Identifying the Missing Digits in Addition

Sometimes an addition problem is written with one digit hidden, and the sum is given instead. Your job is to work backward through the columns of the problem to figure out what that hidden digit must be. This is closely related to the column addition you already use when adding tens and ones, except now one piece of the puzzle is missing instead of the total.

Every addition problem, no matter how large, is really just several one-digit additions happening side by side in the ones, tens, and hundreds columns. If you know two of the three numbers in a column (the two digits being added, or one digit and the total), you can always solve for the third.

Follow these steps every time:

1. Line up the numbers by place value, just as you would for regular addition, so ones sit under ones, tens under tens, and hundreds under hundreds.

2. Start with the ones column. If the ones digit is missing, use subtraction: subtract the known addend's ones digit from the sum's ones digit.

3. Move to the tens column, then the hundreds column, repeating the same idea. Remember to add any digit that was carried over from the column before it.

4. If a column adds up to 10 or more, write down only the ones part of that total and carry a 1 into the next column to the left.

5. Once you have solved for the missing digit, add the full numbers together to check that you get the given sum.

This strategy works the same way whether the numbers are small or go up into the hundreds, as in adding with digits up to 1000. The size of the numbers does not change the method, it only means there may be more columns to check.

Find the missing digit in the problem below.

Start with the ones column: \( 2 + 5 = 7 \), which matches the given sum, so there is no carry into the tens column. Now look at the tens column: an unknown digit plus \( 1 \) must give \( 5 \), since there is nothing carried in. Using subtraction, \( 5 - 1 = 4 \), so the missing digit is \(4\). The hundreds column confirms it: \( 4 + 3 = 7 \), which matches the sum. The completed problem is \( 442 + 315 = 757 \).

Now try a problem where a digit needs to be carried, similar to what you practiced in addition using mental strategies when totals pass 10.

Begin with the ones column: \( 6 + 7 = 13 \). Write the \(3\) in the sum and carry a \(1\) into the tens column. Now the tens column has an unknown digit plus \(8\) plus the carried \(1\), and the sum shows a \(5\) with another \(1\) carried into the hundreds column, so the total for that column is \(15\). Subtracting the known parts, \( 15 - 8 - 1 = 6 \), so the missing digit is \(6\). Checking the hundreds column, \( 3 + 1 + 1 = 5 \), which matches the sum. The completed problem is \( 386 + 167 = 553 \).

Watch out for these errors when identifying a missing digit:

• Forgetting to add a carried \(1\) before subtracting to find the missing digit.

• Working on the wrong column, always start from the ones place and move left.

• Mixing up which number is missing, the top addend, the bottom addend, or the sum itself.

• Skipping the final check, always add your completed numbers to confirm they equal the given sum.

Before tackling a problem with a hidden digit, make sure column addition itself feels automatic. Reviewing strategies such as making 10s or estimating a sum first can help you sanity check whether the missing digit you found makes sense. If your finished addition does not match the given total, recheck each column one at a time rather than starting the whole problem over.

Related lessons

• Math 3