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Numerical Ability lesson

Division: Long Division, Shortcuts and Remainders

Divide step by step without losing a 0 in the quotient, then go faster: simplify first, the shortcuts for 5, 25 and 125, and chunking. Handle remainders the way the word problem needs (round up or down), and check by multiplying back.

30 min read · Professional and Sub-Professional

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What it is

Division splits an amount into equal groups, or asks how many groups fit. In 7,248 ÷ 6 = 1,208, 7,248 is the dividend, 6 is the divisor and 1,208 is the quotient. What does not fit is the remainder. Division undoes multiplication: 1,208 × 6 = 7,248.

Why it matters on the exam

Sharing, averages, rates (per hour, per kilo), unit price, percent and ratios all end in a division. The common slips are a missing 0 in the quotient, cancelling zeros from only one number, and handling the remainder wrongly in word problems (how many vans are needed?).

How to recognize it

Look for ÷ or a fraction bar, and the words per, each (when the total is given), share equally, split, average, how many groups, how many times, ratio.

What to look for

  • End zeros on both numbers: cancel them first.
  • A divisor of 5, 25 or 125: use the shortcut.
  • A dividend that splits into friendly multiples (700 + 252 for ÷ 7).
  • Whether the problem needs the remainder, a rounded-up or a rounded-down answer.
  • How many digits the quotient should have.

Words to know

  • Dividend ÷ divisor = quotient (remainder r).
  • Remainder: what is left over; always smaller than the divisor.
  • Divisible: divides with remainder 0 (see Lesson 11 for the quick tests).
  • Inverse: division undoes multiplication, so multiply to check.
Method 1 of 7

Long division: divide, multiply, subtract, bring down

Start at the left. Take the fewest digits the divisor goes into. Divide and write the digit on top, multiply it back, subtract, then bring down the next digit. Repeat until every digit has been brought down. If the divisor does not go into the number you have, write 0 on top and bring down the next digit.

How to use it

  1. Divide: how many times does the divisor go in?
  2. Multiply that digit by the divisor.
  3. Subtract.
  4. Bring down the next digit; if the divisor does not go in, write 0 and bring down again.
Example
Divide 7,248 ÷ 6.
  1. 7 ÷ 6 = 1, remainder 1.
  2. Bring down 2: 12 ÷ 6 = 2, remainder 0.
  3. Bring down 4: 4 ÷ 6 = 0 → write 0, remainder 4.
  4. Bring down 8: 48 ÷ 6 = 8.
7,248 ÷ 6 on paper, one step at a time
12086)7248−612−124−048−480
  1. Step 1D7 ÷ 6 = 1 (with 1 left) → write 1 on topM1 × 6 = 6S7 − 6 = 1BBring down 2 → 12
  2. Step 2D12 ÷ 6 = 2 → write 2 on topM2 × 6 = 12S12 − 12 = 0BBring down 4 → 4
  3. Step 3D4 ÷ 6 = 0 (with 4 left) → write 0 on topM0 × 6 = 0S4 − 0 = 4BBring down 8 → 48
  4. Step 4D48 ÷ 6 = 8 → write 8 on topM8 × 6 = 48S48 − 48 = 0✓Nothing left to bring down: no remainder
Answer1,208

Check: 1,208 × 6 = 7,248 ✓

Answer: 1,208

Check: 1,208 × 6 = 7,248 ✓

Tandaan: Divide, multiply, subtract, bring down. Every digit brought down gets a digit on top.

Watch out: Not writing the 0 when 6 does not go into 4: the answer becomes 128. Subtracting wrongly and getting a remainder bigger than the divisor.

