Mental Ability · Lesson 2

Operations & Word Problems

Add, subtract, multiply and divide big numbers with confidence, learn clever shortcuts to do it in your head, and crack any word problem with a simple 4-step method.

~45 min read6 quick checksTest: 20 questions

1Addition and subtraction

Every part of a sum has a name. Olympiads sometimes ask for these names, so learn them well.

OperationExampleNames of the parts
Addition250 + 130 = 380250 and 130 are addends; 380 is the sum
Subtraction900 − 356 = 544900 is the minuend; 356 is the subtrahend; 544 is the difference

Adding with carrying

Write the numbers in columns, ones under ones. Add from the right. If a column adds up to 10 or more, write the ones digit and carry the tens digit to the next column.

TTh Th H T O 1 1 1 1 3 8 4 7 6 2 7 8 5 9 + 6 6 3 3 5 carry sum
38,476 + 27,859. Ones: 6 + 9 = 15 → write 5, carry 1. Tens: 7 + 5 + 1 = 13 → write 3, carry 1. Keep going to get 66,335.

Subtracting with borrowing

Subtract from the right. If the top digit is smaller, borrow 1 from the next place on the left (it becomes 10 in your column).

✏️ Worked example — borrowing across zeros

Find 40,000 − 23,657.

  1. The ones place has 0, and so do tens, hundreds and thousands. Borrow from the 4: it becomes 3, and the zeros become 9, 9, 9 and the last one 10.
  2. So 40,000 is treated as 3 | 9 | 9 | 9 | 10.
  3. Subtract column by column: 10 − 7 = 3, 9 − 5 = 4, 9 − 6 = 3, 9 − 3 = 6, 3 − 2 = 1.
  4. Answer: 16,343. Check: 16,343 + 23,657 = 40,000 ✓
💡 Shortcut: subtracting from 1,00,000 or 10,000

What must be added to 47,386 to make 1,00,000? Subtract every digit from 9, but the last non-zero digit from 10: 9−4 = 5, 9−7 = 2, 9−3 = 6, 9−8 = 1, 10−6 = 4 → 52,614. Check: 47,386 + 52,614 = 1,00,000 ✓

📌 Remember

Check subtraction by addition: difference + subtrahend = minuend. To find a missing addend, subtract: ? + 2,480 = 5,000 → ? = 5,000 − 2,480 = 2,520.

Find 60,000 − 27,684.

  1. 33,316
  2. 32,316
  3. 32,426
  4. 43,316
Borrowing across the zeros turns 60,000 into 5 | 9 | 9 | 9 | 10. Then 10−4 = 6, 9−8 = 1, 9−6 = 3, 9−7 = 2, 5−2 = 3 → 32,316. Check: 32,316 + 27,684 = 60,000.

2Properties of the four operations

PropertyAdditionMultiplication
Order (commutative)15 + 20 = 20 + 1512 × 7 = 7 × 12
Grouping (associative)(4 + 6) + 9 = 4 + (6 + 9)(2 × 5) × 7 = 2 × (5 × 7)
IdentityAdding 0 changes nothing: 58 + 0 = 58Multiplying by 1 changes nothing: 58 × 1 = 58
Zero—Any number × 0 = 0
Spreading (distributive)6 × (10 + 3) = 6 × 10 + 6 × 3 = 60 + 18 = 78

Special facts about division

⚠️ Olympiad trap

Order matters for subtraction and division: 20 − 5 ≠ 5 − 20 and 20 ÷ 5 ≠ 5 ÷ 20. Only addition and multiplication can be swapped. And don't mix up "0 ÷ 9" (= 0, fine) with "9 ÷ 0" (not defined).

3Multiplication

In 45 × 12 = 540, 45 is the multiplicand, 12 is the multiplier and 540 is the product. Multiplication is quick repeated addition: 4 × 3 = 3 + 3 + 3 + 3.

Multiplying by 10, 100, 1000 and their multiples

Multiplying by a 2-digit number

✏️ Worked example

Find 436 × 27.

  1. Multiply by the ones digit: 436 × 7 = 3,052.
  2. Multiply by the tens digit (20): 436 × 20 = 8,720. (Write a 0 in the ones place first, then 436 × 2.)
  3. Add the two lines: 3,052 + 8,720 = 11,772.
30 × 20 = 600 80 30 × 6 = 180 24 30 4 20 6 600 + 80 + 180 + 24 = 884
The area model shows why this works: split 34 × 26 into (30 + 4) × (20 + 6) and add the four parts.
✏️ Worked example — 3-digit by 3-digit

Find 254 × 123.

