UGC NET Questions (Paper – 1)

Let the original numbers be 4x and 7x. After adding 8 to each, the new numbers are 4x + 8 and 7x + 8 and their ratio is 5:8. So (4x + 8)/(7x + 8) = 5/8. Cross-multiplying gives 8(4x + 8) = 5(7x + 8), which simplifies to 32x + 64 = 35x + 40 and then to 3x = 24, so x = 8. The original numbers are 32 and 56, and the larger is 56.

Option A:
Option A, 40, does not appear as either 4x or 7x when x = 8 and would not preserve the required ratio 4:7 together with its pair. If we assume 40 as the larger number, the smaller would be less than 40, leading to a ratio different from 4:7 and a different transformed ratio after adding 8. Thus 40 is inconsistent with the given conditions.

Option B:
Option B, 48, might be guessed by partial calculation but does not satisfy the equation (4x + 8)/(7x + 8) = 5/8 when treated as the larger number. With 48 as the larger term, the corresponding smaller term from ratio 4:7 would be non-integral or lead to a different new ratio after adding 8. Hence 48 is not correct.

Option C:
Option C is correct because with x = 8 the original numbers are 4 × 8 = 32 and 7 × 8 = 56, which are in the ratio 4:7. After adding 8, we get 40 and 64, whose ratio is 40:64 = 5:8, exactly matching the condition. This confirms 56 as the unique value for the larger original number.

Option D:
Option D, 64, is the larger number after the increment rather than before it in the correct solution. Treating 64 as the original larger number would produce a much bigger value after adding 8 and the resulting ratio would no longer become 5:8. Thus 64 does not align with the stated transformation.

Let the numbers be 6x and 7x. Their sum is given as 6x + 7x = 13x = 325. Solving this, we get x = 325/13 = 25. Therefore the two numbers are 6 × 25 = 150 and 7 × 25 = 175. The smaller number is 150.

Option A:
Option A is correct because it emerges directly from solving the equation based on the ratio and the sum. Substituting 150 and 175 back into the problem, their sum is 325 and their ratio is 150:175, which reduces to 6:7 on dividing by 25. This confirms the consistency of the solution.

Option B:
Option B, 175, is the larger of the two numbers. The question explicitly asks for the smaller number, so choosing 175 would ignore this requirement. Although 175 is part of the correct pair, it does not answer the query.

Option C:
Option C, 200, would imply that the other number is 125 to maintain the sum 325. The ratio 125:200 simplifies to 5:8, not 6:7, so this pair does not match the stated ratio. Therefore 200 cannot be the smaller number satisfying both conditions.

Option D:
Option D, 225, paired with 100, would give a sum of 325 but a ratio of 100:225, which simplifies to 4:9. This is different from the required 6:7, so 225 is not a valid candidate for the smaller number under the given constraints.

Let the present ages of the father and son be 7x and 3x respectively. After 10 years, their ages will be 7x + 10 and 3x + 10. The future ratio is (7x + 10) : (3x + 10) = 13 : 7. Cross-multiplying gives 7(7x + 10) = 13(3x + 10), i.e., 49x + 70 = 39x + 130. This simplifies to 10x = 60, so x = 6. Therefore, the father’s present age is 7x = 7 × 6 = 42 years.

Option A:
Option A, 35 years, would make the son 15 years old (ratio 7 : 3), and after 10 years their ages would be 45 and 25, giving a ratio of 9 : 5, not 13 : 7.

Option B:
Option B, 42 years, makes the son 18 years old. After 10 years they would be 52 and 28, and 52 : 28 simplifies to 13 : 7, satisfying both the given present ratio and the future ratio.

Option C:
Option C, 49 years, would imply a son aged 21, and after 10 years their ages become 59 and 31, which do not reduce to 13 : 7.

Option D:
Option D, 56 years, would give the son 24 years at present and 66 and 34 after 10 years, with ratio 33 : 17, again not 13 : 7.

Let the original numbers be 2x and 5x. After adding 6 to each, they become 2x + 6 and 5x + 6, and the new ratio is 1:2. Thus, (2x + 6)/(5x + 6) = 1/2. Cross-multiplying gives 2(2x + 6) = 5x + 6, which simplifies to 4x + 12 = 5x + 6 and then x = 6. Therefore, the original numbers are 12 and 30, and the smaller is 12.

Option A:
Option A is correct because it corresponds to 2x with x = 6 in the ratio 2:5. Substituting 12 and 30 back into the problem, we get (12 + 6):(30 + 6) = 18:36 = 1:2, which matches the given transformed ratio. This confirms that 12 is the smaller of the original pair.

