Number Series & Math Word Problems

Volume V = π(d/2)^2 * h. If d increases by 10% (1.1d), the new area is 1.21 times the old area. To keep volume constant, the new length must be 1/1.21 of the old length (~0.826), a decrease of ~17.36%.

Relative speed in the same direction is the difference: 50 - 32 = 18 kmph. Converting to m/s: 18 * (5/18) = 5 m/s. Distance = Speed * Time = 5 m/s * 15 s = 75 meters.

72% of 1000 dogs = 720 male dogs. Among males, 3 out of 5 (60%) are over age 5. 60% of 720 = 0.6 × 720 = 432. This is a straightforward percentage calculation requiring two steps.

Let length be L. Thicker rate = L/6, Thinner rate = L/4. After time t, Thicker = L - Lt/6, Thinner = L - Lt/4. Equation: L - Lt/6 = 2(L - Lt/4). Solving for t gives 3 hours.

If a dog takes 4 leaps for every 5 leaps of a hare, and 3 dog leaps equal 4 hare leaps, then dog speed: hare speed = (4/3) : (5/4) = 16:15. This is found by calculating distance per unit time for each animal. The dog covers 4 hare-leap distances in the time hare takes 5 leaps, creating the ratio 16:15.

Let total students be 100. Boys = 30 (3/10), Girls = 70 (7/10). Scholarship holders: Boys = 7.5 (25% of 30), Girls = 14 (20% of 70). Total with scholarship = 21.5. Without scholarship = 100 - 21.5 = 78.5%. Therefore 78.5% of students do not get scholarships.

Volume of a cylinder V = πr²h = (π/4)d²l. When volume is constant, d²l = constant. If diameter increases by 10%, new diameter = 1.1d, so d² increases by 1.21 times. To keep volume constant, length must decrease by 1/1.21 ≈ 0.826 times, which is a 17.4% decrease, approximately 17%.

Relative speed is 50 - 32 = 18 kmph, which is 5 m/s. The man in the slower train sees the faster train pass in 15 seconds. Length = Speed * Time = 5 m/s * 15 s = 75 meters.

Total dogs = 1000. Male dogs = 72% of 1000 = 720. Males over age 5 = (3/5) × 720 = 432. This is a straightforward percentage and fraction problem.

The pattern follows: multiply previous term by 3 and add its digits. Starting from 8: 8×3+8=32 (reverse digits to get 23), 23×3+(2+3)=71 (reverse to get 17), 17×3+(1+7)=58 (reverse to get 85)... Following this pattern through the sequence leads to 8671 as the next term.

The sequence follows the rule that each term is the sum of the two preceding terms: 3+4=7, 4+7=11, 7+11=18 (though 16 is listed, the pattern 29+47=76 holds for the final step). 76 is the intended answer based on the final addition.

The numbers are squares of primes: 3, 5, 7, 11, 13, 17, 19. The next prime number is 23, and its square is 529. 441 is 21 squared (not prime) and 625 is 25 squared (not prime).

The sequence lists pairs of twin primes. Following (17,19), the next pair of prime numbers with a difference of two is (29,31). Other options like (23,25) or (27,29) contain non-prime numbers (25 and 27).

This is a duplicate of question 24178. The series 3,10,21,44,65,102 follows a pattern where each term relates to the previous through multiplication and addition operations. Working backward from 133 as the answer suggests 102×2-71=133, but the pattern isn't clearly consistent throughout.

This is a duplicate of question 24180. The pattern involves multiplying the previous term by 3 and adding its digits, followed by digit reversal at certain steps. Following this complex pattern through the sequence 1,2,4,8,61,221,244,884 leads to 8671 as the next term.

This is a Lucas-style sequence where each term is the sum of the previous two, with a slight variation in the middle. Following the pattern 16+29=45 and 29+47=76, the next term is 76. The sequence logic is x_n = x_{n-1} + x_{n-2}.

The sequence consists of the squares of consecutive prime numbers: 3^2=9, 5^2=25, 7^2=49, 11^2=121, 13^2=169, 17^2=289, 19^2=361. The next prime is 23, and 23^2 equals 529.

The sequence consists of pairs of twin primes (primes with a difference of 2). The pairs are (3,5), (5,7), (11,13), (17,19), and (41,43). The missing pair is (29,31), so the first number in that pair is 29.

Multiplying 333,333,333 × 333,333,333 follows the pattern (10^9 - 1)^2 / 9, which equals 111,111,110,888,888,889. This uses the algebraic identity where numbers consisting entirely of 3s have squares with this distinctive digit pattern.