Math and Logic Puzzles

Let x be canaries and y be cages. x = y + 1 and x = 2(y - 1). Substituting: y + 1 = 2y - 2, so y = 3. Then x = 4. Thus, 3 cages and 4 canaries satisfy both conditions perfectly.

Original area = 14 * 10 = 140. To make it a square, (14-x) = (10+x), so 2x = 4, x = 2. New dimensions are 12 * 12 = 144. The area changes from 140 to 144, a difference of 4. 144 is the new area, not the change.

Distance = Speed × Time. 54 kph for 5 minutes is 54 × (5/60) = 54 × (1/12) = 4.5 km. The radius of the well is irrelevant to the total distance traveled along the path. 2.5 and 3.5 are incorrect calculations.

Let son be x, father 3x. In 15 years: 3x + 15 = 2(x + 15). Solving gives x = 15. Father's present age is 3x = 45. Distractors like 36 or 42 do not satisfy the 'twice as old' condition after 15 years.

The word CONSULTANCY has 11 letters with C and N repeating twice each. Total permutations are 11!/(2!*2!) = 11!/4. To find 'new' words formed, we subtract the original word itself, resulting in (11!/4) - 1. Options 11! and 9! ignore repetitions and length.

To find the unit digit of 567^4, we only need to consider the unit digit of the base, which is 7. Calculating the powers of 7: 7^1=7, 7^2=49, 7^3=343, 7^4=2401. The unit digit is 1. Distractors like 7, 9, and 3 represent other steps in the power cycle.

Each coin toss has a 1/2 probability of being heads. For 4 independent tosses, the probability of all 4 being heads is (1/2) * (1/2) * (1/2) * (1/2) = 1/16.

For a square inside a circle, the diagonal equals the diameter. Diagonal = 2 * 5sqrt(2) = 10sqrt(2). Side = diagonal/sqrt(2) = 10. Area = 10^2 = 100 sq units.

Let C be classes and M be methods per class. Total = CM. Also Total = (C-2)(M+1). CM = CM + C - 2M - 2 => C - 2M = 2. For C=8, M=3 (Total 24). If C=6, M=4 (Total 24). This fits the condition.

Let train speed be T, cat speed be C. Distance from entrance is 10, tunnel is 30. If cat runs to entrance, it meets train. If it runs to exit (20m), it meets train. This implies the train reaches the entrance in the time the cat runs 10m. The ratio of speeds is 12:36.

We have 5 distinct digits (0,1,2,8,9) and need 4-digit numbers without repetition. First digit cannot be 0: 4 choices. Then arrange remaining 3 from 4 available digits: 4P3 = 4×3×2 = 24. Total = 4×24 = 96. Option D is correct.

Since there are 5 vowels (A,E,I,O,U) and they must be together, we treat the block of 5 vowels as a single unit. However, since there are no other letters mentioned to form a word with, we are just arranging the 5 vowels. The number of arrangements is 5! = 120.

The series alternates between squares and cubes: 1^2, 1^3, 2^2, 2^3, 3^2, 3^3, 4^2, 4^3. The next term should be 5^2, which is 25.

This is a factorial series: 1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120. The next term is 6! = 120 * 6 = 720.

The differences between terms are increasing by 6: 19-7=12, 37-19=18, 61-37=24. The next difference should be 24+6=30. 61 + 30 = 91.

The pattern is n^3 - n: 1^3-1=0, 2^3-2=6, 3^3-3=24, 4^3-4=60, 5^3-5=120. The next term is 6^3-6 = 216 - 6 = 210.