Option B:
Option B, 18, would require the larger number to be 45 to maintain the original ratio 2:5. The pair 18 and 45 has sum 63, and after adding 6 to each we get 24 and 51, whose ratio is 24:51 = 8:17, not 1:2. Hence 18 does not satisfy the transformation condition.

Option C:
Option C, 20, when paired with 50 for the 2:5 ratio, leads to new numbers 26 and 56 after the equal addition of 6. Their ratio 26:56 simplifies to 13:28, which again is not 1:2. Thus, 20 cannot be the smaller original number.

Option D:
Option D, 24, would pair with 60 to give the 2:5 ratio. After adding 6 we obtain 30 and 66, and the ratio 30:66 reduces to 5:11, not 1:2. Therefore, 24 fails to meet the required ratio change after the addition.

Statements A, B and C state the standard time–work model: one-day work as 1/n, combined rates added and inverse relation between time and rate. Statement E is also true since many problems can be set up and solved using simple linear equations in the unknown rates. Statement D is false because if A is twice as fast as B, A will take half the time, not twice. Therefore, the combination A, B, C and E only is correct.

Option A:
Option A is incomplete because it leaves out E and thus fails to mention the useful algebraic viewpoint that rates can be handled by linear equations. Without E, the picture of modelling is not fully captured.

Option B:
Option B is correct since it collects all the true statements and excludes D, which reverses the relationship between speed and time. It reflects exactly how NET aspirants are expected to reason about time and work problems.

Option C:
Option C is incorrect because it includes D, the wrong claim about time for a faster worker, even though it also lists some true statements. Including a false statement invalidates the combination.

Option D:
Option D is also wrong as it retains D and omits A, failing to state the fundamental definition of one-day work while endorsing an incorrect comparison of times.

Let the numbers be 5x and 12x. Their difference is 12x − 5x = 7x, which is given as 133. Solving 7x = 133 gives x = 19. Hence, the numbers are 5 × 19 = 95 and 12 × 19 = 228, and the larger number is 228.

Option A:
Option A, 190, might be formed by doubling 95, but it does not fit the structure 12x with x = 19. If 190 were the larger number, the smaller under the 5:12 ratio would be 190 × (5/12) ≈ 79.17, and the difference would not be 133, so this choice is inconsistent.

Option B:
Option B, 209, cannot be expressed as 12x with integer x that also makes 5x an integer, because 209/12 is not an integer. Even if this were ignored, the implied difference between 5x and 12x would not match 133, so 209 fails both ratio and difference conditions.

Option C:
Option C is correct because it follows from solving the simple linear equation based on the difference of 7x. The pair (95, 228) satisfies both the ratio 5:12 and the difference 133, so 228 is the unique larger number that fits all parts of the problem.

Option D:
Option D, 247, would require x = 247/12 ≈ 20.58, making the smaller number 5x non-integer and leading to a different difference than 133. Therefore, 247 does not satisfy the structure imposed by the ratio 5:12 and the given difference.

Let the present ages of A and B be A and B years respectively. We know A + B = 50. After 10 years, their ages will be A + 10 and B + 10, and the ratio (A + 10) : (B + 10) is given as 4 : 3. This gives the equation (A + 10) / (B + 10) = 4 / 3, which can be solved together with A + B = 50. Solving this system yields A = 30 and B = 20, so A’s present age is 30 years.

Option A:
Option A, 20 years, actually corresponds to B’s present age when the system is solved. It reverses the roles of A and B and therefore does not answer the question correctly.

Option B:
Option B, 25 years, would make the sum of their ages 50 only if B were also 25, but this would not satisfy the future ratio of 4 : 3 after 10 years.

Option C:
Option C satisfies both conditions: the present ages 30 and 20 sum to 50, and after 10 years they become 40 and 30, which indeed are in the ratio 4 : 3.

Option D:
Option D, 35 years, would require B to be 15 to keep the sum at 50, and after 10 years the ratio 45 : 25 is not equal to 4 : 3.

Let the three consecutive even numbers be n − 2, n and n + 2, where n is the middle one. Their sum is (n − 2) + n + (n + 2) = 3n. According to the given condition, 3n = 66, so n = 66 ÷ 3 = 22. Hence, the middle even number is 22.

Option A:
Option A, 20, would give the three numbers 18, 20 and 22, whose sum is 60, not 66. Therefore, it does not satisfy the condition set in the problem.

