3-Digit Subtraction: Decomposing Hundreds - Second Grade Mathematics

In three-digit subtraction, what happens when the tens column does not have enough tens to subtract, such as trying to subtract 7 tens from 2 tens in 625 - 271? Just as you unbundle a ten rod to make 10 ones, you can unbundle a hundred flat to create 10 tens rods! This is three-digit subtraction with decomposing hundreds. In some advanced problems, you will decompose from both tens and hundreds in the very same problem. In this chapter, you will master unbundling hundreds into tens, navigate double-regrouping subtraction, and verify your results using the addition check.

Decomposing 1 Hundred into 10 Tens

A hundred flat is made of 10 ten rods fused together.

The Unbundling Trade:
1 Hundred Flat [   ]  ====== UNBUNDLE ======>  10 Ten Rods | | | | | | | | | |

When Is Decomposing Hundreds Needed?

Look at 625 - 271: Ones column: 5 - 1 = 4. (Plenty of ones! 5 >= 1). Tens column: 2 - 7. Can you take 7 tens from 2 tens? No! You must go to the hundreds column of 625, take 1 hundred from the 6 hundreds, and trade it for 10 tens. 10 tens added to the existing 2 tens makes 12 tens!

The Step-by-Step Algorithm for 625 - 271:

        5   12  <--- Decomposed Hundreds and Tens!
        6    2   5
     -  2    7   1
     -------------
        3    5   4

Step 1: Subtract ones: 5 - 1 = 4. Write 4 in ones place.
Step 2: Unbundle 1 hundred from 6 hundreds:
        Cross out 6 and write 5.
        Give 10 tens to 2 tens: 10 + 2 = 12 tens.
        Cross out 2 and write 12.
Step 3: Subtract tens: 12 tens - 7 tens = 5 tens. Write 5 in tens place.
Step 4: Subtract hundreds: 5 hundreds - 2 hundreds = 3 hundreds. Write 3 in hundreds place.
Final Difference: 354!

Double Regrouping in Subtraction

Double regrouping in subtraction occurs when BOTH the ones column and the tens column need to borrow!

Solving 734 - 286 with Double Regrouping:

        6   12   14  <--- Decomposed Across Both Columns!
        7    3    4
     -  2    8    6
     ---------------
        4    4    8

Step 1: Look at ones: 4 - 6 (not enough ones).
        Go to tens: take 1 ten from 3 tens (leaving 2 tens).
        Give 10 ones to 4 ones: 10 + 4 = 14 ones.
        Now subtract ones: 14 - 6 = 8 ones.

Step 2: Look at tens: we now have 2 tens left, and need to subtract 8 tens!
        Not enough tens (2 < 8)!
        Go to hundreds: take 1 hundred from 7 hundreds (leaving 6 hundreds).
        Give 10 tens to 2 tens: 10 + 2 = 12 tens.
        Now subtract tens: 12 - 8 = 4 tens.

Step 3: Look at hundreds:
        6 hundreds - 2 hundreds = 4 hundreds.
        Final Difference: 448!

One Column at a Time

Double regrouping looks intimidating only if you try to do everything at once. If you focus strictly on one column at a time—first ones, then tens, then hundreds—each step is a simple, familiar trade.

Base-Ten Model of Double Regrouping

Let us trace the blocks during 734 - 286:

Start with 734: 7 Flats, 3 Rods, 4 Cubes.

Trade 1 (Need Ones):
Unbundle 1 Rod: now have 7 Flats, 2 Rods, 14 Cubes.
Subtract 6 Cubes: 14 - 6 = 8 Cubes remain.

Trade 2 (Need Tens):
Unbundle 1 Flat: now have 6 Flats, 12 Rods, 8 Cubes.
Subtract 8 Rods: 12 - 8 = 4 Rods remain.

Subtract Hundreds:
Subtract 2 Flats: 6 - 2 = 4 Flats remain.
Result: 4 Flats (400) + 4 Rods (40) + 8 Cubes (8) = 448!

Practice Exercises

Exercise 1: Solve 715 - 382 vertically. State which column required borrowing and show all crossed-out digits.

Exercise 2: Solve 823 - 467 using double regrouping. Show the new numbers above the ones, tens, and hundreds columns.

Exercise 3: Set up and solve the addition check for 823 - 467 using your answer from Exercise 2.

Exercise 4: A warehouse stored 548 crates of fruit. A shipment of 275 crates was sent to supermarkets. How many crates of fruit remained in the warehouse?

Solutions and Explanations

Solution to Exercise 1: In 715 - 382, subtract ones: 5 - 2 = 3. In tens: 1 ten cannot subtract 8 tens. Unbundle 1 hundred from 7 hundreds (leaving 6 hundreds) to give 10 tens to 1 ten (making 11 tens). Subtract tens: 11 - 8 = 3. Subtract hundreds: 6 - 3 = 3. The difference is 333. The tens column required borrowing from hundreds.

Solution to Exercise 2: Align vertically: 823 over 467. Step 1: Ones: 3 cannot subtract 7. Borrow 1 ten from 2 tens (leaving 1 ten) to make 13 ones. 13 - 7 = 6 ones. Step 2: Tens: 1 ten cannot subtract 6 tens. Borrow 1 hundred from 8 hundreds (leaving 7 hundreds) to make 11 tens. 11 - 6 = 5 tens. Step 3: Hundreds: 7 - 4 = 3 hundreds. The difference is 356. Above the columns, write 7 in hundreds, 11 in tens, and 13 in ones.

Solution to Exercise 3: To check 823 - 467 = 356, add difference (356) + subtrahend (467). Ones: 6 + 7 = 13 (write 3, carry 1 ten). Tens: 1 (attic) + 5 + 6 = 12 tens (write 2, carry 1 hundred). Hundreds: 1 (attic) + 3 + 4 = 8 hundreds. The check sum is 823, which matches the original minuend.

Solution to Exercise 4: To find remaining crates, calculate 548 - 275. Subtract ones: 8 - 5 = 3. In tens, 4 cannot subtract 7. Borrow 1 hundred from 5 (leaving 4 hundreds) to make 14 tens. Subtract tens: 14 - 7 = 7. Subtract hundreds: 4 - 2 = 2. There were 273 crates remaining in the warehouse.