subtract 24 by the flag times the previous quotient digit.

24 – (1 × 2) = 22

Step 4: Divide 22 by the flag pole: 22 ÷ 3 = 7, remainder 1. Put down 7 and prefix the remainder to 8 so it becomes 18.

Step 5: Subtract 18 by the flag times the previous quotient digit.

18 – (1 × 7) = 11

Step 6: Divide 11 by the flag pole: 11 ÷ 3 = 3, remainder 2. Put down 3 and carry over the 2.

Solution: The answer is 27.3.

QUICK TIP

The flag method can be a little hard to pick up at first, but if you devote 15 to 20 minutes of your time to practicing it, you should get it. I must confess, when I first looked at this method, I was totally lost! After 10 minutes, though, I was able to understand and apply it. Just remember the two rules: divide and subtract.

Example 2

Solve the problem 5,576 ÷ 25 to two decimal places.

Step 1: Lay out the problem. Because 5,576 is the dividend, put a line before it. For the divisor 25, put it to the left of the line, with the 2 serving as the flag pole and the 5 serving as the flag. You only have one digit in the flag, so put a line one to the left of the dividend to indicate where the decimal point goes.

Step 2: Divide 5 by the flag pole, which is 2. This gives you 2 as the quotient and 1 as the remainder (because 2 × 2 = 4, which is the highest multiple of 2 that can go into 8). Put down 2 and prefix the remainder of 1 to 5 so it becomes 15.

Step 3: Subtract 15 by the flag times the previous quotient digit.

15 – (5 × 2) = 5

Step 4: Divide 5 by the flag pole: 5 ÷ 2 = 2, remainder 1. Put down 2 and prefix the remainder of 1 to 7 so it becomes 17.

Step 5: Subtract by the flag times the previous quotient digit.

17 – (5 × 2) = 7

Step 6: Divide 7 by the flag pole: 7 ÷ 3, remainder 1. Put 3 down as the quotient and prefix the remainder of 1 to 6 so it becomes 16.

Step 7: Subtract 16 by the flag times the previous quotient digit.

16 – (5 × 3) = 1

Step 8: Divide 1 by the flag pole: 1 ÷ 2 = 0, remainder 1. Put down 0 and add a 0 above so you can prefix the 1 to it, making it 10.

Step 9: Subtract 10 by the flag times the previous quotient digit.

10 – (5 × 0) = 10

Step 10: Divide 10 by the flag pole: 10 ÷ 2 = 5. Put down 5.

Solution: The answer is 223.05.

Example 3

Solve the problem 2,924 ÷ 72 to two decimal places.

Step 1: Lay out the problem. Because 2,924 is the dividend, put a line before it. For the divisor 72, put it to the left of the line, with the 7 serving as the flag pole and the 2 serving as the flag. You only have one digit in the flag, so put a line one to the left of the dividend to indicate where the decimal point goes.

Step 2: Divide 29 by the flag pole, 7. This gives you 4 as a quotient digit and a remainder of 1 (because 7 × 4 = 28, which is the highest multiple of 7 that can go into 29). Put down 4 and prefix the remainder of 1 to 2 so it becomes 12.

Step 3: Subtract 12 by the flag times the previous quotient digit.

12 − (2 × 4) = 12 − 8 = 4

Step 4: Divide 4 by the flag pole: 4 ÷ 7 = 0, remainder 4. Put down 0 and prefix the remainder of 4 to the 4 above so it becomes 44.

Step 5: Subtract 44 by the flag times the previous quotient digit.

44 − (2 × 0) = 44

Step 6: Divide 44 by the flag pole: 44 ÷ 7 = 6, remainder 2. Put down 6 and carry over the remainder of 2.

Step 7: To find out the answer to two decimal places, put 0 after the remainder of 2 and subtract 20 by the flag times the previous quotient digit.

20 − (2 × 6) = 8

Step 8: Divide 8 by the flag pole: 8 ÷ 7 = 1 and 1 remainder.

Solution: The answer is 40.61.

Three-Digit Divisors

Now that you’ve gotten more comfortable with the flag method, let’s step up to three-digit divisors. In this case, you’ll have one digit in the flag pole and two digits in the flag. This also means the decimal point is two digits from the left of the dividend. You’ll continue the division and subtraction cycle, but the way the digits in the flag are used is slightly different.

