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Simplification & Approximation

BODMAS, fractions, decimals, surds and quick approximation.

46 solved questions · 16 easy · 21 medium · 9 hard

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Shortcuts & formulas

  • BODMAS order: Brackets (innermost first), Of, Division, Multiplication, Addition, Subtraction. Division and multiplication have equal rank: go left to right.
  • a^2 - b^2 = (a+b)(a-b); (a+b)^2 = a^2 + 2ab + b^2; a^3 + b^3 = (a+b)(a^2 - ab + b^2).
  • Rationalise: 1/(sqrt(a) + sqrt(b)) = (sqrt(a) - sqrt(b))/(a - b).
  • Telescoping: 1/(n(n+1)) = 1/n - 1/(n+1), so 1/(1x2) + 1/(2x3) + ... + 1/(nx(n+1)) = n/(n+1).
  • Recurring decimal 0.abab... = ab/99; 0.abcabc... = abc/999.
  • Approximation: round each number to a convenient nearby value, compute, then pick the closest option.
  • Decimals: count total decimal places when multiplying; multiply numerator and denominator by 10^k to remove decimals when dividing.

Solved examples

Try each one first, then open the answer and the step-by-step solution.

Easy questions

Q1.

Simplify: 15 + 3 x 4 - 8 ÷ 2

  1. A23
  2. B32
  3. C28
  4. D19
Show answer & solution

Answer: A. 23

Solution

3*4 = 12 8/2 = 4 15+12-4 = 23 Answer: 23

Q2.

Simplify: 48 ÷ 6 x 2 + 5

  1. A9
  2. B56
  3. C21
  4. D11
Show answer & solution

Answer: C. 21

Solution

48/6 = 8 8*2 = 16 16+5 = 21 Answer: 21

Shortcut: Division and multiplication have the same rank, so go left to right.

Q3.

Simplify: 3/4 + 5/6

  1. A0.8
  2. B19/12
  3. C0.625
  4. D2/3
Show answer & solution

Answer: B. 19/12

Solution

9/12 = 0.75 10/12 = 5/6 9/12+10/12 = 19/12 Answer: 19/12

Q4.

Find the value of 0.6 x 0.07

  1. A0.42
  2. B0.0042
  3. C4.2
  4. D0.042
Show answer & solution

Answer: D. 0.042

Solution

6*7/1000 = 0.042 Answer: 0.042

Q5.

Simplify: 5.6 + 3.45 - 2.05

  1. A6
  2. B7.1
  3. C9
  4. D7
Show answer & solution

Answer: D. 7

Solution

5.6+3.45 = 9.05 9.05-2.05 = 7 Answer: 7

Q6.

What is the value of the square root of 784?

  1. A26
  2. B28
  3. C27
  4. D32
Show answer & solution

Answer: B. 28

Solution

28 x 28 = 784 (since 28^2 = (30 - 2)^2 = 900 - 120 + 4 = 784). Answer: 28

Q7.

Find the value of 2/3 of 4/5 of 150

  1. A80
  2. B100
  3. C120
  4. D100.8
Show answer & solution

Answer: A. 80

Solution

4/5*150 = 120 2/3*120 = 80 Answer: 80

Q8.

Simplify: 25^2 - 15^2

  1. A100
  2. B610
  3. C400
  4. D375
Show answer & solution

Answer: C. 400

Solution

(25+15)*(25-15) = 400 Answer: 400

Shortcut: a^2 - b^2 = (a+b)(a-b) = 40 x 10.

Q9.

Simplify: 1/2 + 1/3 + 1/6

  1. A5/6
  2. B3/11
  3. C1
  4. D0.5
Show answer & solution

Answer: C. 1

Solution

3/6+2/6+1/6 = 1 Answer: 1

Q10.

