Simplification and approximation for SBI PO

Simplification asks for an exact value. Approximation asks for the nearest option after sensible rounding. In the SBI PO prelims, where quant gives you 40 seconds a question, these are the questions that should take 20 to 30 seconds and bank time for DI and word problems. When they appear, they are among the quickest marks in the paper.

Both reward the same skills: fast mental arithmetic, a stock of number facts you recall without thinking, and judgement about how much rounding the options allow. This guide covers the methods, why each one works, worked examples and a practice set.

The order of operations

BODMAS: brackets, then orders (powers and roots) and "of", then division and multiplication from left to right, then addition and subtraction from left to right.

"Of" means multiply, and it is done before a plain division. So 1/4 of 80 ÷ 5 is (1/4 × 80) ÷ 5 = 20 ÷ 5 = 4.

Number facts to know cold

Also learn squares to 50 and cubes to 20. Every root question in this topic depends on them.

Methods and why they work

Squares near 50. (50 ± d)² = 2,500 ± 100d + d². So 47² = 2,500 − 300 + 9 = 2,209 and 53² = 2,500 + 300 + 9 = 2,809.

Squares ending in 5. For a number ending in 5, multiply the tens part by the next number and write 25 after it. 65²: 6 × 7 = 42, so 4,225. It works because (10a + 5)² = 100a(a + 1) + 25.

Difference of squares. a² − b² = (a − b)(a + b). So 58² − 42² = 16 × 100 = 1,600, with no squaring at all.

Square roots of perfect squares. Look at the last digit to get two possible last digits of the root, then bracket the number between two squares of tens.

Cube roots of perfect cubes. Every digit 0 to 9 has a unique last digit when cubed, so the last digit of the cube fixes the last digit of the root (1→1, 8→2, 7→3, 4→4, 5→5, 6→6, 3→7, 2→8, 9→9). Then drop the last three digits and find the largest cube not above what is left.

Digit-sum check. The digit sum of a product equals the product of the digit sums, reduced to one digit. This is because every number leaves the same remainder on division by 9 as its digit sum. It will not give you the answer, but it quickly rejects wrong options in a multiplication.

Multiplying by 5, 25 and 125. Multiply by 10, 100 or 1,000 and divide by 2, 4 or 8.

Worked examples: simplification

Example 1. (3/8 of 1,248 + 45% of 780) ÷ 13 = ?

• 1,248 ÷ 8 = 156, and 156 × 3 = 468.
• 45% of 780 = 351 (10% is 78, so 40% is 312, and 5% is 39).
• 468 + 351 = 819. Then 819 ÷ 13 = 63.

Example 2. 18² + √? = 25% of 1,600

• 324 + √? = 400, so √? = 76.
• ? = 76² = (75 + 1)² = 5,625 + 150 + 1 = 5,776.

Example 3. √7,396 = ?

• Last digit 6, so the root ends in 4 or 6.
• 80² = 6,400 and 90² = 8,100, so the root is in the 80s.
• 85² = 7,225, which is less than 7,396, so the root is above 85: 86.
• Check: 86² = 7,396.

Example 4. ∛29,791 = ?

• Last digit 1, so the root ends in 1.
• Drop the last three digits: 29. The largest cube not above 29 is 27 = 3³.
• Root = 31. Check: 31³ = 29,791.

Example 5 (option check). 347 × 28 = ? Options: 9,706, 9,716, 9,726, 9,736.

• Digit sums: 347 → 14 → 5. 28 → 10 → 1. Product → 5.
• 9,716 → 23 → 5. The others give 4, 6 and 7.
• 9,716. (The last digit alone, 7 × 8 = 56 ending in 6, would not separate these options. The digit sum does.)

Worked examples: approximation

Example 6. 24.97% of 1,999.8 + 14.02 × 15.98 − 9.99² ≈ ?

• ≈ 25% of 2,000 + 14 × 16 − 10²
• = 500 + 224 − 100 = 624.

Example 7. 7,199.7 ÷ 59.98 × 4.99 + 3.01³ ≈ ?

• ≈ 7,200 ÷ 60 × 5 + 27
• = 120 × 5 + 27 = 627.

When to be careful with rounding

• Look at the options first. If they are 600, 625, 650 and 700, bold rounding is safe. If they are 620, 624, 628 and 632, keep one more step of precision.
• Round in opposite directions in a product. In 14.02 × 15.98, one number goes down and one up, so the errors partly cancel. Rounding both up adds the errors together.
• Powers magnify error. 11.4² is about 130, not 121. Round the base only when it is very close to a whole number.
• Division by a rounded number. A small change in the divisor moves the answer more than you expect. 5,600 ÷ 68 is about 82, not 80.

A daily drill

Log how long the last block takes each day. The speed you build here carries straight into DI.

Practice set

1. 64% of 450 + 5/9 of 1,080 = ?
2. √5,184 − 8² = ?
3. Approximate: 59.97% of 849.9 − 14.98 × 11.02
4. ? × 18 = 36% of 1,500
5. Approximate: 3,249.8 ÷ 64.98 + 7.01³
6. 58² − 42² = ?
7. 12.5% of 2,568 + 62.5% of 480 = ?
8. ∛10,648 + √1,764 = ?

Answers:

1. 888. 64% of 450 = 288; 1,080 ÷ 9 × 5 = 600.
2. 8. √5,184 = 72 (ends in 4, so 2 or 8; between 70² and 80²; 75² = 5,625 is too big, so 72). 72 − 64 = 8.
3. 345. ≈ 60% of 850 − 15 × 11 = 510 − 165.
4. 30. 36% of 1,500 = 540; 540 ÷ 18 = 30.
5. 393. ≈ 3,250 ÷ 65 + 7³ = 50 + 343.
6. 1,600. (58 − 42)(58 + 42) = 16 × 100.
7. 621. 2,568 ÷ 8 = 321; 480 × 5/8 = 300.
8. 64. ∛10,648: ends in 8, so the root ends in 2; 10 lies between 2³ and 3³, so 22. √1,764 = 42. 22 + 42 = 64.

What to do next

• Learn the fraction table and squares to 50 until recall takes under two seconds.
• Do the ten-minute drill daily for three weeks and watch your times.
• Move on to number series and quadratic comparisons, the other quick prelims types.
• Read the quant and DI plan to see where this topic fits.