When you first subtract, it’s only from the first digit of the flag times the previous quotient digit. However, for subsequent subtraction portions, you do the following:

Digit – [(First Digit of Flag × Previous Quotient Digit) + (Second Digit of Flag × Quotient Digit Before Previous Quotient Digit)]

This probably looks pretty confusing, so let me walk you through some examples so you can fully understand the process.

Example 1

Solve the problem 888 ÷ 672 to two decimal places.

Step 1: Lay out the problem. Because 888 is the dividend, put a line before it. For the divisor 672, put it to the left of the line, with the 6 serving as the flag pole and the 72 serving as the flag. You have two digits in the flag, so put a line two to the left of the dividend to indicate where the decimal point goes.

Step 2: Divide 8 by the flag pole, which is 6. This gives you a quotient digit of 1 and a remainder of 2 (because 6 × 1 = 6, which is the highest multiple of 6 that can go into 8). Put down 1 and prefix the remainder of 2 to the 8 so it becomes 28.

Step 3: Subtract 28 by the first digit of the flag times the previous quotient digit.

28 − (1 × 7) = 28 − 7 = 21

Step 4: Divide 21 by the flag pole: 21 ÷ 6 = 3, remainder 3. Put down 3 and prefix the remainder of 3 to 8 so it becomes 38.

Step 5: Subtract 38 by the first digit of the flag times the previous quotient plus the second digit of the flag times the quotient digit before that. In this case, you have (7 × 3) and (2 × 1) in the brackets.

38 − [(3 × 7) + (1 × 2)] = 38 − 23 = 15

Step 6: Divide 15 by the flag pole: 15 ÷ 6 = 2, remainder 3. Put down 2 and put the remainder of 3 after the 8.

Solution: The answer is 1.32.

Example 2

Solve the problem 70,319 ÷ 823 to two decimal places.

Step 1: Lay out the problem. Because 70,319 is the dividend, put a line before it. For the divisor 823, put it to the left of the line, with the 8 serving as the flag pole and the 23 serving as the flag. You have two digits in the flag, so put a line two to the left of the dividend to indicate where the decimal point goes.

Step 2: Divide 70 by the flag pole, 8. This gives you 8 as the quotient digit with a remainder of 6 (because 8 × 8 = 64, which is the highest multiple of 8 that can go into 70). Put down 8 and prefix the remainder of 6 to 3 so it becomes 63.

Step 3: Subtract 63 by the first digit of the flag times the previous quotient digit.

63 − (2 × 8) = 63 − 16 = 47

SPEED BUMP

In the third step, do not subtract 63 − (23 × 8). Use only the first digit of the flag in the multiplication portion: 63 − (2 × 8).

Step 4: Divide 47 by the flag pole: 47 ÷ 8 = 5, remainder 7. Put down 5 and prefix the remainder of 7 to 1 so it becomes 71.

Step 5: Subtract 71 by the first digit of the flag times the previous quotient plus the second digit of the flag times the quotient digit before that. For this problem, this means you have (2 × 5) and (3 × 8) in the brackets.

71 − [(2 × 5) + (3 × 8)] = 71 − 34 = 37

Step 6: Divide 37 by the flag pole: 37 ÷ 8 = 4, remainder 5. Put down 4 and prefix the remainder of 5 to 9 so it becomes 59.

Step 7: Subtract 59 by the first digit of the flag times the previous quotient plus the second digit of the flag times the quotient digit before that. Here, you put (2 × 4) and (3 × 5) in the brackets.

59 − [(4 × 2) + (5 × 3)] = 59 − 23 = 36

Step 8: Divide 36 by the flag pole: 36 ÷ 8 = 4, remainder 4. Put down 4 and put the remainder of 4 after the 9.

Solution: The answer is 85.44.

Example 3

Solve the problem 10,643 ÷ 743 to two decimal places.

Step 1: Lay out the problem. Because 10,643 is the dividend, put a line before it. For the divisor 743, put it to the left of the line, with the 7 serving as the flag pole and the 43 serving as the flag. You have two digits in the flag, so put a line two to the left of the dividend to indicate where the decimal point goes.

Step 2: Divide 10 by the flag pole, 7. This gives you a quotient of 1 and a remainder of 3 (because 7 × 1 = 7, which is the highest multiple of 7 that can go into 10). Put down 1 and prefix the remainder of 3 to 6 so it becomes 36.

