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πŸ“˜ Chapter 2 Β· The Four Operations

Addition, subtraction, multiplication and division β€” the four power-tools of maths.

1 Meet the four operations & their words β–Ό

The four power-tools of maths β€” and the special word for the answer of each:

OperationSymbolAnswer is called the…Example
Addition+Sum5 + 3 = 8 (sum)
Subtractionβˆ’Difference9 βˆ’ 4 = 5 (difference)
MultiplicationΓ—Product6 Γ— 7 = 42 (product)
DivisionΓ·Quotient20 Γ· 4 = 5 (quotient)
Division words: in 20 Γ· 4 = 5 we say:
β€’ 20 is the dividend (the number being shared)
β€’ 4 is the divisor (how many groups)
β€’ 5 is the quotient (the answer)
β€’ Anything left over is the remainder.
πŸ’‘ If a question says "find the product", you multiply. If it says "find the difference", you subtract. The word tells you the operation!
2 Addition with carrying (regrouping) β–Ό

When you add, line the numbers up by their place value β€” ones under ones, tens under tens β€” and add column by column from right to left.

If a column adds to 10 or more, carry the extra to the next column on the left.

Example: 4,567 + 2,834 
  ΒΉ ΒΉ ΒΉ
    4 5 6 7
  + 2 8 3 4
  ─────────
    7 4 0 1
Ones: 7+4 = 11 β†’ write 1, carry 1.
Tens: 6+3+1 = 10 β†’ write 0, carry 1.
Hundreds: 5+8+1 = 14 β†’ write 4, carry 1.
Thousands: 4+2+1 = 7. Answer: 7,401.
πŸ’‘ Always start at the ones column. Adding from left feels natural in reading, but in maths the carries flow leftward β€” so right-to-left is the safe direction.
3 Subtraction with borrowing β–Ό

Subtract column by column, right to left, the same way as addition. If the top digit is smaller than the bottom, borrow 1 from the next column on the left.

Example: 8,512 βˆ’ 3,789 
    7  14  10  12
    8ΜΆ  5ΜΆ  1ΜΆ  2
  βˆ’ 3  7  8  9
  ──────────────
    4  7  2  3
Ones: 2βˆ’9? Borrow! Tens lend 1 β†’ ones become 12, tens become 0.
12βˆ’9 = 3.
Tens: 0βˆ’8? Borrow! Hundreds lend 1 β†’ tens become 10, hundreds become 4.
10βˆ’8 = 2.
Hundreds: 4βˆ’7? Borrow! Thousands lend 1 β†’ hundreds become 14, thousands become 7.
14βˆ’7 = 7.
Thousands: 7βˆ’3 = 4. Answer: 4,723.
⚠️ Subtraction is NOT commutative: 9 βˆ’ 4 β‰  4 βˆ’ 9. The bigger number must be on top.
4 Multiplication β€” column method β–Ό

To multiply a big number by a small one, write the bigger number on top and multiply each digit, carrying as needed.

Example: 345 Γ— 6 
  Β² Β³
    3 4 5
  Γ—     6
  ─────────
  2 0 7 0
5Γ—6 = 30 β†’ write 0, carry 3.
4Γ—6 = 24, +3 = 27 β†’ write 7, carry 2.
3Γ—6 = 18, +2 = 20. Write 20.
Answer: 2,070.

For 2-digit Γ— 2-digit, multiply by the ones digit, then by the tens digit (shifted one place left), then add.

Example: 24 Γ— 25 
      2 4
  Γ—   2 5
  ─────────
    1 2 0   ← 24 Γ— 5
  + 4 8     ← 24 Γ— 2, shifted left
  ─────────
    6 0 0
Answer: 600.
πŸ’‘ The shift on the second row is because the "2" in 25 is really 20 β€” so 24Γ—2 = 48 sits in the tens, hundreds, thousands columns.
5 Long division β–Ό

Long division has 4 steps that loop until the dividend is gone:

  1. Divide β€” how many times does the divisor go in?
  2. Multiply β€” quotient Γ— divisor.
  3. Subtract β€” bring the difference down.
  4. Bring down the next digit and repeat.

Memory trick: D-M-S-B β†’ "Daddy, Mummy, Sister, Brother".

Example: 936 Γ· 4 
       2 3 4
     β”Œβ”€β”€β”€β”€β”€β”€β”€β”€β”€
   4 β”‚ 9 3 6
       8        ← 4 Γ— 2
       ─
       1 3      ← bring down 3
       1 2      ← 4 Γ— 3
       ─
         1 6    ← bring down 6
         1 6    ← 4 Γ— 4
         ─
           0    ← remainder 0
Answer: 234 with remainder 0.
πŸ’‘ Always check a division: quotient Γ— divisor + remainder should equal the dividend. (234 Γ— 4 + 0 = 936 βœ“)
6 BODMAS β€” order of operations β–Ό

When a question mixes operations, do them in this order:

LetterMeansExample
BBrackets ( )(8 + 4)
OOrder (powers / square roots)3Β²
DDivision Γ·20 Γ· 5
MMultiplication Γ—6 Γ— 7
AAddition +5 + 3
SSubtraction βˆ’9 βˆ’ 4
(8 + 4) Γ— 3 β†’ brackets first: 12, then Γ—3 = 36.
6 Γ— 7 + 3 β†’ Γ— first: 42, then +3 = 45.
15 + 3 Γ— 4 β†’ Γ— first: 12, then 15+12 = 27.
(20 βˆ’ 5) Γ· 3 β†’ brackets first: 15, then Γ·3 = 5.
⚠️ D and M are equal in rank β€” when both appear together (no brackets), do them left to right. Same for A and S.
Example: 8 Γ· 2 Γ— 3 = (8 Γ· 2) Γ— 3 = 4 Γ— 3 = 12. Not 8 Γ· 6.
7 Properties β€” clever shortcuts β–Ό

These rules make calculations easier (and they're great exam questions!).