Method 2 of 7

Simplify first: cancel zeros and common factors

You may divide both numbers by the same number without changing the answer. Cancel the same number of end zeros from both, or divide both by 2, 5, 7 … until the division is easy.
The rulea ÷ b = (a ÷ k) ÷ (b ÷ k)

How to use it

  1. Cancel the end zeros that both numbers share.
  2. Divide both by a common factor.
  3. Divide what is left.
Example
4,200 ÷ 350
  1. Cancel one zero from each: 420 ÷ 35.
  2. Divide both by 7: 60 ÷ 5.
  3. 60 ÷ 5 = 12.
Simplify first
a ÷ b = (a ÷ k) ÷ (b ÷ k)
Dividend 4,200
Divisor 350
4,200 ÷ 350M/DDivide 4,200 by 350=420 ÷ 35M/DDivide 420 by 35=60 ÷ 5M/DDivide 60 by 5=12✓Check: 350 × 12 = 4,200 ✓CHECK
Answer12

Answer: 12

Check: 350 × 12 = 4,200 ✓

Tandaan: Same change to both, same answer.

Watch out: Cancelling zeros from only one number (4,200 ÷ 35 is not 4,200 ÷ 350).

Method 3 of 7

Divide by 5, 25 and 125

To divide by 5, multiply by 2 and divide by 10. To divide by 25, multiply by 4 and divide by 100. To divide by 125, multiply by 8 and divide by 1,000.

How to use it

  1. Multiply the dividend by 2 ( ÷ 5), 4 ( ÷ 25) or 8 ( ÷ 125).
  2. Move the decimal point 1, 2 or 3 places to the left.
Example
1,575 ÷ 25
  1. 1,575 × 4 = 6,300.
  2. 6,300 ÷ 100 = 63.

Answer: 63

Check: 63 × 25 = 1,575 ✓

Tandaan: ÷ 5: × 2 ÷ 10. ÷ 25: × 4 ÷ 100. ÷ 125: × 8 ÷ 1,000.

Watch out: ÷ 25 by multiplying by 2.

Method 4 of 7

Chunking: split the dividend

Split the dividend into parts the divisor goes into easily (like 100 × or 10 × the divisor), divide each part, and add the answers.
The rule(a + b) ÷ d = a ÷ d + b ÷ d

How to use it

  1. Take off a big easy multiple of the divisor.
  2. Divide the rest (split again if needed).
  3. Add the partial quotients.
Example
952 ÷ 7
  1. 700 ÷ 7 = 100; 952 − 700 = 252.
  2. 210 ÷ 7 = 30; 252 − 210 = 42.
  3. 42 ÷ 7 = 6.
  4. 100 + 30 + 6 = 136.
The same division on paper (long division)
1367)952−725−2142−420
  1. Step 1D9 ÷ 7 = 1 (with 2 left) → write 1 on topM1 × 7 = 7S9 − 7 = 2BBring down 5 → 25
  2. Step 2D25 ÷ 7 = 3 (with 4 left) → write 3 on topM3 × 7 = 21S25 − 21 = 4BBring down 2 → 42
  3. Step 3D42 ÷ 7 = 6 → write 6 on topM6 × 7 = 42S42 − 42 = 0✓Nothing left to bring down: no remainder
Answer136

Check: 136 × 7 = 952 ✓

Answer: 136

Check: 136 × 7 = 952 ✓

Tandaan: Split the top, never the bottom.

Watch out: Splitting the divisor.

Method 5 of 7

Remainders in word problems: round up or round down?

Find the quotient and the remainder. Then read the question. If every person or item must be included (vans needed, boxes needed, trips), round UP. If only complete groups count (full boxes, complete sets, how many can you buy), round DOWN. If the question asks what is left, the answer IS the remainder.
The ruleDividend = divisor × quotient + remainder

