Long Division
Long division is a systematic method for dividing large numbers by breaking the process into smaller, manageable steps. The algorithm involves repeatedly dividing, multiplying, subtracting, and bringing down digits until the entire dividend is processed. This method works with any divisor and produces exact quotients with remainders when necessary.
Why it matters
Long division appears in countless real-world calculations, from splitting a $240 restaurant bill among 8 friends ($30 each) to determining how many 12-inch tiles fit along a 156-inch wall (13 tiles). Construction workers use it to calculate material quantities, while bakers divide large recipe batches into smaller portions. The algorithm forms the foundation for polynomial division in algebra, decimal conversions in advanced arithmetic, and division algorithms in computer science. Medical professionals calculate dosages by dividing total medications among multiple doses, and financial planners use it for budget allocations. Students encounter long division requirements in CCSS.4.NBT standards for single-digit divisors and CCSS.5.NBT for two-digit divisors, building computational fluency essential for fraction operations, percentage calculations, and scientific notation in higher mathematics.
How to solve long division
Long division — how to
- See how many times the divisor fits into the first digits of the dividend.
- Multiply, subtract, bring down the next digit.
- Repeat until nothing is left. Express remainder as a decimal.
Example: 728 ÷ 10: 72 r 8 → 72.8.
Worked examples
15 ÷ 3 = _______
Answer: 5
- Understand what division means → 15 ÷ 3 — Division means sharing equally. Imagine splitting 15 sweets among 3 friends so everyone gets the same amount.
- How many times does 3 fit into 15? → 3 × 5 = 15 — We ask: '3 times what equals 15?' The answer is 5, because 3 × 5 = 15.
- Check: no leftovers → 15 - 15 = 0 — There is nothing left over. 15 divides evenly by 3.
- Write the answer → 15 ÷ 3 = 5 — Each friend gets 5. That is our answer!
- Verify by multiplying back → 5 × 3 = 15 ✓ — Multiply the answer by the divisor: 5 × 3 = 15. Correct!
A teacher has 64 pencils to give equally to 8 students. How many each?
Answer: 8
- Understand what division means → 64 ÷ 8 — Division means sharing equally. Imagine splitting 64 sweets among 8 friends so everyone gets the same amount.
- How many times does 8 fit into 64? → 8 × 8 = 64 — We ask: '8 times what equals 64?' The answer is 8, because 8 × 8 = 64.
- Check: no leftovers → 64 - 64 = 0 — There is nothing left over. 64 divides evenly by 8.
- Write the answer → 64 ÷ 8 = 8 — Each friend gets 8. That is our answer!
- Verify by multiplying back → 8 × 8 = 64 ✓ — Multiply the answer by the divisor: 8 × 8 = 64. Correct!
A ribbon is 99 cm long. Cut it into 3 equal pieces. How long is each piece?
Answer: 33
- Understand what division means → 99 ÷ 3 — Division means sharing equally. Imagine splitting 99 sweets among 3 friends so everyone gets the same amount.
- How many times does 3 fit into 99? → 3 × 33 = 99 — We ask: '3 times what equals 99?' The answer is 33, because 3 × 33 = 99.
- Check: no leftovers → 99 - 99 = 0 — There is nothing left over. 99 divides evenly by 3.
- Write the answer → 99 ÷ 3 = 33 — Each friend gets 33. That is our answer!
- Verify by multiplying back → 33 × 3 = 99 ✓ — Multiply the answer by the divisor: 33 × 3 = 99. Correct!
Common mistakes
- Forgetting to bring down the next digit results in incomplete division, such as calculating 156 ÷ 12 as 1 instead of 13
- Misaligning digits during subtraction leads to errors like computing 84 - 72 as 22 instead of 12 in the algorithm
- Estimating incorrectly causes wrong quotient digits, such as using 8 × 15 = 120 when dividing 127 ÷ 15 instead of the correct 8 × 15 = 120, remainder 7