Step 3: Subtract 36 by the first digit of the flag times the previous quotient digit.

36 − (4 × 1) = 32

Step 4: Divide 32 by the flag pole: 32 ÷ 7 = 4, remainder 4. Put down 4 and prefix the remainder of 4 to the 4 above so it becomes 44.

Step 5: Subtract 44 by the first digit of the flag times the previous quotient plus the second digit of the flag times the quotient digit before that. In this case, you have (4 × 4) and (3 × 1) in the brackets.

44 − [(4 × 4) + (3 × 1)] = 44 − 19 = 25

Step 6: Divide 25 by the flag pole: 25 ÷ 7 = 3, remainder 4. Put down 3 and prefix the remainder of 4 to 3 so it becomes 43.

Step 7: Subtract 43 by the first digit of the flag times the previous quotient plus the second digit of the flag times the quotient digit before that. In this case, you have (4 × 3) and (3 × 4) in the brackets.

43 − [(4 × 3) + (3 × 4)] = 43 − 24 = 19

Step 8: Divide 19 by the flag pole: 19 ÷ 7 = 2, remainder 5. Put down 2 and put the remainder of 5 after the 3.

Solution: The answer is 14.32.

Four-Digit Divisors

The flag method for a four-digit divisor is very similar to what you do for the three-digit divisor. The main difference is that you now have two digits in the flag pole, and you use the two digits together to divide the numbers. The following examples show you what you need to do with division problems involving four-digit divisors.

Example 1

Solve the problem 4,213 ÷ 1,234 to two decimal places.

Step 1: Lay out the problem. Because 4,213 is the dividend, put a line before it. For the divisor 1,234, put it to the left of the line, with the 12 serving as the flag pole and the 34 serving as the flag. You have two digits in the flag, so put a line two to the left of the dividend to indicate where the decimal point goes.

Step 2: Divide 42 by the flag pole, 12. This gives you a quotient of 3 and a remainder of 6 (because 12 × 3 = 36, which is the highest multiple of 12 that can go into 42). Put down 3 and prefix the remainder of 6 to 1 so it becomes 61.

Step 3: Subtract 61 by the first digit of the flag times the previous quotient digit.

61 − (3 × 3) = 61 − 9 = 52

Step 4: Divide 52 by the flag pole: 52 ÷ 12 = 4, remainder 4. Put down 4 and prefix the remainder of 4 to 3 so it becomes 43.

Step 5: Subtract 43 by the first digit of the flag times the previous quotient plus the second digit of the flag times the quotient digit before that. In this case, you have (3 × 4) and (4 × 3) in the brackets.

43 − [(4 × 3) + (3 × 4)] = 43 − 24 = 19

Step 6: Divide 19 by the flag pole: 19 ÷ 12 = 1, remainder 7. Put down 1 and put the remainder of 7 after the 3.

Solution: The answer is 3.41.

Altered Remainders

In some cases, you may get a negative when working a division problem using the flag method. What do you do? Let me introduce you something called altered remainders. With altered remainders, you decrease the quotient to increase the remainder.

For example, for 43 ÷ 8, the quotient is 5 and the remainder is 3. If you want to increase the remainder, you drop 1 from the quotient and add 8 to the remainder.

Let’s take the same quotient and remainder and apply them to a different problem, 28 ÷ 5. In this case, you drop 1 from the quotient and add 5 to the remainder:

QUICK TIP

As you can see in the tables, how much you add to the remainder is not equal to how much you take away from the quotient. The change in the remainder depends on the value of the divisor.

The following examples show you how to apply altered remainders.

Example 1

Solve the problem 3,412 ÷ 24 to one decimal place.

Step 1: As you did for previous examples, lay out the problem. Because 3,412 is the dividend, put a line before it. For the divisor 24, put it to the left of the line, with the 2 serving as the flag pole and the 4 serving as the flag. You have one digit in the flag, so put a line one to the left of the dividend to indicate where the decimal point goes.

Step 2: Divide 3 by the flag pole, 2. This gives you a quotient of 1 and a remainder of 1 (because 2 × 1 = 2, which is the highest multiple of 2 that can go into 4). Put down 1 and prefix the remainder of 1 to 4 so it becomes 14.

Step 3: Subtract 14 by the flag times the previous quotient digit.