Simplify: 144 ÷ 12 + 12 x 12 - 100

  1. A56
  2. B156
  3. C-28
  4. D146
Show answer & solution

Answer: A. 56

Solution

144/12 = 12 12*12 = 144 12+144-100 = 56 Answer: 56

Q11.

Find the value of 7.5 ÷ 0.25

  1. A3
  2. B0.3
  3. C300
  4. D30
Show answer & solution

Answer: D. 30

Solution

75/2.5 = 30 750/25 = 30 Answer: 30

Shortcut: Dividing by 0.25 is the same as multiplying by 4: 7.5 x 4 = 30.

Q12.

What is the cube root of 2744?

  1. A12
  2. B14
  3. C16
  4. D18
Show answer & solution

Answer: B. 14

Solution

Unit digit 4 means the cube root ends in 4. 14^3 = 14 x 196 = 2744. Answer: 14

Shortcut: Cube ending in 4 means the root ends in 4; 2744 lies between 13^3 and 15^3, so 14.

Q13.

What is 35% of 240?

  1. A84
  2. B72
  3. C42
  4. D74
Show answer & solution

Answer: A. 84

Solution

240*35/100 = 84 Answer: 84

Q14.

Simplify: 4 3/5 - 2 1/2

  1. A2 1/5
  2. B2.5
  3. C2.1
  4. D1.9
Show answer & solution

Answer: C. 2.1

Solution

4 3/5 = 23/5 and 2 1/2 = 5/2. 23/5 - 5/2 = 46/10 - 25/10 = 21/10 = 2.1. Answer: 2.1

Q15.

Find the value of (12 + 8) x (12 - 8) ÷ 4.

  1. A16
  2. B20
  3. C5
  4. D80
Show answer & solution

Answer: B. 20

Solution

12 + 8 = 20 and 12 - 8 = 4. 20 x 4 = 80, then 80 ÷ 4 = 20. Answer: 20

Q16.

Find the value of (9/4) ÷ (3/8).

  1. A27/32
  2. B3/2
  3. C3
  4. D6
Show answer & solution

Answer: D. 6

Solution

Dividing by a fraction means multiplying by its reciprocal. (9/4) x (8/3) = 72/12 = 6. Answer: 6

Medium questions

Q17.

Simplify: 18 + 12 ÷ 3 x (5 - 2)^2

  1. A90
  2. B30
  3. C166/9
  4. D54
Show answer & solution

Answer: D. 54

Solution

5-2 = 3 12/3 = 4 4*3*3 = 36 18+36 = 54 Answer: 54

Q18.

Simplify: (0.8^3 + 0.5^3) / (0.8^2 - 0.8 x 0.5 + 0.5^2)

  1. A0.3
  2. B1.3
  3. C0.39
  4. D0.637
Show answer & solution

Answer: B. 1.3

Solution

0.512+0.125 = 0.637 0.64-0.4+0.25 = 0.49 0.637/0.49 = 1.3 Answer: 1.3

Shortcut: (a^3 + b^3)/(a^2 - ab + b^2) = a + b = 0.8 + 0.5 = 1.3.

Q19.

Simplify: (12.5^2 - 7.5^2) ÷ 5

  1. A20
  2. B5
  3. C100
  4. D4
Show answer & solution

Answer: A. 20

Solution

12.5*12.5 = 156.25 7.5*7.5 = 56.25 156.25-56.25 = 100 100/5 = 20 Answer: 20

Shortcut: (12.5 + 7.5)(12.5 - 7.5)/5 = 20 x 5 / 5 = 20.

Q20.

Find the value of the square root of 0.0144.

  1. A1.2
  2. B0.012
  3. C0.12
  4. D0.0012
Show answer & solution

Answer: C. 0.12

Solution

0.0144 = 144/10000. sqrt = 12/100 = 0.12. Answer: 0.12

Shortcut: Decimal places in the root = half the decimal places: 4 places -> 2 places; sqrt(144) = 12.

Q21.

