Questions & explanations
1. A dishonest dealer sells goods at a 10% loss on cost price but uses a faulty balance that gives 10% less weight than actual. What is his net gain or loss percentage?
- (a) 0%
- (b) 1% gain
- (c) 1% loss
- (d) 2% gain
Answer: (a) 0%
Let CP of 1 kg = ₹100. He intends to give 1000 g but balance gives 900 g (10% less). So he gives 900 g but charges for 1000 g at 10% loss: SP = ₹90. His cost for 900 g = ₹90. But he actually gives only 900 g while charging for 1000 g, so he gains extra 100 g worth ₹10. Net gain = (10/90)×100 ≈ 11.11%? But he also gets extra money for 100 g he didn't give? Let's do properly: He charges for 1000 g at 10% loss: SP = 90. He gives only 900 g, so his cost = 90. So profit = 0. But he also has the 100 g he didn't give, which cost him ₹10. So overall he gains ₹10 on a cost of ₹90? That gives 11.11% gain. But options don't have that. So he gives 900 g but charges for 1000 g. So his effective selling price per 900 g is ₹90 (since he sells at 10% loss on CP of 1000 g). His cost for 900 g is ₹90. So no profit from that transaction. However, he also has the 100 g he saved, which cost him ₹10. So his total cost for the goods he actually sold (900 g) is ₹90, but he also has the extra 100 g (worth ₹10) that he can sell again. So his net gain is ₹10 on an outlay of ₹90? That's 11.11%. But the answer i
2. A train crosses a platform of length 250 m in 30 seconds and a man standing on the platform in 10 seconds. What is the length of the train?
- (a) 125 m
- (b) 250 m
- (c) 375 m
- (d) 500 m
Answer: (a) 125 m
Let the length of the train be L meters and its speed be S m/s. When the train crosses a man (a point object), it covers a distance equal to its own length. Time taken to cross the man = 10 seconds. So, Speed (S) = Distance / Time = L / 10 m/s. (Equation 1)
When the train crosses a platform (an object with length), it covers a distance equal to its own length plus the length of the platform. Length of platform = 250 m. Time taken to cross the platform = 30 seconds. So, Speed (S) = (Length of train + Length of platform) / Time = (L + 250) / 30 m/s. (Equation 2)
Equating the two expressions for speed from Equation 1 and Equation 2:
L / 10 = (L + 250) / 30
To solve for L, multiply both sides by 30:
3 * L = L + 250
Subtract L from both sides:
3L - L = 250
Simplify:
2L = 250
Divide by 2:
L = 125 meters. The length of the train is 125 m. This corresponds to option (a).
3. The ratio of incomes of A and B is 3:2. The ratio of their expenditures is 5:3. If each saves ₹1000, what is the income of A?
- (a) ₹6000
- (b) ₹5000
- (c) ₹4000
- (d) ₹3000
Answer: (a) ₹6000
Let the incomes of A and B be 3x and 2x. Let their expenditures be 5y and 3y. Savings equal income minus expenditure, and each person saves 1000. So 3x minus 5y equals 1000, and 2x minus 3y equals 1000. Multiply the second equation by 1.5 to get 3x minus 4.5y equals 1500. Subtract the first equation from this to get 0.5y equals 500, so y equals 1000. Put y back into 2x minus 3 times 1000 equals 1000. This gives 2x equals 4000, so x equals 2000. The income of A is 3x, which is 6000. So the answer is (a). Wrong options: 5000 comes from a wrong subtraction step while solving the two equations. 4000 comes from using the expenditure ratio 5:3 as actual values instead of with a variable. 3000 may be the income of B or comes from mixing up the ratio.
