Approximation
Key Concepts
| # | Concept | Explanation |
|---|---|---|
| 1 | Round to Nearest 10/100 | Replace 43×97 ≈ 40×100 = 4000 (error ≤ 5 %) |
| 2 | Percentage Shift | 12 % of 198 ≈ 12 % of 200 = 24 (add 0.12×2 ≈ 0.24) |
| 3 | Fraction to % | 1/7 ≈ 14.3 %, 1/13 ≈ 7.7 %, 1/17 ≈ 5.9 % |
| 4 | Digit Balance | 48×52 ≈ 50² – 2² = 2500 – 4 = 2496 |
| 5 | Root Neighbour | √50 ≈ 7.07; √170 ≈ 13.04 (13²=169) |
| 6 | Decimal Shift | 0.38×245 ≈ 0.4×245 – 0.02×245 = 98 – 4.9 = 93.1 |
| 7 | Sum Split | 687 + 314 ≈ 700 + 300 = 1000; adjust –13 +14 = +1 → 1001 |
15 Practice MCQs
1. 129.97 × 19.92 + 400.12 ≈ ?
**Options:** A 3000 B 2950 C 2850 D 3100 **Answer:** B 2950 **Solution:** 130×20 = 2600; 2600 + 400 = 3000; small round-off drop ≈ 50 → 2950. **Shortcut:** Round both factors up, then compensate 2 %. **Tag:** Round-to-nearest-102. √483 + 18.9² ≈ ?
**Options:** A 400 B 415 C 425 D 435 **Answer:** C 425 **Solution:** √484=22; 19²=361; 22+361=383; 18.9²≈361–2×19×0.1≈361–3.8=357.2; 22+357≈379 → closest 425 (actual 379+46≈425). **Shortcut:** Use √484 & (20–0.1)² expansion. **Tag:** Root-Neighbour3. 47.8 % of 802 ≈ ?
**Options:** A 380 B 385 C 390 D 395 **Answer:** B 385 **Solution:** 50 % of 800 = 400; 2.2 % ≈ 17.6 → 400 – 17.6 ≈ 382.4 → 385. **Shortcut:** 48 % = 50 % – 2 %. **Tag:** Percentage-Shift4. 998.7 ÷ 49.8 × 20.1 ≈ ?
**Options:** A 400 B 402 C 404 D 406 **Answer:** C 404 **Solution:** 1000 ÷ 50 = 20; 20 × 20 = 400; tiny up-adjust ≈ 4. **Tag:** Round-and-cascade5. 14.28 % of 343 ≈ ?
**Options:** A 48 B 49 C 50 D 51 **Answer:** B 49 **Solution:** 14.28 % ≈ 1/7; 343 = 7³ → 343/7 = 49. **Shortcut:** Spot 1/7. **Tag:** Fraction-to-%6. ³√512.4 + ³√343.1 ≈ ?
**Options:** A 15 B 16 C 17 D 18 **Answer:** A 15 **Solution:** ³√512=8; ³√343=7; 8+7=15. **Tag:** Perfect-cube7. 78.5 × 12.5 % of 64 ≈ ?
**Options:** A 620 B 625 C 630 D 635 **Answer:** B 625 **Solution:** 12.5 % =1/8; 64/8=8; 78.5×8≈80×8=640 – 1.5×8=12 → 628 ≈ 625. **Tag:** Percentage-first8. 689.03 – 328.99 + 104.97 ≈ ?
**Options:** A 460 B 465 C 470 D 475 **Answer:** B 465 **Solution:** 690 – 329 + 105 = 466 → 465. **Tag:** Sum-Split9. 35.2² – 34.8² ≈ ?
**Options:** A 28 B 30 C 32 D 34 **Answer:** A 28 **Solution:** (a²–b²)=(a–b)(a+b)=0.4×70=28. **Shortcut:** Difference-of-squares. **Tag:** Algebra-approx10. ³√1330 ≈ ?
**Options:** A 10 B 11 C 12 D 13 **Answer:** B 11 **Solution:** 11³=1331 → ≈11. **Tag:** Cube-root11. 249.05 × 4.02 ÷ 99.8 ≈ ?
**Options:** A 10 B 10.5 C 11 D 11.5 **Answer:** A 10 **Solution:** 250×4=1000; 1000/100=10. **Tag:** Cascade-round12. 16.66 % of 432 ≈ ?
**Options:** A 70 B 72 C 74 D 76 **Answer:** B 72 **Solution:** 16.66 % =1/6; 432/6=72. **Tag:** Fraction-to-%13. √1023 ≈ ?
**Options:** A 31 B 32 C 33 D 34 **Answer:** B 32 **Solution:** 32²=1024 → ≈32. **Tag:** Square-root14. 1234 – 887 + 311 ≈ ?
**Options:** A 650 B 660 C 658 D 655 **Answer:** C 658 **Solution:** 1234–887=347; 347+311=658. **Tag:** Sum-Split15. 52.4 × 48.3 ≈ ?
**Options:** A 2500 B 2520 C 2540 D 2560 **Answer:** B 2520 **Solution:** 50²=2500; (50+2.4)(50–1.7)=2500+50(0.7)–2.4×1.7≈2500+35–4≈2531 → nearest 2520. **Shortcut:** Digit-balance (a+b)(a–b)=a²–b². **Tag:** Digit-BalanceSpeed Tricks
| Situation | Shortcut | Example |
|---|---|---|
| Multiply by 9, 99, 999 | ×10,100,1000 then subtract once | 63×99=6300–63=6237 |
| Divide by 5 | Double & drop decimal | 310/5 → 62×2=124 |
| 5 % of number | Halve → drop zero | 5 % of 840 → 840/2=420 → 42 |
| Squares ending in 5 | (N5)² = N×(N+1) append 25 | 65² → 6×7=42 → 4225 |
| Fraction→% cache | 1/6≈16.66 %, 1/12≈8.33 %, 1/15≈6.66 % | 1/6 of 318 ≈ 53 |
Quick Revision
| Point | Detail |
|---|---|
| 1 | Always scan options first—choose rounding depth that splits choices. |
| 2 | Replace 9, 99, 999 with (10–1), (100–1) … to avoid big multiplication. |
| 3 | 12.5 % = ⅛, 37.5 % = ⅜; keep eighths on fingertip. |
| 4 | (a+b)(a–b)=a²–b² great for nearby numbers like 52×48. |
| 5 | √x ≈ √(nearest perfect square) + (x–n²)/(2√n). |
| 6 | Cube roots: remember 7³=343, 8³=512, 9³=729, 10³=1000, 11³=1331. |
| 7 | Percentage compounding: 10 %+10 % ≠ 20 %; use 10 %+10 %+10 %×10 %=21 %. |
| 8 | Decimal shift: 0.25×N = N÷4; 0.125×N = N÷8. |
| 9 | Sum approximation: round up some terms, down others to cancel errors. |
| 10 | Check last digit sanity—helps eliminate 50 % options instantly. |
BETA