Option B:
Option B fits perfectly: with 22 as the middle number, the three numbers become 20, 22 and 24, and their sum is 20 + 22 + 24 = 66, matching the given total.

Option C:
Option C, 24, would generate the sequence 22, 24 and 26, whose sum is 72, exceeding the required sum.

Option D:
Option D, 26, would produce 24, 26 and 28, summing to 78, which is also inconsistent with the problem statement.

Let the number be x. The statement “three times the number decreased by 5 equals 16” translates to the equation 3x − 5 = 16. Adding 5 to both sides gives 3x = 21, and then dividing by 3 yields x = 7. Substituting back into the original relation, 3 × 7 − 5 equals 21 − 5 = 16, confirming that 7 satisfies the condition.

Option A:
Option A, 5, when substituted gives 3 × 5 − 5 = 10, which is not equal to 16. Thus, it fails to satisfy the equation implied by the verbal statement.

Option B:
Option B fits the equation exactly, turning 3x − 5 into 16 when x is 7. This closes the loop between the word problem and its algebraic representation.

Option C:
Option C, 9, would make 3 × 9 − 5 = 22, overshooting the stated result 16, so it cannot be correct.

Option D:
Option D, 11, yields 3 × 11 − 5 = 28, which is even further from 16 and clearly inconsistent.

To solve 3x − 5 = 2x + 7, we first bring like terms together. Subtracting 2x from both sides gives x − 5 = 7. Then we add 5 to both sides to isolate x, obtaining x = 12. This value satisfies the equation when substituted back, confirming that 12 is the correct solution.

Option A:
Option A, 6, does not satisfy the equation because substituting x = 6 gives 3 × 6 − 5 = 13 and 2 × 6 + 7 = 19, which are not equal. This indicates that 6 cannot be the solution.

Option B:
Option B, 10, also fails on substitution, as 3 × 10 − 5 equals 25 while 2 × 10 + 7 equals 27, showing a mismatch between the two sides.

Option C:
Option C results in equal values on both sides when substituted, with 3 × 12 − 5 giving 31 and 2 × 12 + 7 also giving 31. This equality verifies that x = 12 satisfies the original equation.

Option D:
Option D, 15, leads to 3 × 15 − 5 = 40 and 2 × 15 + 7 = 37, again producing unequal sides and ruling out 15 as a solution.

Let the numbers be 5x and 8x. After subtracting 20 from each, they become 5x − 20 and 8x − 20, with ratio 3:5. So (5x − 20)/(8x − 20) = 3/5. Cross multiplying gives 5(5x − 20) = 3(8x − 20), which simplifies to 25x − 100 = 24x − 60 and hence x = 40. The original numbers are therefore 200 and 320, so the larger is 320.

Option A:
Option A, 200, is the smaller original number when x = 40. If we took 200 as the larger, it would contradict the initial ratio 5:8, in which 8x must exceed 5x. Substituting 200 as the larger in the equations would also fail to reproduce the ratio 3:5 after subtraction.

Option B:
Option B, 240, does not correspond to either 5x or 8x for the solved value x = 40. If we attempted to solve using 240 as one of the numbers, the resulting x would not satisfy the transformed ratio condition. Hence 240 is inconsistent with the problem’s constraints.

Option C:
Option C, 280, again fails to match the form 5x or 8x with x = 40 and does not produce the stated ratio 3:5 after subtracting 20 from both numbers. Any attempt to fit 280 into the ratio structure leads to contradictions with the proportional relationship.

Option D:
Option D is correct because 320 equals 8 × 40, matching the larger term in the original ratio 5:8. With numbers 200 and 320, subtracting 20 gives 180 and 300, and 180:300 simplifies to 3:5. This verifies that 320 is the correct larger original number.

Let the numbers be 9x and 13x. After subtracting 12 from each, they become 9x − 12 and 13x − 12, and their ratio is 3:5. So (9x − 12)/(13x − 12) = 3/5. Cross-multiplying gives 5(9x − 12) = 3(13x − 12), which simplifies to 45x − 60 = 39x − 36 and then to 6x = 24, so x = 4. Hence the original numbers are 36 and 52, and the larger number is 52.

Option A:
Option A, 36, is the smaller of the two numbers for x = 4 and does not answer the question, which asks specifically for the larger number. Though 36 fits the ratio 9:13 with 52, choosing it would ignore the wording about which number is larger.

