math & puzzles Online Quiz - 9

This is a stars and bars combinatorics problem. Distributing 6 identical balls into 9 squares with each of 3 rows getting at least 1 ball means we need to count distributions where balls appear in all 3 rows. The calculation yields 81 valid arrangements.

The pattern is factorial: 1×1=1, 1×2=2, 2×3=6, 6×4=24, 24×5=120. Each term is obtained by multiplying the previous term by an incrementing integer.

First find the cost per gram: Rs. 28 / (16 × 900g) = Rs. 28 / 14400g. For 27 packets of 1kg (1000g) each: 27 × 1000g × (Rs. 28 / 14400g) = Rs. 52.50. This is a unitary method problem where you must adjust for both quantity and weight change.

Let the fractions be x and y with x > y. Given xy = 14/15 and x/y = 35/24. From x/y = 35/24, we get x = (35/24)y. Substituting: (35/24)y² = 14/15, so y² = (14/15) × (24/35) = 16/25, giving y = 4/5. Then x = (35/24) × (4/5) = 7/6.

This is an alternating series with two patterns. Odd positions (1st, 3rd, 5th, 7th): 75, 85, 45, 85 (decreasing by 30, then repeating 85). Even positions (2nd, 4th, 6th): 65, 55, 35 (decreasing by 10 each time). The next two terms are 25 (continue 65→55→35→25) and 85 (repeat pattern).

Let length = x meters and cost per meter = Rs. y. Given xy = 35 and (x + 4)(y - 1) = 35. Expanding: xy - x + 4y - 4 = 35. Substituting xy = 35: 35 - x + 4y - 4 = 35, so 4y - x = 4. Since y = 35/x, we get 4(35/x) - x = 4, which gives x² + 4x - 140 = 0, so (x + 14)(x - 10) = 0. Therefore x = 10 meters.

First 2/3 of 15 miles = 10 miles at 40 mph takes 10/40 = 1/4 hour = 15 minutes. Remaining 1/3 = 5 miles at 60 mph takes 5/60 = 1/12 hour = 5 minutes. Total time = 15 + 5 = 20 minutes.

In a 6x6x6 cube, cubes with at least 2 sides painted are the edge cubes and corner cubes. There are 12 edges, each with (n-2) cubes, and 8 corners. Calculation: 12*(6-2) + 8 = 12*4 + 8 = 48 + 8 = 56.

Find the pattern in differences: 5-1=4, 14-5=9, 30-14=16. These are 2², 3², 4². Next difference should be 5²=25, giving 30+25=55. Verify: 91-55=36=6², continuing the pattern.

The average of first n natural numbers is (n + 1)/2. For n = 40: (40 + 1)/2 = 41/2 = 20.5. This formula comes from the arithmetic series sum n(n+1)/2 divided by n.

Initial alcohol content: 15 L x 20% = 3 L pure alcohol. When 5 L pure water is added, total volume becomes 20 L, but alcohol quantity remains unchanged at 3 L. New percentage = (3 L alcohol / 20 L total) x 100 = 15%. The key insight is that adding water dilutes the concentration without changing the amount of alcohol.

Substituting values: AB = 5 x 0 = 0, EE = 2 x 2 = 4, ED = 10 x 2 = 20, AC = 5 x 2 = 10. The expression becomes: 0 + 4 - (20)^0 + (10)^2 = 0 + 4 - 1 + 100 = 103. Note that any number to the power of 0 equals 1, and any number to the power of 2 equals its square.

Stream speed = (downstream speed - upstream speed) / 2 = (13 - 8) / 2 = 2.5 kmph. This formula works because downstream speed = boat speed + stream speed, while upstream speed = boat speed - stream speed. Subtracting eliminates boat speed and doubles the stream speed.

For 5 consecutive odd numbers, they can be represented as x, x+2, x+4, x+6, x+8. Sum = 5x + 20 = 1075, so 5x = 1055 and x = 211. The numbers are 211, 213, 215, 217, 219. The largest is 219. Alternatively, for 5 consecutive numbers with same difference, the middle number (third) equals average, so middle = 1075/5 = 215, making the largest 215+4 = 219.