  1. 254 × 3 = 762.
  2. 254 × 20 = 5,080.
  3. 254 × 100 = 25,400.
  4. Add: 762 + 5,080 + 25,400 = 31,242.
⚠️ Olympiad trap: the missing zero

The most common mistake in long multiplication is forgetting the placeholder 0 on the second line. Without it, 436 × 27 would come out as 3,052 + 872 = 3,924 — far too small. Quick estimate check: 436 × 27 is about 400 × 30 = 12,000, so 11,772 makes sense.

4Division

In 8,745 ÷ 6: 8,745 is the dividend, 6 is the divisor, the answer 1,457 is the quotient, and what is left over (3) is the remainder.

Long division: Divide, Multiply, Subtract, Bring down

1457 6 8745 −6 27 −24 34 −30 45 −42 3 6 × 1 6 × 4 6 × 5 6 × 7 quotient remainder divisor
8,745 ÷ 6 = 1,457 remainder 3. Repeat the cycle Divide → Multiply → Subtract → Bring down until no digits are left.
Remember the cycle: "Does Mummy Serve Burgers?" → Divide, Multiply, Subtract, Bring down. Then start again.
✏️ Worked example — a zero in the quotient

Find 6,324 ÷ 6.

  1. 6 ÷ 6 = 1. Subtract, remainder 0.
  2. Bring down 3. 3 ÷ 6 = 0 — write 0 in the quotient! Bring down 2 to make 32.
  3. 32 ÷ 6 = 5, remainder 2. Bring down 4 to make 24.
  4. 24 ÷ 6 = 4, remainder 0. Quotient = 1,054.
✏️ Worked example — 2-digit divisor

Find 4,567 ÷ 23.

  1. 45 ÷ 23 = 1 (23 × 1 = 23). 45 − 23 = 22. Bring down 6 → 226.
  2. 226 ÷ 23: try 23 × 9 = 207. 226 − 207 = 19. Bring down 7 → 197.
  3. 197 ÷ 23: try 23 × 8 = 184. 197 − 184 = 13.
  4. Quotient = 198, remainder = 13.

The golden check

📌 Remember

Dividend = Divisor × Quotient + Remainder. Check: 23 × 198 + 13 = 4,554 + 13 = 4,567 ✓. The remainder must always be smaller than the divisor — so when dividing by 12, the largest possible remainder is 11.

✏️ Worked example — find the dividend

A number is divided by 9. The quotient is 47 and the remainder is 5. Find the number.

  1. Use Dividend = Divisor × Quotient + Remainder.
  2. 9 × 47 = 423.
  3. 423 + 5 = 428.

Dividing by 10, 100, 1000: when the number ends in enough zeros, just remove that many zeros. 45,300 ÷ 100 = 453. 72,000 ÷ 1000 = 72.

When a number is divided by 7, the quotient is 36 and the remainder is 4. What is the number?

  1. 252
  2. 288
  3. 247
  4. 256
Dividend = 7 × 36 + 4 = 252 + 4 = 256. (252 forgets the remainder; 247 subtracts it instead of adding.)

5Order of operations and brackets

When a question mixes operations, everyone must follow the same order, or we would all get different answers.

  1. Brackets first — work out whatever is inside ( ).
  2. Then × and ÷, going from left to right.
  3. Then + and −, going from left to right.
Remember BODMAS: Brackets, Of, Division, Multiplication, Addition, Subtraction. D and M are partners (do whichever comes first from the left); A and S are partners too.
✏️ Worked examples
  1. 36 − 4 × 5 + 10 → multiply first: 36 − 20 + 10 → left to right: 16 + 10 = 26.
  2. (36 − 4) × 5 → brackets first: 32 × 5 = 160. Same numbers, very different answer!
  3. 100 − (25 + 15) ÷ 8 → brackets: 100 − 40 ÷ 8 → divide: 100 − 5 = 95.
  4. 48 ÷ 6 × 2 → left to right: 8 × 2 = 16 (not 48 ÷ 12 = 4).
✏️ Worked example — "which statement is true?"