How to use it

  1. Divide: quotient and remainder.
  2. Needed / must include all → quotient + 1.
  3. Complete / can afford → quotient.
  4. Left over → remainder.
Example
250 participants ride vans that hold 12 each. How many vans are needed?
  1. 250 ÷ 12 = 20 remainder 10.
  2. 20 vans carry 240; 10 people are left.
  3. They need one more van: 21.
Round up
Dividend = divisor × quotient + remainder
People 250
Per van 12
250 ÷ 12SETPut the numbers into the formula12 × 20 =240M/DMultiply 12 by 20250 − 240 =10A/SSubtract 240 from 250Remainder 10 > 020 + 1 =21A/SAdd 20 and 1✓Check: 21 × 12 = 252 ✓CHECK
Answer21 vans

Answer: 21 vans

Check: 21 × 12 = 252 ≥ 250, and 20 × 12 = 240 < 250 ✓

Tandaan: Needed → up. Complete → down. Left → remainder.

Watch out: Answering 20 (leaves 10 people behind) or 20.83 (vans cannot be split).

Method 6 of 7

Check by multiplying back and counting digits

Multiply the quotient by the divisor and add the remainder; you must get the dividend. Before dividing, decide how many digits the quotient has: in 7,248 ÷ 6, 6 goes into 7, so the quotient starts in the thousands and has 4 digits.

How to use it

  1. Count the quotient digits before you start.
  2. After dividing: quotient × divisor + remainder.
  3. Compare with the dividend.
Example
Is 128 the right answer for 7,248 ÷ 6?
  1. 6 goes into 7 (thousands), so the quotient has 4 digits.
  2. 128 has 3 digits: wrong.
  3. 1,208 × 6 = 7,248 ✓

Answer: No; it is 1,208.

Check: 128 × 6 = 768, far from 7,248.

Tandaan: Count the digits, then multiply back.

Watch out: Trusting an answer with the wrong number of digits.

Method 7 of 7

Dividing decimals: make the divisor whole

If the divisor has a point, move it to the right until the divisor is a whole number, and move the point in the dividend the same number of places. Then divide as usual: put the point in the answer straight above the point in the dividend, and write zeros after the point to keep dividing until nothing is left.

How to use it

  1. Move both points the same number of places until the divisor is whole.
  2. Divide; put the point in the answer above the point.
  3. If something is left, write a 0 after the point and bring it down.
Example
12.6 ÷ 0.4
  1. Move both points 1 place: 126 ÷ 4.
  2. 12 ÷ 4 = 3; 6 ÷ 4 = 1 r 2.
  3. Write a 0 after the point: 20 ÷ 4 = 5.
  4. Answer: 31.5.
12.6 ÷ 0.4 on paper

MOVEThe divisor must be a whole number: move both points 1 place to the right. 12.6 ÷ 0.4 → 126 ÷ 4

31.54)126.0−126−420−200
  1. Step 1D12 ÷ 4 = 3 → write 3 on topM3 × 4 = 12S12 − 12 = 0BBring down 6 → 6
  2. Step 2D6 ÷ 4 = 1 (with 2 left) → write 1 on topM1 × 4 = 4S6 − 4 = 2•Put the point in the answer, right above the pointBNo digits left: write a 0 after the point and bring it down → 20
  3. Step 3D20 ÷ 4 = 5 → write 5 on topM5 × 4 = 20S20 − 20 = 0✓Nothing left to bring down: no remainder
Answer31.5

Check: 31.5 × 0.4 = 12.6 ✓

Answer: 31.5

Check: 31.5 × 0.4 = 12.6 ✓

Tandaan: Divisor whole first; same move for both.

Watch out: Moving the point in the divisor only (12.6 ÷ 4 = 3.15).

Which division tool do I use?

You see…UseExample
Exact answer, no shortcutLong division (T1)7,248 ÷ 6
End zeros or a common factorSimplify first (T2)4,200 ÷ 350 = 420 ÷ 35 = 12
÷ 5, 25, 125× 2, 4, 8 and move the point (T3)1,575 ÷ 25
A dividend that splits into easy multiplesChunking (T4)952 ÷ 7 = (700 + 252) ÷ 7
Vans, boxes, packs, leftoversRemainder rule (T5)250 people ÷ 12 per van
Close choicesMultiply back; count digits (T6)1,208 vs 128
Which division tool?

