📚 Concept Cheat Sheet · Numbers + Four Operations ·
← 10-day plan
Chapter 1 · Numbers
Putting commas — two systems
- Indian system (3-2-2): from the right, first comma after 3 digits, then every 2 digits. 12345678 → 1,23,45,678
- International system (3-3-3): a comma every 3 digits from the right. 12345678 → 12,345,678
- Indian places: ones, tens, hundreds, thousand, ten-thousand, lakh, ten-lakh, crore, ten-crore.
- International places: ones, tens, hundreds, thousand, ten-thousand, hundred-thousand, million, ten-million.
- Quick links: 1 lakh = 100 thousand, 1 crore = 10 million.
Place value vs face value
- Face value = the digit itself, wherever it sits. (In 53,210 the face value of 3 is just 3.)
- Place value = digit × its place worth. (In 30,210 the place value of 3 is 3 × 10,000 = 30,000.)
- The place value of 0 is always 0.
Successor · predecessor · expanded form
- Successor = +1 (the number just after). Predecessor = −1 (the number just before).
- Successor of 9,999 = 10,000; predecessor of 10,000 = 9,999.
- Expanded form breaks a number by place value. 5,34,067 = 5,00,000 + 30,000 + 4,000 + 60 + 7
- Standard form = add the parts back together.
Comparing, ordering & rounding
- To compare: more digits = bigger. Same digits? Compare from the left, place by place.
- Ascending = smallest → largest. Descending = largest → smallest.
- Rounding rule: look at the digit just to the right of the rounding place. 5 or more → round up; less than 5 → round down (keep it the same).
- Round 7,839 to nearest 10 → look at 9 → up → 7,840. Round 64,560 to nearest 100 → look at 6 → up → 64,600.
Roman numerals
- Add when symbols go big → small: VIII = 5+1+1+1 = 8.
- Subtract when a smaller symbol comes before a bigger one: IV=4, IX=9, XL=40, XC=90, CD=400, CM=900.
- Never repeat a symbol more than three times in a row.
- Examples: 46 = XLVI, 72 = LXXII, 90 = XC, 99 = XCIX.
Chapter 2 · The Four Operations
Vocabulary & special properties
- Addition → answer is the sum. Subtraction → answer is the difference (the number we subtract from is the minuend).
- Multiplication → answer is the product. Division → answer is the quotient, with a remainder left over.
- a + 0 = a (0 is the additive identity). a − 0 = a. a − a = 0.
- a × 1 = a (1 is the multiplicative identity). a × 0 = 0.
- a ÷ 1 = a (the number itself). a ÷ a = 1 (for any non-zero a).
- Order does not matter for + and ×: 432 × 755 = 755 × 432.
Multiplying & dividing by 10, 100, 1000
- × 10 / 100 / 1000: add one / two / three zeros. (476 × 100 = 47,600.)
- ÷ 10 / 100 / 1000: the last one / two / three digits become the remainder; the rest is the quotient. (5,678 ÷ 100 → quotient 56, remainder 78.)
Long multiplication & division
- Multiply by each digit of the second number, shift each row one place left, then add the rows.
- Divide using long division: divide, multiply, subtract, bring down the next digit — repeat.
- The remainder is always smaller than the divisor.
⭐ Divide and check (very important!)
- After any division, verify with: (quotient × divisor) + remainder = dividend.
- Example: 77,218 ÷ 5 → quotient 15,443, remainder 3. Check: (15,443 × 5) + 3 = 77,215 + 3 = 77,218 ✓ — matches the dividend, so the division is correct.
Word-problem checklist
- Read twice. Underline the numbers and the question.
- Decide the operation: altogether / total → + or ×; left / remain / how many more → −; shared equally / each → ÷.
- Show working, write the answer with its unit (₹, buns, children…), and check it makes sense.