Is 7 × 8 = 60 − 4 true?

  1. Work out each side separately. Left: 7 × 8 = 56.
  2. Right: 60 − 4 = 56.
  3. Both sides are 56, so the statement is true. Always calculate both sides fully before deciding.

What is 45 ÷ 5 × 3?

  1. 3
  2. 27
  3. 45
  4. 15
÷ and × are partners, so go left to right: 45 ÷ 5 = 9, then 9 × 3 = 27. Doing 5 × 3 first (45 ÷ 15 = 3) is the trap.

6Mental maths shortcuts and estimation

Olympiad questions reward speed. These tricks turn long sums into quick ones.

× 5

Multiply by 10, then halve. 46 × 5 = 460 ÷ 2 = 230.

× 25

Multiply by 100, then divide by 4. 38 × 25 = 3,800 ÷ 4 = 950.

× 9 and × 99

67 × 9 = 670 − 67 = 603. 45 × 99 = 4,500 − 45 = 4,455.

× 11 (2-digit)

Split the digits and put their sum in the middle. 53 × 11 → 5 (5+3) 3 = 583. If the sum is 10 or more, carry: 78 × 11 → 7 (15) 8 → 858.

Friendly pairs

Look for 2 × 5 = 10, 4 × 25 = 100, 8 × 125 = 1,000. 25 × 47 × 4 = (25 × 4) × 47 = 100 × 47 = 4,700.

Halve and double

Halve one number and double the other. 16 × 25 = 8 × 50 = 4 × 100 = 400.

Round and fix

398 + 257 = 400 + 257 − 2 = 655. 5,432 − 999 = 5,432 − 1,000 + 1 = 4,433.

Spread it out

Distributive trick: 7 × 106 = 7 × 100 + 7 × 6 = 700 + 42 = 742.

Estimation

Round the numbers to easy values, then calculate. Use it to check answers and to eliminate wrong options quickly.

✏️ Worked example
  1. 612 × 48 ≈ 600 × 50 = 30,000. (Exact: 29,376 — close.)
  2. 7,890 ÷ 38 ≈ 8,000 ÷ 40 = 200. (Exact: 207, remainder 24.)
  3. In an MCQ with options 2,937 · 29,376 · 2,93,760 · 293, the estimate 30,000 points straight to 29,376.

Using a shortcut, 125 × 23 × 8 = ?

  1. 23,000
  2. 2,300
  3. 18,400
  4. 230,000
Pair 125 × 8 = 1,000 first. Then 1,000 × 23 = 23,000.

7Solving word problems: the 4-step method

A word problem is a story with a maths question hidden inside. Don't rush to calculate — first understand the story.

  1. Read & understand. What do I know? What is the question asking? Underline the numbers and the question.
  2. Plan. Which operation(s) do I need, and in what order? Drawing a bar model helps.
  3. Solve. Calculate carefully, one step at a time. Write units (₹, m, kg).
  4. Check. Does the answer make sense? Estimate or work backwards.

Keyword clues

OperationClue words
+altogether, in all, total, sum, combined, more than (to find a bigger amount)
−left, remaining, difference, how many more, fewer, less than, gave away
×each (to find a total), every, times, twice, per, rows of
÷shared equally, each (to find one share), per, groups of, how many in each
⚠️ Olympiad trap: keywords can lie

"Riya has 45 stickers. That is 12 more than Sam. How many does Sam have?" The word "more" tempts you to add, but Sam has fewer: 45 − 12 = 33. Always picture the story instead of grabbing a keyword.

The unitary method: find ONE first

✏️ Worked example

7 bags cost ₹364. What is the cost of 12 such bags?

  1. Cost of 1 bag = 364 ÷ 7 = ₹52.
  2. Cost of 12 bags = 52 × 12 = ₹624.
  3. Check: 12 bags is a bit less than double 7 bags (₹728), so ₹624 is sensible.
✏️ Worked example — "for every"

In a class there are 5 girls for every 4 boys. There are 36 boys. How many girls are there?

  1. Make groups of 4 boys: 36 ÷ 4 = 9 groups.
  2. Each group has 5 girls: 9 × 5 = 45 girls.

Multi-step problems

✏️ Worked example

A shopkeeper bought 24 boxes with 36 pencils in each. He sold 578 pencils. How many pencils are left?