Option B:
Option B, 44, does not fit into the pair 9x and 13x when the transformation is imposed. If we tried to use 44 as the larger number, the corresponding smaller number from ratio 9:13 would not be an integer, and the new ratio after subtracting 12 would not become 3:5. Thus 44 does not satisfy the stated conditions.

Option C:
Option C, 48, might appear plausible as a mid-range value but cannot be expressed as 13x with integer x satisfying the ratio change. Any attempt to treat 48 as the larger number yields inconsistencies when we try to reach a ratio of 3:5 after subtracting 12. Therefore, 48 is incorrect.

Option D:
Option D is correct because with x = 4 we get 9x = 36 and 13x = 52, and 36:52 reduces to 9:13. Subtracting 12 from each gives 24 and 40, whose ratio is 24:40 = 3:5, precisely matching the requirement. This confirms 52 as the correct larger original number.

Statement (1) presents a solvable linear equation, 3x + 2 = 20, which has a unique solution. Solving it gives x = 6, so statement (1) by itself fully determines the value of x. Statement (2) only tells us that x is an integer but gives no equation or relation to narrow down x further. Therefore, statement (2) alone does not allow us to find a unique x. In data sufficiency terms, (1) alone is sufficient while (2) alone is not.

Option A:
Option A correctly reflects that statement (1) yields a unique solution for x, and that statement (2) adds no new restrictive information when used alone.

Option B:
Option B reverses the roles of the statements and incorrectly claims that simply knowing x is an integer is sufficient to find a unique value, which is impossible without an equation.

Option C:
Option C assumes that neither statement can independently identify x, which is incorrect because statement (1) clearly does.

Option D:
Option D states that each statement alone is sufficient, but this overstates the power of statement (2), which does not determine any specific numerical value.

Let the numbers be 7x and 8x. The difference between them is 8x − 7x = x, and we are told this difference equals 15. So x = 15. Substituting back, the numbers are 7 × 15 = 105 and 8 × 15 = 120. The larger number is therefore 120.

Option A:
Option A, 105, is the smaller of the two numbers obtained from the correct solution. It correctly fits the ratio but does not match the description “larger number” in the question. Choosing 105 would therefore misinterpret which quantity is being asked for.

Option B:
Option B is correct because it corresponds to the 8x term when x = 15, which is explicitly the larger number in the ratio 7:8. Verifying, the difference 120 − 105 is indeed 15, and the ratio 105:120 simplifies to 7:8. This confirms that 120 satisfies both the ratio and difference conditions.

Option C:
Option C, 135, when paired with 120 (to keep a difference of 15), yields a ratio 120:135 = 8:9, not 7:8. If we reversed the roles, 135 and 120 would differ by 15 but still not simplify to 7:8. Thus, 135 cannot be the larger number under the stated ratio.

Option D:
Option D, 140, would require the smaller number to be 125 to maintain a difference of 15. The ratio 125:140 reduces to 25:28, which is again not equal to 7:8. Therefore, 140 is incompatible with the conditions given in the question.

Let the numbers be 4x and 7x. After adding 5 to each, they become 4x + 5 and 7x + 5, and their ratio is given as 3:5. Thus (4x + 5)/(7x + 5) = 3/5. Cross multiplying gives 5(4x + 5) = 3(7x + 5), which simplifies to 20x + 25 = 21x + 15, and hence x = 10. Therefore the numbers are 40 and 70, and the larger original number is 70.

Option A:
Option A, 40, is the smaller of the two original numbers when x = 10. It satisfies the initial ratio but does not answer the question, which specifically asks for the larger number. Moreover, choosing 40 as the larger number would contradict the ratio 4:7 where 4x must be smaller than 7x.

Option B:
Option B, 45, does not fit into any pair maintaining both the original ratio 4:7 and the new ratio 3:5 after adding 5. If we treated 45 as the larger original number, then the smaller would be 45 × (4/7), which is not an integer and fails the requirement for simple integer ratio terms. Thus 45 is an inconsistent choice.

Option C:
Option C is correct because 70 arises naturally as 7 × 10 when solving the given proportion. Substituting 40 and 70 back into the problem, we get (40 + 5):(70 + 5) = 45:75 = 3:5, which verifies the condition. This confirms that 70 is the true larger original number.

Option D:
Option D, 75, would require the smaller number to be 75 × (4/7) ≈ 42.86 to preserve the original ratio, which is not an integer and also fails to give the new ratio 3:5 after adding 5. Therefore 75 does not satisfy the structural requirements of the problem.