8

Chapter 8 Approximation In mathematical expression, which includes division and multiplication of decimal values of large number, it becomes quite complicated to solve these expressions. So, to reduce this complexity we use approximation method. In approximation method, we need not calculate the exact value of an expression, but we calculate the nearest (round off) values. When we use approximation method, then final result obtained is not equal to the exact result, but it is very close to the final result (either a little less or little more). Basic Rules to Solve the Problems by Approximation RuleO To solve the complex mathematical expression, take the nearest value of numbers given in the expression. For example, 199.03 is approximated to 200; 94.6% is approximated to 95% etc. Ex. 1 89% of (599.88 + 30 x 400) + 50 = ? Rule®

To multiply large number, we can take the approximate value (round off) of numbers by increasing one number and decreasing the other accordingly, so that the calculation is eased For example, 589 x 231 is approximated 590 x 230. Numbers are increased or decreased to the nearest value, so that calculation becomes easier like two numbers ending with zero will be much easier to multiply than number with digits other than ending with zero. Rule® When we divide large number with decimals, then we can increase or decrease both numbers accordingly. For example, 7987.26 ■*■ 3869 is approximated as 8000 ■*■ 40. Ex. 3 (9615.36 +1247.18) - (2435.72 +1937.92) = ? Sol. ? = (9615.36+ 1247.18) ■*■ (2435.72+ 1937.92) = 10862.54 ■*■ 4373.64 = 10860 ■*■ 4370 = 2.48 = 2.5 RuleQ To find the percentage of any number, we can use the following shortcut methods • To calculate 10% of any number, we simply put a decimal after a digit from the right end.

• To calculate 1% of any number, we simply put a decimal after two digits from right end. • To calculate 25% of any number, we simply divide the number by 4. Fast Track Practice Directions (Q. Nos. 1-74) What approximate value should come in place of the question mark(?) in the following questions? (Note that you are not expected to calculate the exact value). 1. (12.999)3 = ? [IBPS Clerk 2011] (a) 1800 (b) 1650 (c) 2000 (d) 2500 (e) 2200 2. 50550-5-50-5-5 = ? [IBPS Clerk 2011] (a) 350 (b) 150 (c) 300 (d) 250 (e) 202 3. 49.0003 + 74.999 = ? [IBPS Clerk 2011] (a) 0.05 (b) 0.2 (c) 1 (d) 0.7 re; 2 4. 23.003 x 22.998 + 100.010 = ? [IBPS Clerk 2011] fa; 630 (b) 550 (c; 700 (d) 720 (e) 510

5. 125.009 + 69.999 + 104.989 = ? [IBPS Clerk 2011] (a) 420 (b) 300 fc; 285 (tfj 415 (e) 325 6. 16.003x27.998-209.010 = ? [IBPS Clerk 2011] (a) 150 (b; 200 (c) 75 (tfj 240 fe; 110 7. 840.003-5-23.999 = ? [IBPS Clerk 2011] fa; 47 (b) 8 fc; 35 (d) 18 fe; 23 8. 6885.009 - 419.999 - 94.989 = ? [IBPS Clerk 2011] (a) 6650 (b) 6830 (c) 6370 (c0 6200 fe; 6450 9. (9.95)2x(2.01)3 = 2x(?)2 [Dena Bank Clerk 2005] (a) 12 (b) 15 (fc; 25 (ft) 20 fe;i6 10.503x201 = ? (a) 101100 (b) 1000000

(c; 110000 (c0 100003 fa; 360 (b) 350 (c) 370 (c0 385 fe; None of the above (a) 440 (b) 435 (cj 410 (d) 425 fe; 400 14.36.98276421 x 21.00002 = ? [RBI 2003] (a) 775 fty 785 (c) 800 (c0 805 fe; 750 15.14.995 x 8.001 x 20.991 = ? [Corporation Bank 2004] fa; 1950 (b) 2520 (c) 2200 (d) 1520 fe; 2760 16.26.823 x 27.923 x 4.5001 = ? [RBI 2008] (a) 3500 (b) 3450 (c) 3100 (of) 3150 fe; 3400 17.3739 + 164x27 = ? [Canara Bank PO 2003] (a) 105400 (b) 4000 (c) 8200 (d; 690