14 − (4 × 1) = 14 − 4 = 10

Step 4: Divide 10 by the flag pole: 10 ÷ 2 = 5, remainder 0. Put down 5 and prefix the remainder of 0 to 1 so it becomes 01.

Step 5: Subtract 01 by the flag times the previous quotient digit: 01 − (4 × 5) = 1 − 20 = −20. Because you get a negative, reduce the previous quotient digit from 5 to 4; this increases the remainder from 0 to 2, which when prefixed to the 1 makes it 21.

Now subtract 21 by the flag times the previous quotient digit, which is now 4.

21 − (4 × 4) = 21 − 16 = 5

Step 6: Divide 5 by the flag pole: 5 ÷ 2 = 2, remainder 1. Put down 2 and prefix the remainder of 1 to 2 so it becomes 12.

Step 7: Subtract 12 by the flag times the previous quotient digit.

12 − (4 × 2) = 12 − 8 = 4

Step 8: Divide 4 by the flag pole: 4 ÷ 2 = 2, remainder 0. Put down 2 and prefix the remainder of 0 to 0 above so it becomes 00.

Step 9: Subtract 00 by the flag times the previous quotient digit: 00 − (4 × 2) = −8. Because you get a negative, reduce the previous quotient to 1, which changes the remainder from 0 to 2.

Solution: The answer is 142.1.

Example 2

Solve the problem 5,614 ÷ 21 to one decimal place.

Step 1: Lay out the problem. Because 5,614 is the dividend, put a line before it. For the divisor 21, put it to the left of the line, with the 2 serving as the flag pole and the 1 serving as the flag. You have one digit in the flag, so put a line one to the left of the dividend to indicate where the decimal point goes.

Step 2: Divide 5 by the flag pole, which is 2. This gives you a quotient of 2 with a remainder of 1 (because 2 × 2 = 4, which is the highest multiple of 2 that can go into 5). Put down 2 and prefix the remainder of 1 to 6 so it becomes 16.

Step 3: Subtract 16 by the flag times the previous quotient digit.

16 − (1 × 2) = 14

Step 4: Divide 14 by the flag pole: 14 ÷ 2 = 7, remainder 0. Put down 7 as the quotient digit and prefix the remainder of 0 to 1 so it becomes 01.

Step 5: Subtract 01 by the flag times the previous quotient digit: 01 − (1 × 7) = −7. Because you get a negative, reduce the previous quotient from 7 to 6; this increases the remainder from 0 to 2, which prefixed to 1 becomes 21.

Now subtract 21 by the flag times the previous quotient digit, which is now 6.

21 − (1 × 6) = 15

Step 6: Divide 15 by the flag pole: 15 ÷ 2 = 7, remainder 1. Put down 7 and prefix the remainder of 1 to 4 so it becomes 14.

Step 7: Subtract 14 by the flag times the previous quotient digit

14 − (1 × 7) = 7

Step 8: Divide 7 by the flag pole: 7 ÷ 2 = 3, remainder 1. Put down 3 and put the remainder of 1 after 4.

Solution: The answer is 267.3.

Example 3

Solve the problem 7,943 ÷ 42 to one decimal place.

Step 1: Lay out the problem. Because 7,943 is the dividend, put a line before it. For the divisor 42, put it to the left of the line, with the 4 serving as the flag pole and the 2 serving as the flag. You have one digit in the flag, so put a line one to the left of the dividend to indicate where the decimal point goes.

Step 2: Divide 7 by the flag pole, 4. This gives you a quotient digit of 1 and a remainder of 3 (because 4 × 1 = 4, which is the highest multiple of 4 that can go into 7). Put down 1 and prefix the remainder of 3 to 9 so it becomes 39.

Step 3: Subtract 39 by the flag times the previous quotient digit.

39 − (2 × 1) = 37

Step 4: Divide 37 by the flag pole: 37 ÷ 4 = 9, remainder 1. Put down 9 and prefix the remainder of 1 to 4 so it becomes 14.

Step 5: Subtract 14 by the flag times the previous quotient digit: 14 − (2 × 9) = 14 − 18 = −4. Because you get a negative, bring down the previous quotient from 9 to 8; this increases the remainder from 1 to 5, which prefixed to 4 becomes 54.

Now subtract 54 by the flag times the previous quotient digit, which is now 8.