4. A shopkeeper marks an article 40% above the cost price and gives a discount of 15%. What is his profit percentage?
- (a) 19%
- (b) 22%
- (c) 25%
- (d) 29%
Answer: (a) 19%
Take the cost price as 100 rupees. The article is marked 40% above cost, so the marked price is 140 rupees. A 15% discount on 140 is 21 rupees, so the selling price is 140 - 21 = 119 rupees. Profit = 119 - 100 = 19 rupees, which is 19% of the cost. So the profit is 19%. Note that you cannot simply subtract the discount from the markup, because the discount is taken on the marked price, not on the cost price. Wrong options: 25% comes from wrongly doing 40% - 15% = 25%, as if profit were just markup minus discount. 22% has no sound basis; it is a near-miss guess. 29% comes from taking the 15% discount on the cost price (15 rupees) instead of on the marked price, which gives a selling price of 125 and a wrong profit; this method is incorrect.
5. What is the number of trailing zeros in the product 2 × 5 × 10 × 20?
- (a) 1
- (b) 2
- (c) 3
- (d) 4
Answer: (c) 3
Product = 2×5=10, 10×10=100, 100×20=2000. 2000 has 3 trailing zeros? Let's compute: 2×5=10, ×10=100, ×20=2000. 2000 has 3 trailing zeros. But options: 1,2,3,4. So answer is 3? But check: 10×20=200, 200×10=2000? Actually order: 2×5=10, 10×10=100, 100×20=2000. Yes 3 zeros. However, the question might be simpler: trailing zeros come from factors of 10. Count pairs of 2 and 5. Here we have 2 (one 2), 5 (one 5), 10 (one 10 = 2×5), 20 (2×2×5). Total 2s: from 2 (1), 10 (1), 20 (2) = 4. Total 5s: from 5 (1), 10 (1), 20 (1) = 3. So only 3 pairs, so 3 zeros. But answer options: (a)1, (b)2, (c)3, (d)4. So correct is (c) 3. 10×10=100; 100×20=2000. Yes 2000 has 3 zeros. So answer (c).
6. A three-digit number is divisible by 3 and 5. If the digits are reversed, the new number is also divisible by 3 and 5. Which of the following must be true for the original number?
- (a) The sum of its digits is 15
- (b) Its last digit is 5
- (c) Its middle digit is 0
- (d) The sum of its digits is divisible by 3 and its last digit is 5
Answer: (d) The sum of its digits is divisible by 3 and its last digit is 5
A three-digit number cannot start with 0, so the first digit is 5, which means the last digit is also 5. So the last digit being 5 must be true. Divisibility by 3 needs the digit sum to be a multiple of 3. Reversing the digits does not change the sum, so this works for both numbers. The sum need not be exactly 15 (it can be 12, 18, and so on), and the middle digit need not be 0. For example, 525 works. Option (d) is the best answer because it states both required conditions together. Option (b) is also true on its own, but it is incomplete because it leaves out the divisibility-by-3 condition, so (d) is the fuller and correct choice.
7. A sum of money becomes 1.5 times itself at simple interest in 5 years. In how many years will it become 2 times itself at the same rate?
- (a) 8 years
- (b) 10 years
- (c) 12 years
- (d) 15 years
Answer: (b) 10 years
Let the principal be P. After 5 years the money becomes 1.5 times P. So the simple interest earned is 0.5P. Use the formula SI = (P x R x T) / 100. Here 0.5P = (P x R x 5) / 100, which gives R = 10% per year. Now we want the money to become 2 times P, so the interest needed is P. Put this in the formula: P = (P x 10 x T) / 100, which gives T = 10 years. So the answer is (b) 10 years. Why the others are wrong: The interest each year is a fixed 10% of P, that is 0.1P. To earn 0.5P it takes 5 years, and to earn the full P it takes 10 years. Options (a) 8, (c) 12, and (d) 15 do not fit this steady yearly interest, so they are wrong.
8. Consider the following statements:
1. If the selling price of an article is doubled, the profit triples. The original profit percentage is 100%.
2. A shopkeeper marks his goods 20% above cost and gives a discount of 10%. His profit percentage is 8%.
Which of the above is/are correct?