Division table

÷ 2÷ 3÷ 4÷ 5÷ 6÷ 7÷ 8÷ 9÷ 10
2 ÷ 2 = 13 ÷ 3 = 14 ÷ 4 = 15 ÷ 5 = 16 ÷ 6 = 17 ÷ 7 = 18 ÷ 8 = 19 ÷ 9 = 110 ÷ 10 = 1
4 ÷ 2 = 26 ÷ 3 = 28 ÷ 4 = 210 ÷ 5 = 212 ÷ 6 = 214 ÷ 7 = 216 ÷ 8 = 218 ÷ 9 = 220 ÷ 10 = 2
6 ÷ 2 = 39 ÷ 3 = 312 ÷ 4 = 315 ÷ 5 = 318 ÷ 6 = 321 ÷ 7 = 324 ÷ 8 = 327 ÷ 9 = 330 ÷ 10 = 3
8 ÷ 2 = 412 ÷ 3 = 416 ÷ 4 = 420 ÷ 5 = 424 ÷ 6 = 428 ÷ 7 = 432 ÷ 8 = 436 ÷ 9 = 440 ÷ 10 = 4
10 ÷ 2 = 515 ÷ 3 = 520 ÷ 4 = 525 ÷ 5 = 530 ÷ 6 = 535 ÷ 7 = 540 ÷ 8 = 545 ÷ 9 = 550 ÷ 10 = 5
12 ÷ 2 = 618 ÷ 3 = 624 ÷ 4 = 630 ÷ 5 = 636 ÷ 6 = 642 ÷ 7 = 648 ÷ 8 = 654 ÷ 9 = 660 ÷ 10 = 6
14 ÷ 2 = 721 ÷ 3 = 728 ÷ 4 = 735 ÷ 5 = 742 ÷ 6 = 749 ÷ 7 = 756 ÷ 8 = 763 ÷ 9 = 770 ÷ 10 = 7
16 ÷ 2 = 824 ÷ 3 = 832 ÷ 4 = 840 ÷ 5 = 848 ÷ 6 = 856 ÷ 7 = 864 ÷ 8 = 872 ÷ 9 = 880 ÷ 10 = 8
18 ÷ 2 = 927 ÷ 3 = 936 ÷ 4 = 945 ÷ 5 = 954 ÷ 6 = 963 ÷ 7 = 972 ÷ 8 = 981 ÷ 9 = 990 ÷ 10 = 9
20 ÷ 2 = 1030 ÷ 3 = 1040 ÷ 4 = 1050 ÷ 5 = 1060 ÷ 6 = 1070 ÷ 7 = 1080 ÷ 8 = 1090 ÷ 9 = 10100 ÷ 10 = 10
Division facts: each column divides by the same number
Worked example 1 of 4Easy: ÷ 25 with the shortcut
What is 1,575 ÷ 25?
  1. 63
  2. 630
  3. 6.3
  4. 31.5
Step 1 · Use the shortcut÷ 25 = × 4 ÷ 100.
Step 2 · Multiply by 41,575 × 4 = 6,300.
Step 3 · Divide by 1006,300 ÷ 100 = 63.
Step 4 · Check63 × 25 = 1,575 ✓

Answer: A. 63

Why not the others:

  • 630 and 6.3 move the decimal point the wrong number of places.
  • 31.5 multiplies by 2 instead of 4 (that is ÷ 50).
Worked example 2 of 4Medium: a 0 in the quotient
A ₱7,248 bill is shared equally by 6 offices. How much does each office pay?
  1. ₱128
  2. ₱1,208
  3. ₱1,280
  4. ₱1,028
Step 1 · How many digits?6 goes into 7 (the thousands), so the answer has 4 digits: ₱128 is out.
Step 2 · Divide7 ÷ 6 = 1 r 1 → 12 ÷ 6 = 2 → 4 ÷ 6 = 0 (write 0) r 4 → 48 ÷ 6 = 8.
Step 3 · Read the quotient1, 2, 0, 8 → 1,208.
Step 4 · Check1,208 × 6 = 7,248 ✓

