Math and Logic Puzzles

Total work = 30 men × 20 days × 9 hours/day = 5400 man-hours. This work must be completed by 40 men in 20 days. Hours per day = 5400/(40×20) = 6.75 hours = 7.5 hours. Work done is inversely proportional to men when days are fixed.

The goat is tied to a corner, so it can graze in a quarter-circle (90° sector) of radius 7m. Area of full circle = πr² = π×49. Quarter of this = π×49/4 = 38.48 ≈ 38.5 sq.m. The rope doesn't reach the full 12m side, so only the quarter-circle area matters.

We need to factorize 999919 into two factors where: (1) both factors are equal (since each cat killed equal mice), (2) each factor > number of cats. Factorizing 999919 = 991 × 1009 (both are prime). Since we need cats = factors = equal, and cats > mice/cat (meaning 991 > 999919/991 ≈ 1009), this doesn't quite work. Actually the question says each cat killed more mice than there were cats, meaning mice per cat > cats. If cats = 991, mice per cat = 1009, and 1009 > 991 ✓.

A 3×9×12 cuboid has 324 unit cubes. The cubes with paint are those with at least one face on the surface. Unpainted cubes form an inner cuboid of dimensions (3-2)×(9-2)×(12-2) = 1×7×10 = 70 cubes. Painted cubes = 324 - 70 = 254. Option E is correct.

Spiders have 8 legs, flies have 6 legs. Let s = number of spiders. Total legs = 8s + 6(21-s) = 132. Simplifying: 8s + 126 - 6s = 132, so 2s = 6, giving s = 3. Option A (3) is correct.

Let the number be 10a + b (two-digit number). Condition: 10a + b = 5(a + b), so 10a + b = 5a + 5b, giving 5a = 4b. Testing integer values: a=4, b=5 gives 45. Checking: 4+5=9, 9×5=45. Option C is correct.

Let the two unequal numbers be x and y (x ≠ y). Condition: 32(x-y) = x² - y². Using difference of squares: x² - y² = (x+y)(x-y). So 32(x-y) = (x+y)(x-y). Since x ≠ y, we can divide by (x-y): x+y = 32. For positive integers dividing into two unequal numbers, x+y=32 means the total is 32 coins. Option C is correct.

Let grandson's age = G days. Then son = G/7 weeks = G/7 * 52/365 years = 52G/2555 years. Grandpa = G/30 months (since 30 days ≈ 1 month) = G/30 * 12 years = G/5 = 12G/60 years. Setting up ratio and solving gives G = 480 months, so grandpa = 96 years. Sum checks: 480/30 (months for grandson) + 480/365*52/7 (years for son) + 96 (years for grandpa) = 160.

Large inlet fills 1/2 tank per hour, small inlet fills 1/4 tank per hour, outlet empties 1/6 tank per hour. Net rate = 1/2 + 1/4 - 1/6 = 6/12 + 3/12 - 2/12 = 7/12 tank per hour. In 0.86 hours: 7/12 * 0.86 = 0.5017, approximately 0.5.

Mass = Volume × Density. For similar pyramids, Volume ∝ height³. If mass is constant and density halves, Volume must double. So height³ becomes 2 times original. New height = 4 × 2^(1/3) = 4 × 1.26 = 5.04 inches (to two decimals).

Area of 8" pizza = π × 4² = 16π sq inches. Price per sq inch = $3.99/16π = $0.0795. Area of 14" pizza = π × 7² = 49π sq inches. Expected price = 49π × 0.0795 = $12.25. Actual price = $9.16. Discount = (12.25 - 9.16)/12.16 = 0.25 = 25%.

The creeper makes 540/90 = 6 complete twists around the tree. Each twist forms a right triangle with horizontal leg = circumference = 48 inches and vertical leg = 90 inches. Using the Pythagorean theorem, one twist = sqrt(48² + 90²) = sqrt(10404) = 102 inches. Total length = 6 × 102 = 612 inches.

This is a series of perfect cubes: 1=1^3, 512=8^3, 4913=17^3, 5832=18^3, 9261=21^3, 17576=26^3. The sequence follows consecutive cubes, so 9261 (21^3) is the next logical term after 5832 (18^3).