(e) 6300 18.21 + 3.7 x 2.9 = ? [Indian Bank PO 2003] fa; 74 (b) 70 (c) 27 (d) 32 fe; 44 19. (21 + 99) x (30 - 19.02) = ? [Indian Bank PO 2003] (a) 3581 (b) 131 (c) 1290 (d) 1600 fe; None of the above 20.198.001 x 25 + 112.05 x 24.998 = ? [Indian Bank PO 2003] fa; 7570 (b) 7550 (c) 7500 (d) 7750 fe; 7250 21.17.995 x 16.005 + 15.999 x 15.001 = ? [Vijaya Bank PO 2004] fa; 513 (b) 528 (c) 440 (d) 818 (e) 314 22.127.007 x 7.998 + 6.05 x 4.001 = ? [SBI PO 2005] (a) 1090 (b) 1200 (c) 1120 (c0 1040 fe; 1160 23.198.995 x 12.005 + 16.25 x 6.95 = ? [OBC PO 2005]

fa; 2580 (b) 2550 (c) 2400 (d) 1450 (e) 2500 24. 421 x 0.9 + 130 x 101 + 10000 = ? [IBPS Clerk 2012] (a) 33500 (b) 23500 (c) 225000 (d) 24500 (e) None of the above 25.30.9 x 3000 - 10.1 x 1100 + 8298 - 4302 = ? [IBPS Clerk 2012] (a) 80000 (b) 90000 (C) 105000 (d) 85000 (e) None of the above 26.56.001 x ? - 1000.999 = 231 [Vijaya Bank PO 2003] (a) 22 (b) 45 (c) 37 fd; 16 (ej32 27.1010 -5- 36 + 187 x 20.05 = ? [SBI PO 2005] (a) 3650 (b) 3770 (c) 3825 fdj 3800 (e) 3700 28.724 * 25 x 31.05 + 101 = ? [United Bank of India PO 2005] (a) 900 (b) 1000 (c) 950 fd; 1050 fe) 1010 29.2375.85 +18.01 - 4.525 x 8.05 = ? [Corporation Bank PO 2004] (a) 103 (b) 96 (c) 88 (d) 90 (ej 112 30.2508 + 15.02 + ? x 11 = 200 [Corporation Bank PO 2004]

(a) 13 (b) 8 fc; 3 fd; 4 fej6 31.6.39 x 128.948 + 5.215 +12.189 + 25.056 = ? [IBPS PO 2011] (a) 800 (b) 900 fc; 850 (d) 950 (e) None of the above 32.1559.999+24.001 + 11.005 x 6.999 = ? [RBI 2003] (a) 137 (b) 132 (c) 152 fd; 149 (e) 142 33.3.2x8.1 + 3185+4.95 = ? [Corporation Bank PO 2004] (a) 670 (b) 660 (c) 645 fdj 690 (e) 685 34.1205 + 2.5 = ? [SBI PO 2005] fa; 3000 (b) 4800 fc; 300 (d) 480 fe; None of the above 35.22020+0.011 = ? (a) 20020 (b) 2000000 fc; 200200 fd; 200002 36.4182.395+20.886 = ? [IBPS Clerk 2012] fa; 300 (b) 200 fc; 150 (d) 250 fej None of the above 37.99.999 + 0.99 + 0.00991234 = ? [RBI 2003] fa; 100 (b) loooofc; 1000 fd; 100000

fe; 99999 38.98743 +198 = 800 - ? [IBPS Clerk 2012] (a) 200 (b) 250 fc; 300 fd; 350 (e) None of the above 39.0.0004 + 0.0001 x 36.000009 = ? [Canara Bank PO 2003] fa; 0.10 (b) 1.45 fc; 145 fd; 14.5 (e) 1450 40.4.999 +1.999 x 6.001 = ? [Corporation Bank 2004] (a) 24 (b) 20 fc; 15 fd) 10 r^ is 41.1679 + 14.95x5.02 = ? [Syndicate Bank PO 2004] fa; 540 (b) 525 fc; 545 (d) 565 (e) 520 42.447.75+28x4.99 = ? [Syndicate Bank PO 2004] (a) 44 (b) 70 fc; 32 fd; 80 fe;75 (a) 3.2 (b) 3.5 fc; 2.5 (d) 3 ^ 2 44.425 +16.95 x ? = 225 [OBC PO 2004] (a) 11 (b) 0.8 (c) 9 fd) 19 (e) 0.9 45.26.003 - 154.001 + 6.995 = ?

[Corporation Bank 2004] (a) 4 (b) 18 fc; 9 fd; 10 re; 14 46.9876 + 24.96 + 215.005 - ? = 309.99 [SBI Associates PO 2002] (a) 395 (b) 295 (c) 300 (d) 315 fe; 51 47.953/7+950 " 89 = 95? [Canara Bank PO 2003] (a) 1.9 (b) 3 fc; 2.99 (d) 3.6 fe; 2.7 48.183 ' 5 +2T3 ' 5 x 6s 5 = 2? [RBI 2003] (a) 7 (b) 3.5 (c) 6 (d) 4.5 fe; None of the above 49.37 - 5 +27L5x92 = 3? [Bank of Maharashtra PO 2003] (a) 5 (b) 4.5 (c) 7 (d) 6.5 fe; None of the above 50. 750.0003 + 19.999 = ? [IBPS Clerk 2011] (a) 49 (b) 18 (c) 22 fd; 45 (e) 38

