Fractions – Hard Level Questions – UGC NET Paper 1

Q1): Simplify: (3/5 + 7/10) ÷ (1/2 − 1/5)
A) 13/4
B) 10/3
C) 13/3
D) 14/3

Answer: C) 13/3

Explanation:

  • Step 1: 3/5 + 7/10 = 6/10 + 7/10 = 13/10
  • Step 2: 1/2 − 1/5 = 5/10 − 2/10 = 3/10
  • Step 3: (13/10) ÷ (3/10) = (13/10) * (10/3)
  • Step 4: Cancel 10 with 10
  • Final: 13/3, so option C is correct

Q2): Simplify: (5/6) ÷ (1/3 + 1/4)
A) 5/7
B) 7/10
C) 10/7
D) 8/7

Answer: C) 10/7

Explanation:

  • Step 1: 1/3 + 1/4 = 4/12 + 3/12 = 7/12
  • Step 2: (5/6) ÷ (7/12) = (5/6) * (12/7)
  • Step 3: Cancel 12 with 6 → 12/6 = 2
  • Step 4: Value = (5*2)/7 = 10/7
  • Final: 10/7, so option C is correct

Q3): If (x − 1/3) / (x + 1/6) = 3/4, then x = ?
A) 5/6
B) 11/6
C) 13/6
D) 11/5

Answer: B) 11/6

Explanation:

  • Step 1: Cross-multiply: 4(x − 1/3) = 3(x + 1/6)
  • Step 2: 4x − 4/3 = 3x + 1/2
  • Step 3: 4x − 3x = 4/3 + 1/2
  • Step 4: 4/3 + 1/2 = 8/6 + 3/6 = 11/6
  • Final: x = 11/6, so option B is correct

Q4): Solve: (5/12)x + (3/8)x − (7/24)x = 10
A) 18
B) 20
C) 22
D) 24

Answer: B) 20

Explanation:

  • Step 1: Take LCM of 12, 8, 24 = 24
  • Step 2: 5/12 = 10/24, 3/8 = 9/24, 7/24 stays 7/24
  • Step 3: Coefficient = (10 + 9 − 7)/24 = 12/24 = 1/2
  • Step 4: (1/2)x = 10 → x = 20
  • Final: 20, so option B is correct

Q5): A fraction when multiplied by 3/5 becomes 9/20. The fraction is:
A) 9/25
B) 3/4
C) 5/12
D) 9/4

Answer: B) 3/4

Explanation:

  • Step 1: Let fraction be f
  • Step 2: f * (3/5) = 9/20
  • Step 3: f = (9/20) ÷ (3/5) = (9/20) * (5/3)
  • Step 4: Cancel 5 with 20 → 1/4, so f = (91)/(43) = 3/4
  • Final: 3/4, so option B is correct

Q6): If a/b = 4/7 and b/c = 9/5, then a : c = ?
A) 35 : 36
B) 36 : 35
C) 20 : 21
D) 18 : 35

Answer: B) 36 : 35

Explanation:

  • Step 1: a/c = (a/b) * (b/c)
  • Step 2: a/c = (4/7) * (9/5) = 36/35
  • Step 3: If a/c = 36/35, then a : c = 36 : 35
  • Final: 36 : 35, so option B is correct

Q7): Simplify: [1 3/4 − (2/3 of 3/8)] ÷ (5/6)
A) 5/9
B) 6/5
C) 9/5
D) 3/5

Answer: C) 9/5

Explanation:

  • Step 1: 1 3/4 = 7/4
  • Step 2: 2/3 of 3/8 = (2/3) * (3/8) = 1/4
  • Step 3: 7/4 − 1/4 = 6/4 = 3/2
  • Step 4: (3/2) ÷ (5/6) = (3/2) * (6/5) = 9/5
  • Final: 9/5, so option C is correct