  1. Total pencils = 24 × 36 = 864.
  2. Pencils left = 864 − 578 = 286.
✏️ Worked example — money

Ravi had ₹1,000. He bought 3 notebooks at ₹85 each and a bag for ₹420. How much money is left?

  1. Notebooks: 3 × 85 = ₹255.
  2. Total spent: 255 + 420 = ₹675.
  3. Money left: 1,000 − 675 = ₹325.

What to do with a remainder

✏️ Worked example — round up or round down?
  1. 250 children go on a trip. Each van carries 12. 250 ÷ 12 = 20 remainder 10. The 10 extra children still need a van, so we need 21 vans (round up).
  2. A 100 cm ribbon is cut into 15 cm pieces. 100 ÷ 15 = 6 remainder 10. The 10 cm piece is too short, so we get only 6 pieces (round down).

300 children are going to a match. Each bus holds 14 children. What is the least number of buses needed?

  1. 21
  2. 22
  3. 20
  4. 23
300 ÷ 14 = 21 remainder 6. Those 6 children also need a seat, so 21 + 1 = 22 buses.

8Olympiad problem types

Sum and difference

Big 120 ? Small ? 540 Remove 120 → 420 = two equal bars Small = 210 · Big = 330
Two numbers add up to 540 and differ by 120. Remove the extra 120 and split the rest equally.
💡 Shortcut

Larger number = (sum + difference) ÷ 2. Smaller number = (sum − difference) ÷ 2. For 540 and 120: larger = 660 ÷ 2 = 330, smaller = 420 ÷ 2 = 210.

Posts, trees and gaps

1 5 m 2 5 m 3 5 m 4 5 m 5
A 20 m fence with a post every 5 m: 20 ÷ 5 = 4 gaps, but 5 posts.
✏️ Worked example

Poles are put up every 5 m along a straight 60 m road, with a pole at both ends. How many poles?

  1. Number of gaps = 60 ÷ 5 = 12.
  2. With a pole at both ends, poles = gaps + 1 = 13.
  3. If there are no poles at the ends: gaps − 1 = 11. Around a closed shape (a circle or a park boundary): poles = gaps = 12.
⚠️ Olympiad trap

Answering 12 instead of 13 is the classic mistake. Draw a tiny version (like the 20 m fence) to see the "+1".

Shape puzzles (early algebra)

✏️ Worked example

★ + ★ = 18, ★ × ● = 36, ● − ■ = 1. Find ■.

  1. Start with the line that has only one shape: ★ + ★ = 18, so ★ = 18 ÷ 2 = 9.
  2. 9 × ● = 36, so ● = 36 ÷ 9 = 4.
  3. 4 − ■ = 1, so ■ = 3.

Multiplying with mixed units

✏️ Worked example

A father is 3 times as old as his son. The son is 11 years 8 months old. How old is the father?

  1. Multiply each unit separately: 3 × 11 years = 33 years, 3 × 8 months = 24 months.
  2. 24 months = 2 years (12 months = 1 year).
  3. Father = 33 years + 2 years = 35 years.

The sum of two numbers is 800 and their difference is 140. What is the smaller number?

  1. 470
  2. 400
  3. 330
  4. 660
Smaller = (800 − 140) ÷ 2 = 660 ÷ 2 = 330. The larger is 330 + 140 = 470; check 330 + 470 = 800 ✓.

9Watch and explore

10Quick recap

Flip cards

QuestionIn 725 − 318 = 407, which is the minuend?
Answer725 — the number we subtract from.
QuestionLargest possible remainder when dividing by 15?
Answer14 — the remainder is always less than the divisor.
Question56 × 25 in your head?
Answer5,600 ÷ 4 = 1,400.
Question64 × 11?
Answer6 (6+4) 4 = 704.
Question20 + 30 ÷ 5 = ?
AnswerDivide first: 20 + 6 = 26.
Question5 pens cost ₹60. Cost of 9 pens?
Answer1 pen = ₹12, so 9 pens = ₹108.
QuestionTrees every 10 m along a 100 m road, both ends?
Answer100 ÷ 10 = 10 gaps, so 11 trees.
QuestionWhat must be added to 2,468 to make 10,000?
Answer7,532 (9−2, 9−4, 9−6, 10−8).