54 − (2 × 8) = 54 − 16 = 38

Step 6: Divide 38 by the flag pole: 38 ÷ 4 = 9, remainder 2. Put down 9 and prefix the remainder of 2 to 3 so it becomes 23.

Step 7: Subtract 23 by the flag times the previous quotient digit.

23 − (2 × 9) = 23 − 18 = 5

Step 8: Divide 5 by the flag pole: 5 ÷ 4 = 1, remainder 1. Put down 1 and put the remainder of 1 after 3.

Solution: The answer is 189.1.

Auxiliary Fractions

Auxiliary fractions help you more easily find the decimal value of a fraction than doing long division. There are two types of auxiliary fractions: ones with a denominator ending in 9 or a series of 9s, and ones with a denominator ending in 1 or a series of 1s. The following sections go through each type. Once you have the process down, you’ll have no trouble zipping through fractions!

Type 1: Fractions with a Denominator Ending in 9 or a Series of 9s

To find the auxiliary fraction for fractions in which the denominator ends in a 9 or a series of 9s, you drop the 9 or 9s from the denominator and increase the remaining number by 1; you then divide the numerator by 10. After that, you begin dividing.

For each remainder, you prefix it to the previous quotient and then divide that by the denominator until you get the answer to the number of decimal places you require.

Example 1

Solve the problem to four decimal places.

Step 1: Find the auxiliary fraction. For , drop the 9 from the denominator and increase 4 by 1 so the denominator is now 5. Finish by dividing the numerator by 10: 6 ÷ 10 = 0.6. This gives you an auxiliary fraction of .

Step 2: Divide 0.6 by 5. This gives you a quotient of 0.1 and a remainder of 1. Put down the 0.1 and put the remainder of 1 in front of it so it becomes 1.1. This is the next dividend.

Step 3: Ignoring the decimal, divide 1.1 by 5: 11 ÷ 5 = 2, remainder 1. Put down 2 and put the remainder of 1 in front of it so it becomes 12. This is the next dividend.

Step 4: Divide 12 by 5: 12 ÷ 5 = 2, remainder 2. Put down 2 and put the remainder of 2 in front of it so it becomes 22. This is the next dividend.

Step 5: Divide 22 by 5: 22 ÷ 5 = 4, remainder 2. Put down the 4 and put the remainder of 2 in front of it. Because you’re only finding the answer to four decimal places, you don’t need to continue.

Solution: The answer is 0.1224.

QUICK TIP

You’re probably getting tired of reading “This is the next dividend” at the end of each step. However, I think it’s a good way to emphasize that every number you get is used to get the next one. If you have other tricks to keep that idea in your head, feel free to use them. It’s all about making this process easier for you!

Example 2

Solve the problem to four decimal places.

Step 1: Find the auxiliary fraction. For , drop the 9 from the denominator and increase 14 by 1 so the denominator is now 15. Finish by dividing the numerator by 10: 11 ÷ 10 = 1.1. This gives you an auxiliary fraction of .

Step 2: Ignoring the decimal, divide 1.1 by 15. This gives you a quotient of 0 and a remainder of 11. Put down 0 and put the remainder of 11 before it so it becomes 110. This is the next dividend.

Step 3: Divide 110 by 15: 110 ÷ 15 = 7, remainder 5. Put down 7 and put the remainder of 5 before it so it becomes 57. This is the next dividend.

Step 4: Divide 57 by 15: 57 ÷ 15 = 3, remainder 12. Put down 3 and put the remainder of 12 before it so it becomes 123. This is the next dividend.

Step 5: Divide 123 by 15: 123 ÷ 15 = 8, remainder 3. Put down 8 and put the remainder of 3 before it. Because you’re only finding the answer to four decimal places, you don’t need to continue.

Solution: The answer is 0.0738.

Example 3

Solve the problem to four decimal places.

Step 1: Find the auxiliary fraction. For , drop the 9 from the denominator and increase 1 by 1 so the denominator is now 2. Finish by dividing the numerator by 10: 16 ÷ 10 = 1.6. This gives you an auxiliary fraction of .

Step 2: Ignoring the decimal, divide 1.6 by 2. This gives you a quotient of 8 and a remainder of 0. Put down 8 and put the remainder of 0 before it so it becomes 08. This is the next dividend.