Q8): Convert 0.272727… into a fraction in simplest form.
A) 27/100
B) 3/11
C) 27/90
D) 11/3

Answer: B) 3/11

Explanation:

  • Step 1: Let x = 0.272727…
  • Step 2: 100x = 27.272727…
  • Step 3: Subtract: 100x − x = 27.2727… − 0.2727…
  • Step 4: 99x = 27 → x = 27/99 = 3/11
  • Final: 3/11, so option B is correct

Q9): Simplify: (1/2 + 2/3) ÷ (3/4 − 1/6)
A) 3/2
B) 2
C) 5/2
D) 4/3

Answer: B) 2

Explanation:

  • Step 1: 1/2 + 2/3 = 3/6 + 4/6 = 7/6
  • Step 2: 3/4 − 1/6 = 9/12 − 2/12 = 7/12
  • Step 3: (7/6) ÷ (7/12) = (7/6) * (12/7)
  • Step 4: Cancel 7, and 12/6 = 2
  • Final: 2, so option B is correct

Q10): Simplify: (3/7 − 2/5) ÷ (1/2 + 1/3)
A) 6/175
B) 12/175
C) 6/125
D) 3/175

Answer: A) 6/175

Explanation:

  • Step 1: 3/7 − 2/5 = (15 − 14)/35 = 1/35
  • Step 2: 1/2 + 1/3 = (3 + 2)/6 = 5/6
  • Step 3: (1/35) ÷ (5/6) = (1/35) * (6/5)
  • Step 4: = 6/175
  • Final: 6/175, so option A is correct

Q11): If 1/x + 1/6 = 1/4, then x = ?
A) 8
B) 10
C) 12
D) 14

Answer: C) 12

Explanation:

  • Step 1: 1/x = 1/4 − 1/6
  • Step 2: 1/4 − 1/6 = (3 − 2)/12 = 1/12
  • Step 3: So 1/x = 1/12
  • Final: x = 12, so option C is correct

Q12): Simplify: (2/3) ÷ (5/6 − 1/4)
A) 7/8
B) 8/7
C) 6/7
D) 7/6

Answer: B) 8/7

Explanation:

  • Step 1: 5/6 − 1/4 = 10/12 − 3/12 = 7/12
  • Step 2: (2/3) ÷ (7/12) = (2/3) * (12/7)
  • Step 3: Cancel 12 with 3 → 12/3 = 4
  • Step 4: Value = (2*4)/7 = 8/7
  • Final: 8/7, so option B is correct

Q13): If A = (3/4) of B and B = (5/6) of C, then A is what fraction of C?
A) 3/8
B) 5/8
C) 5/6
D) 3/4

Answer: B) 5/8

Explanation:

  • Step 1: A = (3/4)B and B = (5/6)C
  • Step 2: Substitute: A = (3/4) * (5/6) * C
  • Step 3: A/C = (15)/(24)
  • Step 4: 15/24 = 5/8
  • Final: 5/8, so option B is correct

Q14): A tank is first emptied by 2/5 of its water. Then 1/3 of the remaining water is removed. If 24 L is left, capacity is:
A) 48 L
B) 54 L
C) 60 L
D) 72 L

Answer: C) 60 L

Explanation:

  • Step 1: After removing 2/5, remaining = 3/5 of capacity
  • Step 2: Then removed 1/3 of remaining → left = 2/3 of remaining
  • Step 3: Final left = (3/5) * (2/3) = 2/5 of capacity
  • Step 4: 2/5 of capacity = 24 → capacity = 24 * (5/2) = 60
  • Final: 60 L, so option C is correct

Q15): In a class, 3/8 are girls. If 15 girls leave, girls become 1/4 of the class. Original class strength is:
A) 80
B) 90
C) 96
D) 120

Answer: B) 90

Explanation:

