CHAPTER
5

Checking Your Answers with Digit Sums

In This Chapter

• Digit sums explained
• Casting out nines to further simplify your calculations
• Creating a nine-point circle and understanding how it works
• Checking your addition, subtraction, multiplication, and division answers using digit sums

In previous chapters, you learned speedier ways to solve addition, subtraction, multiplication, and division problems. But how do you know if they’re right?

In this chapter, I talk about the concept of digit sums, which can help you check your calculations in a simple—not to mention quick—manner.

What Are Digit Sums?

The word digit means numbers like 1, 2, 3, 4, and so on, while and the word sum means to “add.” So basically, to find the digit sum of a number, you simply have to add the digits in the numbers until you get a single-digit number.

For example, if you have to find the digit sum of 61, you simply add each digit of the number together: 6 + 1 = 7. So the digit sum of 61 is 7.

If you end up with a double-digit number, you continue to add the individual digits until you get a single-digit number. For example, when you add the digits for the number 538, you end up with 16 (5 + 3 + 8). To get a single-digit number, add again: 1 + 6 = 7. For 538, the digit sum is 7.

The following are some other examples of digit sums.

As you can see, this method is simple and very easy to understand. However, you can use some shortcuts to make the process even quicker.

Casting Out Nines

To make the digit-sum process even simpler, you can use a method called casting out nines. For this method, when trying to find the digit sum of a number, you first cast out or ignore any 9s and numbers adding up to 9. After that, you add up and reduce the remaining digits to a single-digit number to get the digit sum.

Let’s look at an example. Say you have to find the digit sum of 8,154,912,320. Instead of adding all those digits together, you can cast out the 9s and any numbers that add to 9. For this number, you cast out the 9, the 8 and 1, and the 5 and 4, because they are or add up to 9. After they’re taken out, the number looks like this:

8154912320

You then add the remaining digits to get the digit sum:

1 + 2 + 3 + 2 + 0 = 8

The digit sum of 8,154,912,320 is 8.

Let’s try this method again using the number 970,230,612. In this number, you can cast out the 9, the 7 and 2, and the 3 and 6, as all of these are or add to 9. Afterward, the number looks like the following:

970230612

You now add the remaining digits:

0 + 0 + 1 + 2 = 3

The digit sum for 970,230,612 is 3.

The Nine-Point Circle

The nine-point circle is a valuable aid when reducing numbers to their digit sums. To create a nine-point circle, you need to first list the numbers, followed by their digit sum. The digit cycles 1 through 9, as anything higher than 9 would have to be reduced. In the following table, I’ve listed 1 to 18 and the digit sums associated with them.

Here’s how these values look when placed on a nine-point circle.

The nine-point circle.

The point to note here is that numbers on the same branch of a nine-point circle have the same digit sum—1 and 10 have a digit sum of 1, 2 and 11 have a digit sum of 2, and so on. Also, if you add or take away 9 from any of the numbers, the digit sum remains the same.

You can build out the branches as far as you need to in order to help you remember what certain numbers reduce to. For example, for a 1 branch, you could expand out beyond 1 and 10 to include 19, 28, 37, and on up.

QUICK TIP

In the nine-point circle, the digit sums 9 and 0 are considered to be equivalent. This is because if you add 9 to 9, you get 18, which has a digit sum of 9; if you take away 9 from 9, it gives you 0. So you can say that in 8,136, the digit sum is 9 or 0. Both answers would be correct!

Using Digit Sums to Check Your Answers

You know what digit sums are and the different ways you can find them. Now, with the help of this concept, you can check your addition, subtraction, multiplication, and division answers. I show the long form for calculating the digit sums in the following sections, but feel free to use casting out nines and the nine-point circle to shorten the work.

Checking Addition Answers

To check your addition using digit sums, you simply reduce each number and then make sure the digit sums of the problem numbers added together are equal to the digit sum of the answer.

Example 1

Verify that the answer for the addition problem 734 + 352 is 1,086.

Step 1: Find the digit sum of 734.

7 + 3 + 4 = 14

1 + 4 = 5

Step 2: Find the digit sum of 352.

3 + 5 + 2 = 10

1 + 0 = 1

Step 3: Add the two digit sums together.

5 + 1 = 6

Step 4: Find the digit sum of 1,086.

1 + 0 + 8 + 6 = 15

1 + 5 = 6

Step 5: Check the digit sums against each other.

Solution: Both are 6, meaning the answer is correct.

Example 2

Verify the answer for the addition problem 2,344 + 6,235 is 8,579.

Step 1: Find the digit sum of 2,344.

2 + 3 + 4 + 4 = 13

1 + 3 = 4

Step 2: Find the digit sum of 6,235.

6 + 2 + 3 + 5 = 16

1 + 6 = 7

Step 3: Add the two digit sums.

4 + 7 = 11

1 + 1 = 2

Step 4: Find the digit sum of 8,579.

8 + 5 + 7 + 9 = 29

2 + 9 = 11

1 + 1 = 2

Step 5: Make sure the combined digit sums of the problem numbers match the digit sum of the answer.

Solution: Both are 2, meaning the answer is absolutely correct!

Checking Subtraction Answers

Checking your subtraction answers using digit sums is very similar to how you checked your addition answers, except you subtract instead of add.

QUICK TIP

Remember, if you don’t get a single-digit number after you add the digits, continue adding until you do. For example, if you get 15 in the first round of addition, add 1 and 5.