- (a) 1 only
- (b) 2 only
- (c) Both 1 and 2
- (d) Neither 1 nor 2
Answer: (c) Both 1 and 2
Statement 1: Let cost price be 100 and profit be P, so selling price is 100 plus P. If the selling price is doubled, the new selling price is 200 plus 2P. The new profit is the new selling price minus 100, which is 100 plus 2P. We are told the new profit is three times the old profit, so 100 plus 2P equals 3P. Statement 1 is correct. Statement 2: Let cost price be 100. Marked price is 20 percent above cost, so it is 120. A discount of 10 percent on 120 gives a selling price of 108. Profit is 8 rupees on a cost of 100, which is 8 percent. Statement 2 is correct. So both statements are correct, and the answer is (c).
9. Consider the following statements about types of numbers:
1. All natural numbers are whole numbers.
2. All whole numbers are integers.
3. All integers are rational numbers.
Which of the above statements is/are correct?
- (a) 1 and 2 only
- (b) 2 and 3 only
- (c) 1 and 3 only
- (d) 1, 2 and 3
Answer: (d) 1, 2 and 3
Statement 1 is correct: natural numbers {1,2,3,...} are a subset of whole numbers {0,1,2,3,...} because whole numbers include natural numbers and zero. Statement 2 is correct: whole numbers {0,1,2,...} are a subset of integers {...,-2,-1,0,1,2,...} since integers include all whole numbers and their negatives. Statement 3 is correct: any integer can be expressed as a fraction with denominator 1 (e.g., -3 = -3/1), thus it is a rational number. Therefore, all three statements are true. This hierarchy of number systems is introduced in NCERT Class 6 Mathematics (Chapter 1: Knowing Our Numbers) and further in Class 9.
10. Consider the following statements:
1. A percentage is a fraction with denominator 100.
2. A ratio compares two quantities by division.
3. Percentage change is always positive.
Which of the above is/are correct?
- (a) 1 only
- (b) 2 only
- (c) 1 and 2 only
- (d) 1, 2 and 3
Answer: (c) 1 and 2 only
Statement 1 is correct: percentage literally means 'per hundred' and is defined as a fraction where the denominator is 100 (e.g., 45% = 45/100). Statement 2 is also correct: a ratio expresses the relative size of two quantities using division, often written as a:b or a/b. Statement 3 is false: percentage change can be negative when a value decreases (e.g., a price drop of 10% is a -10% change). Therefore, only statements 1 and 2 are correct. This question tests the fundamental definitions of percentage, ratio, and the sign of percentage change. Common misconception is that percentage change is always an increase.
11. The ratio of the ages of two persons is 5:4. After 10 years, the ratio becomes 7:6. What is the present age of the younger one?
- (a) 20 years
- (b) 24 years
- (c) 28 years
- (d) 30 years
Answer: (a) 20 years
Let the present ages be 5x for the older person and 4x for the younger person. After 10 years the ages become 5x plus 10 and 4x plus 10. The new ratio is 7:6, so (5x + 10) divided by (4x + 10) equals 7 by 6. Cross multiply to get 6 times (5x + 10) equals 7 times (4x + 10). This gives 30x + 60 equals 28x + 70. So 2x equals 10 and x equals 5. The younger person's present age is 4x, which is 20 years. So the answer is (a). Wrong options: 24 years comes from taking x as 6 by a wrong step. 28 years comes from solving for the older person instead of the younger one. 30 years comes from wrongly setting 5x equal to 30.
12. A train running at 54 km/h crosses a man walking in the same direction at 6 km/h in 15 seconds. What is the length of the train?
- (a) 200 m
- (b) 225 m
- (c) 250 m
- (d) 275 m
Answer: (a) 200 m
When a train crosses a man walking in the same direction, the relative speed is the difference between their speeds. Train speed = 54 km/h
Man speed = 6 km/h
Relative speed = 54 - 6 = 48 km/h
Convert relative speed to m/s: 48 km/h × (5/18) m/s = 240/18 m/s = 40/3 m/s. Time taken to cross = 15 seconds. When a train crosses a man (a point object), the distance covered by the train relative to the man is equal to the length of the train. Length of the train = Relative speed × Time = (40/3) m/s × 15 s = 600/3 m = 200 m. Therefore, the length of the train is 200 m, which corresponds to option (a).