Calculation Speed, Simplification and Approximation

This chapter answers the Simplification and Approximation items of the Quantitative Aptitude section of the Prelims of IBPS Clerk, SBI Clerk, IBPS PO, SBI PO and IBPS RRB, where, in the IBPS Clerk, SBI Clerk, IBPS PO and SBI PO Prelims, the quantitative section has commonly carried about 35 questions in 20 minutes, roughly 35 seconds each (the IBPS RRB pattern differs, and every notification should be checked for the current pattern). It teaches the tables worth memorising (squares to 35, cubes to 15, fraction-decimal-percentage equivalents), multiplication and division shortcuts and the reason each works, the order of operations in long simplification chains, the nearest-value method of approximation and when rounding is unsafe, and quick answer checks by units digit and digit sum. Every item is numerical, so the marks come from speed with accuracy: one-fourth of a mark is lost for each wrong answer.

The tables to know without thinking

A bank Prelims is won in the first five seconds of each question: if you must work out 27² on paper, the question is already lost. So learn three tables once: squares to 35, cubes to 15 and fraction-decimal-percentage equivalents. Read the squares table below as follows: the row label gives the first number a, and the columns are a, a+1, … a+4.

Squares of 11 to 35
Rangeaa + 1a + 2a + 3a + 4
11–15121144169196225
16–20256289324361400
21–25441484529576625
26–30676729784841900
31–359611024108911561225

Cubes: 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 6³ = 216, 7³ = 343, 8³ = 512, 9³ = 729, 10³ = 1000, 11³ = 1331, 12³ = 1728, 13³ = 2197, 14³ = 2744, 15³ = 3375.

Fractions, decimals and percentages
FractionDecimalPercentage (%)
1/20.550
1/30.333333.33
1/40.2525
1/50.220
1/60.166716.67
1/70.142914.29
1/80.12512.5
1/90.111111.11
1/100.110
1/110.09099.09
1/120.08338.33
1/130.07697.69
1/140.07147.14
1/150.06676.67
1/160.06256.25
1/200.055
1/250.044
🧠 Build the rest from the few you know
You need not memorise every entry separately. Multiples follow from the unit fraction: 1/8 is 12.5%, so 3/8 is 37.5% and 5/8 is 62.5%; 1/16 is 6.25%, so 5/16 is 31.25%. A fraction such as 3/7 is three times 14.29%, about 42.86%. For squares above 25, use (30 + k)² = 900 + 60k + k²: 33² = 900 + 180 + 9 = 1089.

Multiplication and division shortcuts, and why they work

Every shortcut is an algebraic identity or a rewriting of a factor. Knowing the reason tells you when it applies.

  • Squaring a number ending in 5: (10a + 5)² = 100a(a + 1) + 25, so write a × (a + 1) and then 25. 65² = (6 × 7) | 25 = 4225; 35² = (3 × 4) | 25 = 1225.
  • Product of numbers equidistant from a base: (b − d)(b + d) = b² − d². 97 × 103 = 100² − 3² = 9991; 997 × 1003 = 1000² − 9 = 999991.
  • Same tens digit, units digits adding to 10: (10a + u)(10a + 10 − u) = 100a(a + 1) + u(10 − u). 47 × 43 = (4 × 5) | (7 × 3) = 20 | 21 = 2021; 68 × 62 = (6 × 7) | (8 × 2) = 42 | 16 = 4216.
  • Multiplying by 11: put the sum of neighbouring digits between the two ends, carrying when it exceeds 9. 63 × 11 = 6 | 9 | 3 = 693; 78 × 11 = 7 | (7 + 8) | 8 = 7 | 15 | 8 = 858.
  • Multiplying by 5, 25, 125 and by 9, 99: × 5 = × 10 ÷ 2; × 25 = × 100 ÷ 4; × 125 = × 1000 ÷ 8; × 99 = × 100 − the number. 36 × 25 = 3600 ÷ 4 = 900; 48 × 125 = 48000 ÷ 8 = 6000; 47 × 99 = 4700 − 47 = 4653.

Division reverses these: ÷ 5 = × 2 ÷ 10; ÷ 25 = × 4 ÷ 100; ÷ 12.5 = × 8 ÷ 100. So 340 ÷ 25 = 1360 ÷ 100 = 13.6. For a square near 50, write 47² = (50 − 3)² = 2500 − 300 + 9 = 2209.

