Q1): Simplify the ratio 24 : 36.
A) 2 : 3
B) 3 : 2
C) 4 : 5
D) 6 : 7
Answer: A) 2 : 3
Explanation:
- Step 1: Find GCD of 24 and 36.
- Step 2: GCD(24, 36) = 12.
- Step 3: 24 ÷ 12 = 2.
- Step 4: 36 ÷ 12 = 3.
- Step 5: Simplified ratio = 2 : 3.
Q2): If 5 : x = 15 : 24, find x.
A) 6
B) 8
C) 10
D) 12
Answer: B) 8
Explanation:
- Step 1: Convert ratio form to fraction form: 5/x = 15/24.
- Step 2: Cross multiply: 5 × 24 = 15 × x.
- Step 3: 120 = 15x.
- Step 4: x = 120 ÷ 15.
- Step 5: x = 8.
Q3): Divide ₹840 in the ratio 2 : 5 : 7. What is the middle share?
A) ₹280
B) ₹300
C) ₹320
D) ₹360
Answer: B) ₹300
Explanation:
- Step 1: Total parts = 2 + 5 + 7 = 14.
- Step 2: Value of 1 part = 840 ÷ 14 = 60.
- Step 3: Middle share = 5 parts.
- Step 4: 5 × 60 = 300.
- Step 5: Middle share = ₹300.
Q4): If A : B = 3 : 4 and B : C = 6 : 5, then A : C = ?
A) 9 : 10
B) 10 : 9
C) 18 : 20
D) 3 : 5
Answer: A) 9 : 10
Explanation:
- Step 1: Make B common in both ratios.
- Step 2: A : B = 3 : 4 and B : C = 6 : 5.
- Step 3: LCM of 4 and 6 = 12.
- Step 4: Convert A : B to B = 12 → multiply by 3 → 9 : 12.
- Step 5: Convert B : C to B = 12 → multiply by 2 → 12 : 10.
- Step 6: So A : C = 9 : 10.
Q5): The ratio 7 : 9 is equivalent to 28 : x. Find x.
A) 32
B) 34
C) 36
D) 40
Answer: C) 36
Explanation:
- Step 1: 7 becomes 28 means ×4.
- Step 2: Multiply 9 also by 4.
- Step 3: x = 9 × 4 = 36.
Q6): If 60 km is covered in 5 hours, how many hours will it take to cover 96 km at the same speed?
A) 7 hours
B) 8 hours
C) 9 hours
D) 10 hours
Answer: B) 8 hours
Explanation:
- Step 1: Same speed means distance ∝ time (direct proportion).
- Step 2: 60 km : 5 hr = 96 km : x hr.
- Step 3: Write proportion: 60/5 = 96/x.
- Step 4: Cross multiply: 60x = 96 × 5 = 480.
- Step 5: x = 480 ÷ 60 = 8 hours.
Q7): A and B share ₹540 in the ratio 5 : 4. How much does B get?
A) ₹200
B) ₹220
C) ₹240
D) ₹260
Answer: C) ₹240
Explanation:
- Step 1: Total parts = 5 + 4 = 9.
- Step 2: B’s share = (4/9) of 540.
- Step 3: 540 ÷ 9 = 60.
- Step 4: 60 × 4 = 240.
- Step 5: B gets ₹240.
Q8): Sugar : Flour = 2 : 3. If total mixture is 25 kg, how much sugar is there?
A) 8 kg
B) 9 kg
C) 10 kg
D) 15 kg
Answer: C) 10 kg
Explanation:
- Step 1: Total parts = 2 + 3 = 5.
- Step 2: 1 part = 25 ÷ 5 = 5 kg.
- Step 3: Sugar = 2 parts.
- Step 4: 2 × 5 = 10 kg.
Q9): In a class, Boys : Girls = 7 : 5 and total students are 48. How many girls are there?
A) 18
B) 20
C) 22
D) 28
Answer: B) 20
Explanation:
- Step 1: Total parts = 7 + 5 = 12.
- Step 2: 1 part = 48 ÷ 12 = 4.
- Step 3: Girls = 5 parts.
- Step 4: 5 × 4 = 20.
Q10): If a : b = 4 : 9, then (a + b) : b = ?
A) 13 : 9
B) 9 : 13
C) 4 : 13
D) 13 : 4
Answer: A) 13 : 9
Explanation:
- Step 1: Let a = 4k and b = 9k.
- Step 2: a + b = 4k + 9k = 13k.
- Step 3: (a + b) : b = 13k : 9k.
- Step 4: Cancel k → 13 : 9.
Q11): The ratio of ages of Ram and Shyam is 5 : 6. If Ram is 25 years old, Shyam is?
A) 28
B) 30
C) 32
D) 35
Answer: B) 30
Explanation:
- Step 1: 5 parts = 25 years.
- Step 2: 1 part = 25 ÷ 5 = 5 years.
- Step 3: Shyam = 6 parts = 6 × 5 = 30 years.
Q12): Two numbers are in the ratio 3 : 5 and their difference is 28. Find the larger number.