PropertyMeansExample
CommutativeSwap order, same answer (+ and Γ—)5+3 = 3+5 βœ“   6Γ—7 = 7Γ—6 βœ“
AssociativeGroup differently, same answer (+ and Γ—)(2+3)+4 = 2+(3+4)
DistributiveΓ— spreads over (+ or βˆ’)5Γ—(3+4) = (5Γ—3) + (5Γ—4)
Identity for ++0 keeps the number78 + 0 = 78
Identity for Γ—Γ—1 keeps the number78 Γ— 1 = 78
Zero of Γ—Γ— 0 always = 078 Γ— 0 = 0
⚠️ Subtraction and division are NOT commutative: 10 βˆ’ 3 β‰  3 βˆ’ 10, and 20 Γ· 4 β‰  4 Γ· 20.
⚠️ You cannot divide by zero β€” it's undefined. But 0 Γ· any number (not zero) = 0.
πŸ’‘ Distributive in action: 7 Γ— 102 = 7 Γ— (100 + 2) = 700 + 14 = 714. Much easier than long multiplication!
8 Estimation β€” quick checks with rounding β–Ό

Before doing a big calculation, estimate the answer by rounding each number first. If your final answer is far from the estimate β€” recheck!

Estimate 487 + 312 by rounding to the nearest 100:
487 β‰ˆ 500   Β·   312 β‰ˆ 300   β†’ estimate = 800.
Actual: 487 + 312 = 799. Very close β€” answer is reasonable. βœ“
Estimate 218 Γ— 9:
218 β‰ˆ 200 β†’ 200 Γ— 9 = ~1,800.
Actual = 1,962. Estimate is in the right ballpark. βœ“
πŸ’‘ Estimation is your "sanity check" superpower in exams. If you write 487 + 312 = 7,990 and the estimate says ~800, you immediately know something's wrong.
9 Word problems β€” a 4-step plan β–Ό

Word problems hide the operation in a story. Use this 4-step plan:

  1. Read the problem twice.
  2. Find the numbers and the question.
  3. Decide which operation: keywords help.
  4. Solve and check.
Keywords β†’ operation
+ Addition: total, altogether, sum, combined, in all, more
βˆ’ Subtraction: difference, left, remain, fewer, less, change
Γ— Multiplication: times, of, each, product, per, double/triple
Γ· Division: share equally, each, split, per, average, group of
Problem: Crispin scored 124, 138 and 159 in three football matches. How many goals altogether?
Numbers: 124, 138, 159. Keyword: "altogether" β†’ +.
Solve: 124 + 138 + 159 = 421 goals.
Problem: 5 friends share 85 stickers equally. How many does each get?
Keyword: "share equally" β†’ Γ·.
Solve: 85 Γ· 5 = 17 stickers each.
⚠️ Two-step problems combine operations. Read carefully β€” the question often asks for the final result, not just one step.
10 Mental-math tricks ⚑ β–Ό

A few shortcuts that make you look like a wizard.

  • Γ— 10, 100, 1000: just stick zeros on the end. 47 Γ— 100 = 4,700.
  • Γ· 10, 100, 1000: remove zeros (or move the decimal point left). 5,200 Γ· 100 = 52.
  • Γ— 5: multiply by 10, then halve. 86 Γ— 5 = 860 Γ· 2 = 430.
  • Γ— 9: multiply by 10, then subtract the number. 47 Γ— 9 = 470 βˆ’ 47 = 423.
  • Γ— 11 (2-digit): add the digits and place between them. 36 Γ— 11 β†’ 3 _ 6 with (3+6)=9 in the middle β†’ 396.
  • + near multiples of 10: 47 + 19 = 47 + 20 βˆ’ 1 = 66.
  • βˆ’ 99: subtract 100 then add 1. 250 βˆ’ 99 = 250 βˆ’ 100 + 1 = 151.
πŸ’‘ These shortcuts do not replace the column methods β€” they're for quick mental work. Always show working in big problems.

✍️ Practice

Auto-scored. Hover wrong items to see the correct answer.

1. Addition

Numbers only β€” no commas needed.

4,567 + 2,834 =
3,500 + 5,500 =
1,250 + 5,070 =
1,234 + 3,087 =

2. Subtraction

8,512 βˆ’ 3,789 =
1,000 βˆ’ 437 =
9,000 βˆ’ 6,514 =
9,999 βˆ’ 2,650 =

3. Multiplication

345 Γ— 6 =
24 Γ— 25 =
126 Γ— 24 =
15 Γ— 7 =

4. Division

936 Γ· 4 =
725 Γ· 5 =
468 Γ· 6 =
1,000 Γ· 8 =

5. Order of Operations (BODMAS)

(8 + 4) Γ— 3 =
20 βˆ’ 6 + 4 =
6 Γ— 7 + 3 =
8 Γ· 2 Γ— 3 =

6. Word Problems

Crispin scored 124, 138, 159 in three games. Total =
A box has 24 chocolates. 8 boxes =
5 friends share 85 stickers. Each gets =
Tickets cost β‚Ή50. 17 sold = β‚Ή

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