⚠️ The shortcut needs its condition
The tens-and-units shortcut works only when the tens digits are the same and the units digits add to exactly 10. Applying it to 47 × 44 gives 20 | 28 = 2028, but the true product is 2068. Also remember the product of the two parts is written in two-digit blocks: 7 × 3 = 21 fills two places, but 9 × 1 = 9 must be written as 09 (49 × 41 = 20 | 09 = 2009).

Simplification chains

Long chains test discipline, not difficulty. Use BODMAS: Brackets (in the order ( ), { }, [ ]), Orders (powers and roots), then Division and Multiplication together from left to right, then Addition and Subtraction together from left to right. Division and multiplication have equal rank, so neither one goes first by habit.

Worked chain 1: (45% of 640) + (3/8 of 480) − 17². Here 45% of 640 = 0.45 × 640 = 288; 3/8 of 480 = 180; 17² = 289. The value is 288 + 180 − 289 = 179.

Worked chain 2: (36² − 24²) ÷ 12 + √1156 − 2.5 × 16. Use a² − b² = (a − b)(a + b): 36² − 24² = 12 × 60 = 720, and 720 ÷ 12 = 60. Next √1156 = 34 (since 34² = 1156) and 2.5 × 16 = 40. The value is 60 + 34 − 40 = 54.

Missing-value form. In '? × 18 = 40% of 2250 − 3/5 of 450 − 12²', evaluate the right side first: 900 − 270 − 144 = 486, then ? = 486 ÷ 18 = 27. Work toward the unknown by inverse operations only after the known side is a single number. When a chain has alternating signs of equal-sized squares, pair them: 15² − 14² + 13² − 12² = (15 + 14) + (13 + 12) = 54, since each pair a² − (a − 1)² equals a + (a − 1).

⚠️ Left to right, not 'multiplication first'
48 ÷ 6 × 2 is 8 × 2 = 16, not 48 ÷ 12 = 4. So 48 ÷ 6 × 2 + 3 × 4 − 8 ÷ 2 = 16 + 12 − 4 = 24. The wrong habit of doing multiplication before division gives 4 + 12 − 4 = 12, which is usually offered as a distractor. A minus sign before a bracket flips every sign inside it: 20 − (5 − 3) = 18, not 12.

Approximation: the nearest-value method

An approximation question gives an expression with awkward decimals and asks for the value nearest to the result, or '≈ ?'. The options are spaced widely, so you only need the answer to about 1 to 2 per cent. Round each number to a friendly one with one or two significant figures, compute, and choose the closest option.

Example 1: 49.8% of 1203 + 24.9% of 799. Take 50% of 1200 = 600 and 25% of 800 = 200, giving about 800. The exact value is 599.09 + 198.95 = 798.05, so the nearest option is 800. Example 2: 7990 ÷ 39.8 ≈ 8000 ÷ 40 = 200 (exact 200.75), and √2400 ≈ 49 because 49² = 2401, so the sum 200.75 + 48.99 is about 250.

  • Roots: pick the nearest perfect square. √1450 ≈ 38 because 38² = 1444; √5000 ≈ 70.7 because 70² = 4900 and 71² = 5041.
  • Percentages: use the fraction table: 16.67% of 480 is 1/6 of 480 = 80; 12.5% of 3996 ≈ 4000 ÷ 8 = 500.
  • Errors: a 1% error in each of two factors gives about a 2% error in their product; in a quotient the errors partly cancel if both numbers are rounded in the same direction.
⚠️ Never round before a subtraction of near-equal numbers
(1004.8 − 998.7) × 25 = 6.1 × 25 = 152.5, so the nearest value is about 150. If you round both numbers to 1000 first, the bracket becomes 0 and the answer 0, which is far wrong. Rounding removes the very digits that survive after the subtraction. Keep full decimals inside any bracket that subtracts nearly equal numbers, and round only after it.

Speed discipline and quick checks

In the common 35-questions-in-20-minutes pattern there are about 35 seconds for each question. Take the questions you can do in one step first, mark the long chains and return to them, and never spend two minutes on one item that is worth the same quarter-mark risk as any other.

