UGCNET Questions

The average of three numbers is the sum of the numbers divided by 3. If the average is 25, then the total sum must be 25 × 3 = 75. The two known numbers are 20 and 30, whose sum is 50. Subtracting 50 from 75 gives 25, so the third number must be 25.

Option A:
Option A, 25, exactly matches the required third number, because 20 + 30 + 25 = 75 and 75 ÷ 3 = 25. It is therefore the correct choice.

Option B:
Option B, 20, would give a total of 20 + 20 + 30 = 70, and 70 ÷ 3 does not equal the required average of 25, so this choice fails the given condition.

Option C:
Option C, 30, leads to 20 + 30 + 30 = 80, giving an average of 80 ÷ 3, which is greater than 25 and inconsistent with the stated mean.

Option D:
Option D, 35, produces a total of 20 + 30 + 35 = 85, resulting in an average greater than 25, again contradicting the problem’s information.

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.

The average of four numbers being 26 means that their total sum is 26 × 4 = 104. The sum of the three known numbers 20, 25 and 29 is 20 + 25 + 29 = 74. The fourth number must make the overall sum 104, so it is 104 − 74 = 30. Therefore, the missing number is 30, which is Option B.

Option A:
Option A, 25, would lead to a total of 20 + 25 + 29 + 25 = 99, and 99 ÷ 4 is 24.75, which is less than 26, so the average condition fails.

Option B:
Option B, 30, makes the sum of the four numbers 104, which divided by 4 gives exactly 26, perfectly satisfying the definition of the average.

Option C:
Option C, 32, would produce a total of 106, and 106 ÷ 4 equals 26.5, which is higher than the stated average.

Option D:
Option D, 34, makes the total 108, giving an average of 27, again inconsistent with the given value.

To solve 5x + 3 = 18, we first subtract 3 from both sides to isolate the term with x. This gives 5x = 15. Dividing both sides by 5, we get x = 15 ÷ 5 = 3. Therefore, the solution of the equation is x = 3.

Option A:
Option A, 2, does not satisfy the equation because substituting x = 2 yields 5 × 2 + 3 = 13, which is not equal to 18. So this value fails to make both sides equal.

Option B:
Option B, 3, when substituted gives 5 × 3 + 3 = 15 + 3 = 18, which matches the right side and confirms that x = 3 is the correct solution.

Option C:
Option C, 4, produces 5 × 4 + 3 = 23, which is greater than 18 and so cannot be the solution.

Option D:
Option D, 5, gives 5 × 5 + 3 = 28, which again does not equal 18, and therefore cannot solve the equation.