Example 1

Verify that the answer to the subtraction problem 74,637 − 24,267 is 50,370.

Step 1: Find the digit sum of 74,637.

7 + 4 + 6 + 3 + 7 = 27

2 + 7 = 9

Step 2: Find the digit sum of 24,267.

2 + 4 + 2 + 6 + 7 = 21

2 + 1 = 3

Step 3: Subtract the two digit sums.

9 − 3 = 6

Step 4: Find the digit sum of 50,370.

5 + 3 + 7 = 15

1 + 5 = 6

Step 5: Make sure the two digit sums are the same.

Solution: Both are 6, so you can safely say the answer is correct.

Example 2

Verify that the answer for the subtraction problem 4,321 − 1,786 is 2,535.

Step 1: Find the digit sum of 4,321.

4 + 3 + 2 + 1 = 10

1 + 0 = 1

Step 2: Find the digit sum of 1,786.

1 + 7 + 8 + 6 = 22

2 + 2 = 4

Step 3: Subtract the two digit sums.

1 − 4 = −3

Step 4: The digit sum for the two values can’t be negative, because the answer’s digit sum won’t be a negative number; to make it positive, subtract −3 from 9.

9 − 3 = 6

Step 5: Find the digit sum of 2,535.

2 + 5 + 3 + 5 = 15

1 + 5 = 6

Step 6: Check the two digit sums against each other to see if they match.

Solution: Both are 6, which means the answer is correct.

Checking Multiplication Answers

Using digit sums to check multiplication answers involves multiplying the digit sums of the problem numbers to see if they match the answer. If you need to reduce the number, though, you add the digits together.

Example 1

Verify that the answer for the multiplication problem 62 × 83 is 5,146.

Step 1: Find the digit sum of 62.

6 + 2 = 8

Step 2: Find the digit sum of 83.

8 + 3 = 11

1 + 1 = 2

Step 3: Multiply the two digit sums. In this case, you have to reduce, which means you add the digits of the answer to get the single-digit number.

8 × 2 = 16

1 + 6 = 7

Step 4: Find the digit sum of 5,146.

5 + 1 + 4 + 6 = 16

1 + 6 = 7

Step 5: Check the digit sums against each other to see if they match.

Solution: Both are 7; therefore, you can conclude your answer is correct.

Example 2

Verify that the answer for the multiplication problem 726 × 471 is 341,946.

Step 1: Find the digit sum of 726.

7 + 2 + 6 = 15

1 + 5 = 6

Step 2: Find the digit sum of 471.

4 + 7 + 1 = 12

1 + 2 = 3

Step 3: Multiply the two digit sums and add the answer to reduce it to a single digit.

6 × 3 = 18

1 + 8 = 9

Step 4: Find the digit sum of 341,946.

3 + 4 + 1 + 9 + 4 + 6 = 27

2 + 7 = 9

Step 5: Check the digit sums to see if they’re the same.

Solution: Both are 9, so the answer is absolutely correct!

SPEED BUMP

The digit-sum method is a great tool for checking your answers, but it has its limitations. For example, if you think the sum of 12 × 34 is 804 instead of 408, you’ll get the same digit sum (3), but the answer will be wrong.

So keep this in mind and be a little careful when using this checking tool.

Checking Division Answers

When doing a digit sums check for a division problem, the digit sum of the divisor times the digit sum of the answer should be equal to the digit sum of the dividend.

Example 1

Verify that the answer for 112 ÷ 7 is 16.

Step 1: Find the digit sum of the divisor, 7. In this case, it’s simply 7.

Step 2: Find the digit sum of the answer.

1 + 6 = 7

Step 3: Multiply the digit sum of the divisor and the answer, and reduce to a single digit.

7 × 7 = 49

4 + 9 = 13

1 + 3 = 4

Step 4: Find the digit sum of the dividend, 112.

1 + 1 + 2 = 4

Step 5: Compare the digit sum of the combined answer and divisor with the digit sum of the dividend to see if they’re the same.

Digit sum of divisor and answer: 4

Digit sum of dividend: 4

Solution: Both are 4, so the answer is correct.

Example 2

Verify that 464 ÷ 8 is 58.

Step 1: Find the digit sum of the divisor, 8. In this case, it’s simply 8.

Step 2: Find the digit sum of the answer.

5 + 8 = 13

1 + 3 = 4

Step 3: Multiply the digit sum of the divisor and the answer, and reduce to a single digit.

8 × 4 = 32

3 + 2 = 5

Step 4: Find the digit sum of the dividend, 464.

4 + 6 + 4 = 14

1 + 4 = 5

Step 5: Compare the digit sum of the combined answer and divisor with the digit sum of the dividend to see if they’re the same.

Digit sum of divisor and answer: 5

Digit sum of dividend: 5

Solution: Because the two digit sums match, the answer is correct.

The Least You Need to Know

• To find the digit sum, add and reduce the digits of a number until you get a single digit.
• Casting out nines and using a nine-point circle as guidance can help you eliminate unnecessary steps when calculating digit sums.
• For addition, subtraction, and multiplication problems, an answer is correct if the digit sum of the problem numbers combined is equal to the digit sum of the answer.
• For division problems, the answer is correct if the digit sum of the combined divisor and answer is equal to the digit sum of the dividend.