  • Units-digit check: the last digit of a product depends only on the last digits of the factors. 487 × 63 ends in 7 × 3 = 21, so in 1.
  • Digit-sum (casting out nines) check: a number leaves the same remainder on division by 9 as its digit sum. 487 has digit sum 19 → 1, and 63 → 0, so the product must have remainder 0, a digit sum divisible by 9.
  • Use the options: the options 30681, 30781, 31681 and 30581 all end in 1, so the units digit cannot decide and the digit sum must: 3 + 0 + 6 + 8 + 1 = 18 passes, while 30781 and 31681 give 19 and 30581 gives 17. Hence 487 × 63 = 30681.
🎯 Why casting out nines works
Each place value 10, 100, 1000 … leaves remainder 1 when divided by 9, so a number and its digit sum differ by a multiple of 9. The check catches most slips but cannot see swapped digits (31 and 13 both leave 4), so pair it with a magnitude check: 487 × 63 must be near 500 × 60 = 30000.

Key takeaways

  • Memorise squares to 35, cubes to 15 and the fraction table to 1/16, 1/20 and 1/25; build multiples (3/8, 5/16) from the unit fraction.
  • Each shortcut has a condition: endings in 5, numbers equidistant from a base, equal tens with units summing to 10. Check the condition before using it.
  • In a chain apply BODMAS: division and multiplication together left to right, then addition and subtraction left to right; a minus before a bracket flips the signs inside.
  • Pair equal-sized squares with alternating signs using a² − b² = (a − b)(a + b); for consecutive numbers a² − (a − 1)² = a + (a − 1).
  • For nearest-value questions round to one or two significant figures and pick the closest option, but never round inside a subtraction of nearly equal numbers.
  • Use the options: units digit and digit sum (casting out nines) eliminate wrong answers in seconds, with a magnitude check against swapped digits.

Practice questions (17)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

  1. What is the value of 28² + 12³?

    1. 2412
    2. 2502
    3. 2512
    4. 2612
    Show answer

    Answer: C — 2512

    28² = 784 and 12³ = 1728, so the sum is 784 + 1728 = 2512. The value 2412 comes from writing 12³ as 1628 by a slip in the cube, and 2502 from taking 28² as 774.

  2. Expressed as a percentage, 5/16 equals

    1. 31.25%
    2. 32.5%
    3. 33.25%
    4. 30.75%
    Show answer

    Answer: A — 31.25%

    1/16 = 6.25%, so 5/16 = 5 × 6.25% = 31.25%. The value 33.25% is tempting because 1/3 is 33.33%, but 5/16 is smaller than 1/3.

  3. Consider these equivalents. P: 1/7 is about 14.29%. Q: 1/9 is about 11.11%. R: 1/12 is about 8.33%. S: 1/13 is about 6.67%. Which statements are correct?

    1. Q, R and S only
    2. P, R and S only
    3. P, Q and S only
    4. P, Q and R only
    Show answer

    Answer: D — P, Q and R only

    P, Q and R are correct. S is wrong: 1/13 is about 7.69%, while 6.67% is 1/15. The pair 13 and 15 is easily swapped, which is why S is the tempting inclusion.

  4. What is the value of 65²?

    1. 4125
    2. 4225
    3. 4325
    4. 4215
    Show answer

    Answer: B — 4225

    For a number ending in 5, multiply the tens digit by its successor and append 25: 6 × 7 = 42, so 65² = 4225. A reply such as 4215 forgets that the ending is always 25.

  5. What is 997 × 1003?

    1. 999901
    2. 1000009
    3. 999991
    4. 999909
    Show answer

    Answer: C — 999991

    Both numbers are 3 away from 1000, so the product is 1000² − 3² = 1000000 − 9 = 999991. The value 1000009 comes from wrongly adding 3² instead of subtracting it, since one factor is below the base.

  6. What is 68 × 62?

    1. 4216
    2. 4116
    3. 4196
    4. 4236
    Show answer

    Answer: A — 4216

    The tens digits are equal (6) and the units digits 8 and 2 add to 10, so the product is (6 × 7) | (8 × 2) = 42 | 16 = 4216. The value 4116 results from writing 8 × 2 as 6 and shifting the digits.

  7. Four shortcut results are given. P: 48 × 125 = 6000. Q: 75² = 5525. R: 73 × 11 = 803. S: 52 × 99 = 5148. Which one is NOT correct?

    1. P
    2. Q
    3. R
    4. S
    Show answer

    Answer: B — Q

    75² = (7 × 8) | 25 = 5625, so Q is wrong. P is right (48000 ÷ 8), R is right (7 | 10 | 3 carries to 803) and S is right (5200 − 52). R is the tempting wrong pick because of the carry.