A) 56
B) 70
C) 84
D) 98
Answer: B) 70
Explanation:
- Step 1: Let numbers be 3k and 5k.
- Step 2: Difference = 5k − 3k = 2k.
- Step 3: Given difference = 28 → 2k = 28.
- Step 4: k = 28 ÷ 2 = 14.
- Step 5: Larger number = 5k = 5 × 14 = 70.
Q13): If x : y = 2 : 7 and y : z = 3 : 4, then x : z = ?
A) 6 : 28
B) 2 : 4
C) 3 : 14
D) 14 : 3
Answer: C) 3 : 14
Explanation:
- Step 1: Make y common in both ratios.
- Step 2: x : y = 2 : 7 and y : z = 3 : 4.
- Step 3: LCM of 7 and 3 = 21.
- Step 4: Convert x : y to y = 21 → multiply by 3 → 6 : 21.
- Step 5: Convert y : z to y = 21 → multiply by 7 → 21 : 28.
- Step 6: So x : z = 6 : 28.
- Step 7: Simplify 6 : 28 by 2 → 3 : 14.
Q14): On a map, 1 cm represents 5 km. If the distance on the map is 8 cm, what is the actual distance?
A) 35 km
B) 40 km
C) 45 km
D) 50 km
Answer: B) 40 km
Explanation:
- Step 1: 1 cm = 5 km.
- Step 2: 8 cm = 8 × 5 km.
- Step 3: 8 × 5 = 40 km.
Q15): A can do a work in 12 days and B can do it in 18 days. Their working rates are in the ratio (A : B) = ?
A) 2 : 3
B) 3 : 2
C) 12 : 18
D) 18 : 12
Answer: B) 3 : 2
Explanation:
- Step 1: Rate is inversely proportional to time.
- Step 2: A’s rate = 1/12 and B’s rate = 1/18.
- Step 3: Rate ratio A : B = (1/12) : (1/18).
- Step 4: Convert by cross method: = 18 : 12.
- Step 5: Simplify 18 : 12 by 6 → 3 : 2.
Q16): If 4 pens cost ₹30, what is the cost of 10 pens at the same rate?
A) ₹70
B) ₹75
C) ₹80
D) ₹85
Answer: B) ₹75
Explanation:
- Step 1: Cost ∝ number of pens (direct proportion).
- Step 2: 4 pens : ₹30 = 10 pens : x.
- Step 3: Cross multiply: 4x = 30 × 10 = 300.
- Step 4: x = 300 ÷ 4 = 75.
- Step 5: Cost of 10 pens = ₹75.
Q17): If 3 kg of apples cost ₹210, how many kg of apples can be bought for ₹350?
A) 4 kg
B) 5 kg
C) 6 kg
D) 7 kg
Answer: B) 5 kg
Explanation:
- Step 1: Quantity ∝ money (direct proportion).
- Step 2: 3 kg : ₹210 = x kg : ₹350.
- Step 3: x = (3 × 350) ÷ 210.
- Step 4: 3 × 350 = 1050.
- Step 5: 1050 ÷ 210 = 5.
- Step 6: Apples = 5 kg.
Q18): Two numbers are in the ratio 8 : 11. If the first number is increased by 6 and the second is decreased by 6, the numbers become equal. Find the sum of the original numbers.
A) 68
B) 72
C) 76
D) 80
Answer: C) 76
Explanation:
- Step 1: Let numbers be 8k and 11k.
- Step 2: Condition: 8k + 6 = 11k − 6.
- Step 3: Move terms: 6 + 6 = 11k − 8k.
- Step 4: 12 = 3k.
- Step 5: k = 12 ÷ 3 = 4.
- Step 6: Numbers are 8×4 = 32 and 11×4 = 44.
- Step 7: Sum = 32 + 44 = 76.
Q19): If A : B = 1 : 2 and B : C = 4 : 5, then A : B : C = ?
A) 1 : 2 : 5
B) 2 : 4 : 5
C) 4 : 2 : 5
D) 2 : 5 : 4
Answer: B) 2 : 4 : 5
Explanation:
- Step 1: A : B = 1 : 2.
- Step 2: B : C = 4 : 5.
- Step 3: Make B same in both ratios.
- Step 4: B is 2 in first ratio and 4 in second ratio.
- Step 5: Multiply first ratio by 2 → A : B = 2 : 4.
- Step 6: Now combine with B : C = 4 : 5.
- Step 7: So A : B : C = 2 : 4 : 5.
Q20): If 45% of x equals 30% of y, then x : y = ?
A) 2 : 3
B) 3 : 2
C) 3 : 5
D) 5 : 3
Answer: A) 2 : 3
Explanation:
- Step 1: 45% of x = 30% of y.
- Step 2: Write as equation: 45x/100 = 30y/100.
- Step 3: Cancel 100: 45x = 30y.
- Step 4: Divide both sides by 15: 3x = 2y.
- Step 5: Bring to ratio form: x/y = 2/3.
- Step 6: So x : y = 2 : 3.