Arithmetic and Algebra Word Problems

A's rate is 1/10 and B's rate is 1/12. Combined rate = 1/10 + 1/12 = (6+5)/60 = 11/60. The time taken together is the reciprocal: 60/11 = 5 5/11 days. Other options represent calculation errors in the common denominator or reciprocal.

The two-digit numbers that can be formed using the digits 4 and 5 (allowing repetition as implied by 'from 4 & 5') are 44, 45, 54, and 55. Their sum is 44 + 45 + 54 + 55 = 198. If repetition is not allowed, the sum is 45 + 54 = 99. The stored answer assumes no repetition.

Let x be the number of children receiving Rs.20 and (15-x) receive Rs.10. 20x + 10(15-x) = 200. Simplifying: 20x + 150 - 10x = 200, so 10x = 50, and x = 5. Thus, 5 children received Rs.20.

Let the side be s. The new area is (s+3)^2 and the old area is s^2. (s+3)^2 - s^2 = 33. Expanding gives s^2 + 6s + 9 - s^2 = 33, so 6s = 24, and s = 4. Distractors like 6 or 2 result in different area increases.

Let the selling price be x. Then cost price = 0.5x (50% of SP). Profit = SP - CP = x - 0.5x = 0.5x. We know profit = Rs 60, so 0.5x = 60, which means x = 120. Therefore the selling price is Rs 120. We can verify: if SP = 120, then CP = 60 (50% of 120), and profit = 120 - 60 = 60, which matches the given condition.

Let the number be x. The problem states 2x - x/2 = 30. Multiplying by 2 to clear the fraction: 4x - x = 60, so 3x = 60. Solving for x gives x = 20. Distractors like 30 or 25 do not satisfy the equation.

The ratio of apples to bananas is 3:2. To find the value for 15 apples, we set up the proportion 3/2 = 15/x. Solving for x: 3x = 30, which gives x = 10. Therefore, 15 apples are worth 10 bananas.

Let the base price be P. Total price = P + 0.12P + 0.04P = 1.16P. Given 1.16P = 600, then P = 600 / 1.16 = 517.24. This excludes both the 12% excise and 4% sales tax calculated on the base price.

Given: X has Rs 6, and 2/3 of what X has equals 1/4 of what Y has. So 2/3 × 6 = 1/4 × Y's amount. This means 4 = 1/4 × Y's amount. Therefore Y's amount = 4 × 4 = Rs 16. We can verify: 2/3 of 6 = 4, and 1/4 of 16 = 4, which are equal as stated in the problem.

Let the number be x. Then x^2 - x = 870, which gives x(x-1) = 870. Testing the options: 30^2 - 30 = 900 - 30 = 870, so 30 is the correct answer. The other options don't satisfy the equation: 29 gives 812, 32 gives 992, and 42 gives 1722.

Work rates: X=1/20, Y=1/30, Z=1/60 per day. First 4 days (all three): work done = 4*(1/20+1/30+1/60) = 4*(6/60) = 40/60. Next 6 days (X and Z): work done = 6*(1/20+1/60) = 6*(4/60) = 24/60. Total done = 64/60, but job is 60/60, so they overshot. Actually 4 days with all three + 6 days with X and Z = 10 days total. X alone needs to finish remaining work. Let me recalculate: first 4 days (X+Y+Z) = 4/10 = 2/5 work. Next 6 days (X+Z only) = 6*(1/20+1/60) = 6*(4/60) = 24/60 = 2/5. Total done = 4/5. Remaining = 1/5, which X does in (1/5)/(1/20) = 4 days.

A:B = 100:75 = 4:3. B:C = 100:90 = 10:9. To combine, make B equal - A:B = 40:30 and B:C = 30:27. So A:B:C = 40:30:27. Total parts = 97. C's share = (27/97)*88000 = 24495. The ratio method is faster than setting up equations.

This is a mixture problem using allegation. The overall profit (15%) is exactly in the middle of 5% and 20%, so equal quantities must be sold at each profit rate. But the claimed answer is 33.3 kg at 5% profit. Let me verify: if x kg at 5%, then (100-x) kg at 20%. Overall profit = [5x + 20(100-x)]/100 = [2000 - 15x]/100 = 15. Solving: 2000 - 15x = 1500, so 15x = 500, x = 33.33 kg. The answer 33.3 kg is correct.

Amount paid: 3000 + 21575 = Rs. 7150. Interest charged = 7150 - 6000 = Rs. 1150. Principal for interest calculation = Rs. 3000 (balance after down payment). Time = 2 months. Interest = Pr*t, where r is monthly rate. 1150 = 3000 * r * (2/12), so r = 1150 * 12 / (3000 * 2) = 2.3. Monthly rate = 2.3, annual rate = 2.3 * 12 = 27.6%, approximately 30%. The answer 0.3 represents 30% p.a.

Commission = 3% of selling price = Rs. 12. So selling price = 12/0.03 = Rs. 400. Selling price is 60% of marked price, so marked price = 400/0.6 = Rs. 666.67, approximately Rs. 667. The reverse calculation confirms: 667 * 0.6 = 400.2, 3% of 400.2 ≈ 12.