  8. Find the value of (45% of 640) + (3/8 of 480) − 17².

    1. 189
    2. 169
    3. 279
    4. 179
    Show answer

    Answer: D — 179

    45% of 640 = 288, 3/8 of 480 = 180 and 17² = 289, so the value is 288 + 180 − 289 = 179. The result 279 comes from taking 17² as 389 or from a slip of 100 in the sum.

  9. Find the value of 48 ÷ 6 × 2 + 3 × 4 − 8 ÷ 2.

    1. 12
    2. 24
    3. 20
    4. 28
    Show answer

    Answer: B — 24

    Division and multiplication rank equally and go left to right: 48 ÷ 6 × 2 = 16, 3 × 4 = 12 and 8 ÷ 2 = 4, so 16 + 12 − 4 = 24. The value 12 comes from doing 6 × 2 first (48 ÷ 12 = 4), which breaks the left-to-right rule.

  10. Find the value of (36² − 24²) ÷ 12 + √1156 − 2.5 × 16.

    1. 44
    2. 64
    3. 54
    4. 74
    Show answer

    Answer: C — 54

    36² − 24² = (36 − 24)(36 + 24) = 720 and 720 ÷ 12 = 60; √1156 = 34 and 2.5 × 16 = 40, so 60 + 34 − 40 = 54. The value 74 arises from adding 40 instead of subtracting it.

  11. If ? × 18 = 40% of 2250 − 3/5 of 450 − 12², then ? equals

    1. 27
    2. 31
    3. 24
    4. 36
    Show answer

    Answer: A — 27

    The right side is 900 − 270 − 144 = 486, so ? = 486 ÷ 18 = 27. A tempting slip is to add 144 instead of subtracting it, which gives 774 ÷ 18 = 43; only 27 satisfies 27 × 18 = 486.

  12. Find the value of 15² − 14² + 13² − 12² + 11² − 10².

    1. 65
    2. 75
    3. 85
    4. 49
    Show answer

    Answer: B — 75

    Each pair a² − (a − 1)² equals a + (a − 1): 29 + 25 + 21 = 75, the sum of 15 down to 10. The value 49 treats the six terms as a single alternating sum of squares and loses the pairing.

  13. The value of 49.8% of 1203 + 24.9% of 799 is nearest to

    1. 760
    2. 850
    3. 900
    4. 800
    Show answer

    Answer: D — 800

    Rounding gives 50% of 1200 + 25% of 800 = 600 + 200 = 800; the exact value 798.05 confirms it. The values 760 and 850 lie 38 and 52 away, far from the true result.

  14. The value of (1004.8 − 998.7) × 25 is nearest to

    1. 0
    2. 100
    3. 150
    4. 200
    Show answer

    Answer: C — 150

    The bracket is 1004.8 − 998.7 = 6.1, and 6.1 × 25 = 152.5, nearest to 150. The reply 0 comes from rounding both numbers to 1000 first, which wipes out the difference; so keep full decimals inside a bracket of near-equal numbers.

  15. The value of 7990 ÷ 39.8 + √2400 is nearest to

    1. 250
    2. 240
    3. 260
    4. 230
    Show answer

    Answer: A — 250

    7990 ÷ 39.8 is about 200 (exactly 200.75) and √2400 is about 49 (since 49² = 2401), so the sum is about 250. The value 240 would need √2400 to be near 40, which is the root of 1600.

  16. Using the digit-sum (casting-out-nines) check, which of these can be the product 487 × 63?

    1. 30781
    2. 30681
    3. 31681
    4. 30581
    Show answer

    Answer: B — 30681

    487 leaves remainder 1 (digit sum 19) and 63 leaves 0, so the product must leave 0. Only 30681 has digit sum 18; 30781 and 31681 have 19 and 30581 has 17. All four end in 1, so the units digit alone could not decide.

  17. In which situation is rounding the numbers first to one or two significant figures most unsafe?

    1. Finding a percentage of a number, such as 24.9% of 799
    2. Dividing a number by another, such as 7990 ÷ 39.8
    3. Adding two numbers of similar size, such as 1203 + 799
    4. Subtracting two nearly equal numbers, such as 1004.8 − 998.7
    Show answer

    Answer: D — Subtracting two nearly equal numbers, such as 1004.8 − 998.7

    A subtraction of near-equal numbers keeps only the small difference, so rounding errors of the size of the difference ruin it. Percentage, division and addition keep the relative error small, which is why rounding is safe there.