All skills Math

Learn Index · Arithmetic and Operations Grade 3–5

Multi-digit arithmetic

Adding and subtracting numbers with many digits by place value.

Read

Big numbers, same rules

Multi-digit arithmetic is adding, subtracting, multiplying and dividing numbers with two or more digits. The secret is place value: work with ones, tens, hundreds and thousands separately, and regroup when a column overflows.

Always estimate first

Before any calculation, estimate the answer. If you get something far away, you’ll know to check.

  • 3,487 + 2,659 ≈ 3,500 + 2,700 = 6,200

Adding with the column method

The column method is the standard algorithm for adding:

   3 4 8 7
 + 2 6 5 9
 ---------
   6 1 4 6
  1. Ones: 7 + 9 = 16. Write 6, carry 1 ten.
  2. Tens: 8 + 5 + 1 = 14. Write 4, carry 1 hundred.
  3. Hundreds: 4 + 6 + 1 = 11. Write 1, carry 1 thousand.
  4. Thousands: 3 + 2 + 1 = 6.

Answer: 6,146 (close to our estimate of 6,200 ✓).

Subtracting with the column method

   5 2 0 3
 − 1 8 4 7
 ---------
   3 3 5 6
  1. Ones: 3 − 7 can’t be done. The tens column is 0, so go to the hundreds: regroup 1 hundred into 10 tens, then 1 ten into 10 ones. Ones: 13 − 7 = 6.
  2. Tens: 9 − 4 = 5.
  3. Hundreds: 1 − 8 can’t be done. Regroup 1 thousand: 11 − 8 = 3.
  4. Thousands: 4 − 1 = 3.

Answer: 3,356. Check: 3,356 + 1,847 = 5,203 ✓.

Multiplying and dividing

For multiplying and dividing big numbers, use long multiplication and long division (see those lessons). The same ideas apply: line up place values, work one place at a time, and regroup.

Tips for neat work

  • Write digits in straight columns. Graph paper helps.
  • Line up numbers on the right (the ones place), not the left.
  • Write carried digits small and clearly.
  • Check subtraction by adding, and multiplication by dividing.

Worked examples

Example 1: 4,568 + 1,975

Estimate: 4,600 + 2,000 = 6,600.
Ones 8 + 5 = 13 (write 3, carry 1); tens 6 + 7 + 1 = 14 (write 4, carry 1); hundreds 5 + 9 + 1 = 15 (write 5, carry 1); thousands 4 + 1 + 1 = 6. Answer: 6,543.

Example 2: 8,000 − 2,436

Regroup across the zeros: 8,000 becomes 7 thousands, 9 hundreds, 9 tens, 10 ones.
10 − 6 = 4; 9 − 3 = 6; 9 − 4 = 5; 7 − 2 = 5. Answer: 5,564.

Example 3: 12,305 + 7,895

Estimate: 12,000 + 8,000 = 20,000. Exact: 20,200.

Common mistakes

  • Lining numbers up on the left, mixing up places.
  • Forgetting to add the carried digit.
  • Subtracting the smaller digit from the bigger one in every column, no matter which number it belongs to.
  • Skipping the estimate, so big errors go unnoticed.

Try it yourself

  1. 2,748 + 3,586
  2. 6,421 − 3,975
  3. 10,000 − 4,567
  4. 15,089 + 9,911

Answers: 1) 6,334 2) 2,446 3) 5,433 4) 25,000

Quick recap

  • Line up multi-digit numbers by place value.
  • Use the column method and regroup when needed.
  • Estimate first and check at the end.