Answer with Solutions

1. (e) (12.999)3 = ? Using rule 1, here 12.999 is approximated to 13 which is the nearest whole number So, ?=(13)3 => 7 = 2197 7=2200 3. (d) 49.0003 -s- 74.999 = 7 By rule 3 for division 49.0003 is decreased to 49 and 74.999 is increased to 75 So, 49-5-75 = ? 0.653 = ? => 7 = 0.7 (CO 23.003 x 22.998 + 100.010 = ? => 7=23x23+100 => = 529 + 100 = 629 ? = 630 5. (6) 125.009 + 69.999 + 104.989 = ? Using rule 1, each value is approximated to nearest whole number => ? =125 + 70 + 105 => ? = 300 6. (d) ? = 16.003 X 27.998 - 209.010 ? = 16 X 28 - 210 = 448 - 210 ? = 238 = 240 7. (C) 7 = 840.003+23.999 Using rule 3, here one number is increased and other is decreased to their nearest whole number = 840 +24 = 35 8. (C) ? = 6885.009 - 419.999 - 94.989 = 6885-420 -95=6370 9. (d) (9.95)Z X (2.01)3 = 2 X (?)2

Here, 9.95 is approximated to 10. 2.01 is approximated to 2 because calculating the squares or cubes of whole number is much o easier than to calculate the square or cube of decimal numbers => 2x(?)2 =(10)z x(2)3 = 100x8 =800 => (?)z = 400 7 = 20 10. (a) 503x201 = 101103 = 101100 14. (a) 36.98276421 X 21.00002 Using rule 2, nearest value in whole number for 36.98276421 is 37 and 21.0002 is 21 = 37X21 = 777 = 775 15. (6) ? = 14.995 X 8.001 X 20.991 = 15X8X21 = 2520 16. (e) 26.823 X 27.923 X 4.5001 = 27 X 28 X 4.5000 = 756 X 4.5000 = 3402 = 3400 17. (c) 3739 + 164 X 27 = ? ? = 3739 + 4428 = 8167 = 8200 18. (d) 21 + 3.7 X 2.9 = ? => 7 = 21 + 4 X3=21 + 12 = 33 = 32 19. (e) (21 + 99) x (30 - 19.02)

(21 + 99) X (30 - 19) = 120X11 = 1320 20. (d) ? = 198.001 X 25 + 112.05 X 24.998 = 198X25 + 112X25 = 25(198 + 112) = 25X310 = 7750 21. (6) ? = 17.995 x 16.005 + 15.999 X 15.001 = 18X16+16X15=16X(18 + 15) = 16x33 = 528 22. (d) ? = 127.007 X 7.998 + 6.05 X 4.001 = 127X8 + 6X4 = 1016 + 24 = 1040 23. (e) ? = 198.995 x 12.005 + 16.25 x 6.95 = 199x12+16x7=2388 + 112 = 2500 24. (6)? = 421 x 0.9 + 130 x 101 + 10000 = 378.9 + 13130 + 10000 = 23508.9 = 23500 25. (d) ? = 30.9 x 3000 - 10.1 x 1100 + 8298-4302 = 92700 - 11110 + 8298 - 4302 = 85586 = 85000 26. (a) 56.001 x 7 1000.999 = 231 => 56x7-1001 = 231 27. (6) ? = 1010 -f- 36 + 187 x 20.05 = 28.0555 + 3749.35 = 3777.40 = 3770 28. (6) 724 +25X31.05+ 101 = ? => 7 = 725+25x31 + 101 = 29 X 31 + 101 = 899 + 101 = 1000 29. (6) ? = 2375.85 +18.01 - 4.525 X 8.05

= 2376 +18 -4.5x8 = 132-36 =96 30. (C) 2508 +15.02+ ?Xll = 200 31. (C) ? = 6.39 X 128.948 + 5.215 = 823.97772 + 0.427844778 + 25.056 = 849.4615648 = 850 32. (e) 1559.999 +24.001 + 11.005 x 6.999 = ? => ? = 1560 +24 + 11 X7=65+ 77 = 142 33. (a) ? = 3.2 X 8.1 + 3185+4.95 = 25.92 + 643.43 = 669.35 = 670 34. (d) 1205 +2.5 = 482 = 480 35. (6)22020 +0.011 = 2001818.181 = 2000000 37. (6)99.999 +0.99 +0.00991234 = 100+1 +0.01 = 10000 38. (C) 98743 + 198 = 800 - ? 39. (C) 0.0004 +0.0001 X 36.000009 = ? 7 = 4+1x36 = 144 = 145

40. (C) 4.999 +1.999 X 6.001 = ? 41. (d) ? = 1679 +14.95 X 5.02 = 1680 +15X5 = 560 =565 42. (d) ? = 447.75+28 X 4.99