Simplify: (1 - 1/2)(1 - 1/3)(1 - 1/4)(1 - 1/5)(1 - 1/6)

  1. A1/12
  2. B5/6
  3. C1/6
  4. D0.2
Show answer & solution

Answer: C. 1/6

Solution

1/2*2/3 = 1/3 1/2*2/3*3/4*4/5*5/6 = 1/6 Answer: 1/6

Shortcut: The product telescopes to 1/n, here 1/6.

Q22.

Simplify: 1/(1x2) + 1/(2x3) + 1/(3x4) + ... + 1/(9x10)

  1. A0.9
  2. B0.1
  3. C10/9
  4. D8/9
Show answer & solution

Answer: A. 0.9

Solution

(1-1/2)+(1/2-1/3) = 2/3 1-1/10 = 0.9 Answer: 0.9

Shortcut: Sum = n/(n+1) = 9/10.

Q23.

Express 0.272727... (27 recurring) as a fraction in lowest terms.

  1. A27/100
  2. B27/999
  3. C6/11
  4. D3/11
Show answer & solution

Answer: D. 3/11

Solution

0.2727... = 27/99. HCF(27, 99) = 9, so 27/99 = 3/11. Answer: 3/11

Q24.

What is the value of 0.333... + 0.666... (both recurring)?

  1. A0.999
  2. B1
  3. C0.99
  4. D9/10
Show answer & solution

Answer: B. 1

Solution

0.333... = 1/3 and 0.666... = 2/3. 1/3 + 2/3 = 1. Answer: 1

Q25.

If ? ÷ 15 x 9 = 72, what is the missing number?

  1. A120
  2. B108
  3. C135
  4. D90
Show answer & solution

Answer: A. 120

Solution

? ÷ 15 x 9 = 72 ? x 9 / 15 = 72 ? = 72 x 15 / 9 = 120. Answer: 120

Q26.

Simplify: (sqrt5 + sqrt3)(sqrt5 - sqrt3)

  1. A8
  2. Bsqrt5 - sqrt3
  3. C2
  4. D2sqrt3
Show answer & solution

Answer: C. 2

Solution

Use (a + b)(a - b) = a^2 - b^2. (sqrt5)^2 - (sqrt3)^2 = 5 - 3 = 2. Answer: 2

Q27.

Rationalise the denominator of 1 / (sqrt5 - 2).

  1. Asqrt5 + 4
  2. Bsqrt5 + 2
  3. C(sqrt5 + 2)/9
  4. D3 - sqrt5
Show answer & solution

Answer: B. sqrt5 + 2

Solution

Multiply numerator and denominator by (sqrt5 + 2). Denominator = (sqrt5)^2 - 2^2 = 5 - 4 = 1. Result = (sqrt5 + 2)/1. Answer: sqrt5 + 2

Q28.

Simplify: (sqrt3 + sqrt2)^2

  1. A5
  2. B5 + sqrt6
  3. C1 + 2sqrt6
  4. D5 + 2sqrt6
Show answer & solution

Answer: D. 5 + 2sqrt6

Solution

(a + b)^2 = a^2 + 2ab + b^2. (sqrt3)^2 + 2 x sqrt3 x sqrt2 + (sqrt2)^2 = 3 + 2sqrt6 + 2 = 5 + 2sqrt6. Answer: 5 + 2sqrt6

Q29.

Simplify: 6 / sqrt3

  1. A3sqrt2
  2. Bsqrt3
  3. C6sqrt3
  4. D2sqrt3
Show answer & solution

Answer: D. 2sqrt3

Solution

Multiply top and bottom by sqrt3: 6 x sqrt3 / 3 = 2sqrt3. Answer: 2sqrt3

Q30.

What approximate value should come in place of the question mark? 4.98 x 19.97 + 29.9% of 801 = ?