Answer: B. ₱1,208

Why not the others:

  • ₱128 skips the 0 in the tens.
  • ₱1,280 puts the 0 in the wrong place.
  • ₱1,028 writes the 0 too early.
Worked example 3 of 4Hard: how many vans are needed?
A seminar has 250 participants. Each van holds 12 participants. How many vans are needed so that everyone can ride?
  1. 20
  2. 21
  3. 22
  4. 10
Step 1 · Divide250 ÷ 12 = 20 remainder 10 (12 × 20 = 240).
Step 2 · Read the question"So that everyone can ride": the 10 left over still need a van. Round up.
Step 3 · Answer20 + 1 = 21 vans.
Step 4 · Check21 × 12 = 252, enough for 250 ✓; 20 × 12 = 240 is not enough.

Answer: B. 21

Why not the others:

  • 20 leaves 10 people behind.
  • 22 is one van too many.
  • 10 is the remainder, not the number of vans.
Worked example 4 of 4Hard: a big division with a remainder
What are the quotient and the remainder when 9,876 is divided by 23?
  1. 429 r 9
  2. 429 r 19
  3. 430 r 9
  4. 42 r 9
Step 1 · How many digits?23 does not go into 9 but goes into 98, so the quotient has 3 digits (42 r 9 is out).
Step 2 · Chunk23 × 400 = 9,200; 9,876 − 9,200 = 676. 23 × 20 = 460; 676 − 460 = 216. 23 × 9 = 207; 216 − 207 = 9.
Step 3 · Add the chunks400 + 20 + 9 = 429, remainder 9.
Step 4 · Check23 × 429 + 9 = 9,867 + 9 = 9,876 ✓

Answer: A. 429 r 9

Why not the others:

  • 429 r 19: a subtraction slip; check 23 × 429 + 19 = 9,886 ≠ 9,876.
  • 430 r 9: 23 × 430 + 9 = 9,899 ≠ 9,876.
  • 42 r 9 has too few digits.

Exam techniques

Count the digits first

Before dividing, decide where the first quotient digit sits; that tells you how many digits the answer has.

When: Every division with choices.

Careful: Then still check with multiplication.

Use the choices: multiply back

Multiply each choice by the divisor and see which gives the dividend. Start with the choice whose last digit fits.

When: When long division would take longer than two multiplications.

Careful: Use the last digit to pick which choice to try first.

Divisibility tests

Use the quick tests from Lesson 11 (digit sum for 3 and 9, last two digits for 4) to see if a division comes out exact.

When: Choosing between exact and non-exact answers, simplifying fractions.

Careful: A test tells you IF it divides, not the answer.

The traps

  • Leaving out a 0 in the quotient. Every digit you bring down gets a digit on top, even 0 (T1). Count the digits first (T6).
  • Cancelling zeros from only one number. Cancel the same number of zeros from both (T2).
  • Splitting the divisor (952 ÷ 7 = 952 ÷ 5 + 952 ÷ 2). Split only the dividend (T4).
  • Rounding the wrong way in a word problem. Needed → round up; complete or can afford → round down (T5).
  • A remainder bigger than the divisor. If the remainder is too big, the quotient digit was too small (T1).

Tandaan

  • Divide, multiply, subtract, bring down; every digit brought down gets a digit on top (even 0).
  • Cancel the same zeros or factors from both numbers first.
  • ÷ 5: × 2 ÷ 10. ÷ 25: × 4 ÷ 100. ÷ 125: × 8 ÷ 1,000.
  • Split the dividend, never the divisor.
  • Needed → round up; complete → round down; left over → remainder.
  • Check: quotient × divisor + remainder = dividend.

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