Step 3: Divide 08 by 2: 08 ÷ 2 = 4, remainder 0. Put down 4 and put the remainder of 0 before it so it becomes 04. This is the next dividend.

Step 4: Divide 04 by 2: 04 ÷ 2 = 2, remainder 0. Put down 2 and put the remainder of 0 before it so it becomes 02. This is the next dividend.

Step 5: Divide 02 by 2: 02 ÷ 2 = 1, remainder 0. Put down 1 and put the remainder of 0 before it. Because you’re only finding the answer to four decimal places, you don’t need to continue.

Solution: The answer is 0.8421.

Type 2: Fractions with a Denominator Ending in 1 or a Series of 1s

To find the auxiliary fraction for fractions in which the denominator ends in a 1 or a series of 1s, the numerator and denominator are both reduced by 1 and the top and bottom are divided by 10.

You get the next dividend by first writing down the remainder and then finding the difference between 9 and the quotient. The first part of the dividend is the remainder, while the second part is the difference.

Example 1

Solve the problem to four decimal places.

Step 1: Find the auxiliary fraction. For , reduce the numerator and denominator by 1, making it . Finish by dividing the numerator and denominator by 10: 3 ÷ 10 = 0.3; 20 ÷ 10 = 2. This gives you an auxiliary fraction of .

Step 2: Ignoring the decimal, divide 0.3 by 2. This gives you a quotient of 1 and a remainder of 1. Put down 1 and put the remainder of 1 before it.

Step 3: Subtract the quotient from 9: 9 − 1 = 8. Put the 8 on top of the 1. The remainder and this answer are the next dividend: 18.

Step 4: Divide 18 by 2: 18 ÷ 2 = 9, remainder 0. Put down 9 and put the remainder of 0 before it.

Step 5: Subtract the quotient from 9: 9 − 9 = 0. Put the 0 on top of the 9. The remainder and this answer are the next dividend: 00.

Step 6: Divide 00 by 2: 00 ÷ 2 = 0, remainder 0. Put down 0 and put the remainder of 0 before it.

Step 7: Subtract the quotient from 9: 9 − 0 = 9. Put the 9 on top of 0. The remainder and this answer are the next dividend: 09.

Step 8: Divide 09 by 2: 09 ÷ 2 = 4, remainder 1. Put down 4 and put the remainder of 1 before it.

Step 9: Subtract the quotient from 9: 9 − 4 = 5. Put the 5 on top of the 4.

Solution: The answer is 0.1904.

Example 2

Solve the problem to four decimal places.

Step 1: Find the auxiliary fraction. For , reduce the numerator and denominator by 1, making it . Finish by dividing the numerator and denominator by 10: 7 ÷ 10 = 0.7; 30 ÷ 10 = 3. This gives you an auxiliary fraction of .

Step 2: Ignoring the decimal, divide 0.7 by 3. This gives you a quotient of 2 and a remainder of 1. Put down 2 and put the remainder of 1 before it.

Step 3: Subtract the quotient from 9: 9 − 2 = 7. Put the 7 on top of the 2. The remainder and this answer are the next dividend: 17.

Step 4: Divide 17 by 3: 17 ÷ 3 = 5, remainder 2. Put down 5 and put the remainder of 2 before it.

Step 5: Subtract the quotient from 9: 9 − 5 = 4. Put the 4 on top of the 5. The remainder and this answer are the next dividend: 24.

Step 6: Divide 24 by 3: 24 ÷ 3 = 8, remainder 0. Put down 8 and put the remainder of 0 before it.

Step 7: Subtract the quotient from 9: 9 − 8 = 1. Put the 1 on top of the 8. The remainder and this answer are the next dividend: 01.

Step 8: Divide 01 by 3: 01 ÷ 3 = 0, remainder 1. Put down 0 and put the remainder of 1 before it.

Step 9: Subtract the quotient from 9: 9 − 0 = 9. Put the 9 on top of the 0.

Solution: The answer is 0.2580.

The Least You Need to Know

• To do the flag method, split the divisor into a “flag pole” and “flag,” and then follow two simple rules: divide and subtract.
• Altered remainders help you avoid a negative number. You simply decrease the quotient and increase the remainder.
• If you want to find the decimal value of a fraction, you can use an auxiliary fraction. You alter the fraction based on whether it ends in 1 or 9 and solve.