  1. A300
  2. B340
  3. C360
  4. D320
Show answer & solution

Answer: B. 340

Solution

Round: 5 x 20 + 30% of 800. 5 x 20 = 100 30% of 800 = 240 100 + 240 = 340 The exact value is about 338.9, closest to 340. Answer: 340

Q31.

What approximate value should come in place of the question mark? 36.01 x 14.98 - 211.9 = ?

  1. A330
  2. B320
  3. C340
  4. D350
Show answer & solution

Answer: A. 330

Solution

Round: 36 x 15 - 212. 36 x 15 = 540 540 - 212 = 328, nearest option 330. The exact value is about 327.5, closest to 330. Answer: 330

Q32.

What approximate value should come in place of the question mark? 7999 ÷ 39.9 + 24.98 x 6.02 = ?

  1. A300
  2. B400
  3. C350
  4. D330
Show answer & solution

Answer: C. 350

Solution

Round: 8000 ÷ 40 + 25 x 6. 8000 ÷ 40 = 200 25 x 6 = 150 200 + 150 = 350 The exact value is about 350.9, closest to 350. Answer: 350

Q33.

Simplify: (3/4 ÷ 5/8) x (10/9)

  1. A5/6
  2. B3/2
  3. C4/3
  4. D9/8
Show answer & solution

Answer: C. 4/3

Solution

(3/4) ÷ (5/8) = (3/4) x (8/5) = 6/5. (6/5) x (10/9) = 60/45 = 4/3. Answer: 4/3

Q34.

Simplify: 3/5 of 250 ÷ 5 + 0.25 x 80

  1. A50
  2. B6
  3. C70
  4. D170
Show answer & solution

Answer: A. 50

Solution

3/5*250 = 150 150/5 = 30 0.25*80 = 20 30+20 = 50 Answer: 50

Q35.

Find the value of (2.5^3 - 1.5^3) / (2.5^2 + 2.5 x 1.5 + 1.5^2)

  1. A4
  2. B2.5
  3. C3.5
  4. D1
Show answer & solution

Answer: D. 1

Solution

15.625-3.375 = 12.25 6.25+3.75+2.25 = 12.25 12.25/12.25 = 1 Answer: 1

Shortcut: (a^3 - b^3)/(a^2 + ab + b^2) = a - b = 1.

Q36.

Find the value of (1/2 + 1/3) ÷ (1/2 - 1/3)

  1. A30
  2. B5
  3. C0.2
  4. D5/6
Show answer & solution

Answer: B. 5

Solution

5/6 = 5/6 1/6 = 1/6 5/6/(1/6) = 5 Answer: 5

Q37.

Simplify: 1/(sqrt2 + 1) + 1/(sqrt3 + sqrt2)

  1. Asqrt3 - 1
  2. Bsqrt3 + 2
  3. Csqrt2 - 1
  4. D1
Show answer & solution

Answer: A. sqrt3 - 1

Solution

1/(sqrt2 + 1) = (sqrt2 - 1)/(2 - 1) = sqrt2 - 1. 1/(sqrt3 + sqrt2) = (sqrt3 - sqrt2)/(3 - 2) = sqrt3 - sqrt2. Sum = sqrt2 - 1 + sqrt3 - sqrt2 = sqrt3 - 1. Answer: sqrt3 - 1

Hard questions

Q38.

Simplify: 1 / (1 + 1 / (1 + 1 / (1 + 1/2)))

  1. A3/5
  2. B8/5
  3. C5/8
  4. D2/3
Show answer & solution

Answer: C. 5/8

Solution

Work from the inside out. 1 + 1/2 = 3/2, so 1/(3/2) = 2/3. 1 + 2/3 = 5/3, so 1/(5/3) = 3/5. 1 + 3/5 = 8/5, so 1/(8/5) = 5/8. Answer: 5/8

Q39.