  • Step 1: Let total students = N, girls = (3/8)N
  • Step 2: After 15 girls leave: girls = (3/8)N − 15, total = N − 15
  • Step 3: Given: [(3/8)N − 15] / (N − 15) = 1/4
  • Step 4: Cross-multiply: 4[(3/8)N − 15] = (N − 15)
  • Step 5: (3/2)N − 60 = N − 15 → (1/2)N = 45 → N = 90
  • Final: 90, so option B is correct

Q16): Simplify: (1/2 + 1/3 + 1/4) − (1/5 + 1/6)
A) 41/60
B) 43/60
C) 47/60
D) 13/20

Answer: B) 43/60

Explanation:

  • Step 1: 1/2 + 1/3 + 1/4 = 6/12 + 4/12 + 3/12 = 13/12
  • Step 2: 1/5 + 1/6 = (6 + 5)/30 = 11/30
  • Step 3: 13/12 − 11/30 → LCM of 12 and 30 = 60
  • Step 4: 13/12 = 65/60 and 11/30 = 22/60
  • Final: 65/60 − 22/60 = 43/60, so option B is correct

Q17): If (x/3) : (5/6) = 4 : 7, then x = ?
A) 7/10
B) 5/7
C) 10/7
D) 20/7

Answer: C) 10/7

Explanation:

  • Step 1: (x/3) / (5/6) = 4/7
  • Step 2: (x/3) * (6/5) = 4/7
  • Step 3: x = (4/7) * 3 * (5/6)
  • Step 4: 3 cancels with 6 → 1/2, so x = (45)/(72) = 10/7
  • Final: 10/7, so option C is correct

Q18): If 3/(x+1) + 2/(x−1) = 1, then the sum of all solutions is:
A) 0
B) 5
C) −5
D) 25

Answer: B) 5

Explanation:

  • Step 1: Take LCM (x+1)(x−1)
  • Step 2: 3(x−1) + 2(x+1) = (x+1)(x−1)
  • Step 3: 3x − 3 + 2x + 2 = x^2 − 1
  • Step 4: 5x − 1 = x^2 − 1 → x^2 − 5x = 0
  • Step 5: x(x − 5) = 0 → x = 0 or x = 5 (valid since x ≠ 1, −1)
  • Final: Sum = 0 + 5 = 5, so option B is correct

Q19): Solve for x: (2x − 1)/3 + (x + 2)/4 = 5/2
A) 27/11
B) 28/11
C) 29/11
D) 14/11

Answer: B) 28/11

Explanation:

  • Step 1: Take LCM of 3 and 4 = 12
  • Step 2: Multiply both sides by 12
  • Step 3: 4(2x − 1) + 3(x + 2) = 12*(5/2)
  • Step 4: (8x − 4) + (3x + 6) = 30 → 11x + 2 = 30
  • Step 5: 11x = 28 → x = 28/11
  • Final: 28/11, so option B is correct

Q20): Simplify: (1 − 2/3)/(1 − 3/4) + (1 − 3/5)/(1 − 5/6)
A) 56/15
B) 15/56
C) 41/15
D) 11/15

Answer: A) 56/15

Explanation:

  • Step 1: First part: (1 − 2/3) = 1/3 and (1 − 3/4) = 1/4
  • Step 2: (1/3) ÷ (1/4) = (1/3) * 4 = 4/3
  • Step 3: Second part: (1 − 3/5) = 2/5 and (1 − 5/6) = 1/6
  • Step 4: (2/5) ÷ (1/6) = (2/5) * 6 = 12/5
  • Step 5: Add: 4/3 + 12/5 = 20/15 + 36/15 = 56/15
  • Final: 56/15, so option A is correct

This Tutorial is Related to: Compare Fractions, Complex Fractions, Fraction Equations, Fraction Simplification, Fractions, Fractions Questions, Hard Level, Mixed Numbers, Nested Fractions, Pyq Pattern, Ratio With Fractions, Recurring Decimals, Ugc Net Paper 1, Unit 5

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