Simplify: sqrt(7 + 4sqrt3)

  1. A3 + sqrt2
  2. B2 + sqrt3
  3. Csqrt7 + sqrt3
  4. D4 + sqrt3
Show answer & solution

Answer: B. 2 + sqrt3

Solution

Write 7 + 4sqrt3 as (a + b)^2 = a^2 + b^2 + 2ab with a^2 + b^2 = 7 and 2ab = 4sqrt3. Try a = 2 and b = sqrt3: 4 + 3 = 7, 2 x 2 x sqrt3 = 4sqrt3. Matches. sqrt(7 + 4sqrt3) = 2 + sqrt3. Answer: 2 + sqrt3

Q40.

Find the value of 1/(1 + sqrt2) + 1/(sqrt2 + sqrt3) + 1/(sqrt3 + sqrt4) + ... + 1/(sqrt8 + sqrt9).

  1. A3
  2. Bsqrt9 + 1
  3. C4
  4. D2
Show answer & solution

Answer: D. 2

Solution

Rationalise each term: 1/(sqrt(k) + sqrt(k+1)) = sqrt(k+1) - sqrt(k). The sum telescopes: (sqrt2 - 1) + (sqrt3 - sqrt2) + ... + (sqrt9 - sqrt8) = sqrt9 - 1 = 3 - 1 = 2. Answer: 2

Q41.

Find the value of sqrt(12 + sqrt(12 + sqrt(12 + ...))) (infinite nesting).

  1. A3
  2. B6
  3. Csqrt12
  4. D4
Show answer & solution

Answer: D. 4

Solution

Let x = sqrt(12 + x). Then x^2 = 12 + x, so x^2 - x - 12 = 0. (x - 4)(x + 3) = 0, so x = 4 (x cannot be negative). Answer: 4

Shortcut: Find two consecutive-type numbers n(n-1)=12 -> 4 x 3, answer is the larger: 4.

Q42.

Simplify: (3sqrt2 - 2sqrt3)^2

  1. A6
  2. B30 - 12sqrt6
  3. C30 - 6sqrt6
  4. D18 - 12sqrt6
Show answer & solution

Answer: B. 30 - 12sqrt6

Solution

(a - b)^2 = a^2 - 2ab + b^2. a^2 = 9 x 2 = 18; b^2 = 4 x 3 = 12; 2ab = 2 x 3sqrt2 x 2sqrt3 = 12sqrt6. 18 + 12 - 12sqrt6 = 30 - 12sqrt6. Answer: 30 - 12sqrt6

Q43.

Find the value of (487^2 + 2 x 487 x 13 + 13^2) ÷ 500^2 x 40

  1. A40
  2. B20000/487
  3. C20
  4. D10
Show answer & solution

Answer: A. 40

Solution

487*487 = 237169 487+13 = 500 500*500/(500*500) = 1 Answer: 40

Shortcut: The numerator is (487 + 13)^2 = 500^2, so the fraction is 1.

Q44.

Find the value of (999^2 - 1) ÷ (998 x 1000) x 7

  1. A6.993
  2. B7.007
  3. C7
  4. D14
Show answer & solution

Answer: C. 7

Solution

998001-1 = 998000 998000/998000 = 1 Answer: 7

Shortcut: 999^2 - 1 = (999 - 1)(999 + 1) = 998 x 1000.

Q45.

Simplify: 5/12 + 7/18 - 3/8

  1. A9/38
  2. B15/72
  3. C31/72
  4. D37/72
Show answer & solution

Answer: C. 31/72

Solution

LCM of 12, 18, 8 = 72. 5/12 = 30/72; 7/18 = 28/72; 3/8 = 27/72. 30/72 + 28/72 - 27/72 = 31/72. Answer: 31/72

Q46.

Simplify: 2^10 - 2^9 - 2^8

  1. A256
  2. B512
  3. C128
  4. D0
Show answer & solution

Answer: A. 256

Solution

Take 2^8 common: 2^8 (2^2 - 2 - 1) = 256 x (4 - 2 - 1) = 256 x 1 = 256. Answer: 256

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