Algebra Medium To Advanced Level Questions

Explanation
The resistance of a cable varies directly with its length and inversely with the square of its diameter. In this case, we are comparing two pieces of wire made of the same material. The first wire is 100 m long and has a diameter of 5 mm, while the second wire is 50 m long and has a diameter of 3 mm.

To compare the resistance, we can use the formula R = k * (L / d^2), where R is the resistance, L is the length, d is the diameter, and k is a constant.

For the first wire, R1 = k * (100 / (5^2)) = k * (100 / 25) = 4k.
For the second wire, R2 = k * (50 / (3^2)) = k * (50 / 9) = 5.56k.

Therefore, R1 = 0.72 * R2.

Explanation
Let's assume the original speed of the boat is x kph. The time taken to travel the first 2 km at this speed is 2/x hours. The time taken to travel the remaining 6 km at a speed of x+1/2 kph is 6/(x+1/2) hours.

According to the given information, the boat arrives 10 minutes (1/6 hours) earlier than if the original speed had been maintained. So, the total time taken at the original speed would be 2/x + 6/x + 1/6 hours.

The boat arrives 10 minutes earlier, so the total time taken at the increased speed would be 2/x + 6/(x+1/2) hours.

Setting these two expressions equal to each other and solving for x, we get x = 4 kph.

Therefore, the original speed of the boat is 4 kph.

Explanation
The group has P100 to spend for lunch. They decided to give a tip of 20% before tax. The sales tax is 7 ½ %. To calculate the amount they should spend, we first find the tip amount by multiplying P100 by 20% which gives P20. Then, we add the tip amount to the original P100 to get P120. Next, we find the tax amount by multiplying P120 by 7 ½ % which gives P8.40. Finally, we subtract the tax amount from P120 to get the final amount they should spend, which is P111.60. Therefore, the correct answer is P78.43.

Explanation
The un-experienced worker can unload a truck in 1 hour and 40 minutes, which is equivalent to 100 minutes. Together with the trainee, they can work for 1 hour, which is 60 minutes. This means that the trainee can work for 60 minutes - 100 minutes = -40 minutes alone. Since it is not possible for someone to work for negative minutes, the trainee cannot work alone. Therefore, the answer is not available.

Explanation
The arithmetic sequence starts with 11 and has a common difference of 4. To find the sixth term, we can use the formula: an = a1 + (n-1)d. Plugging in the values, we get a6 = 11 + (6-1)4 = 11 + 20 = 31. Therefore, the sixth term of the arithmetic sequence is 31.

Explanation
To find the amount of 16% drug that should be added, we can use the concept of weighted averages. Let's assume x mL of the 16% drug is added. The amount of drug in the 16% solution is 0.16x and the amount of drug in the 28% solution is 0.28(15 - x) since the total volume is 15 mL.

To obtain a 24% mixture, the total amount of drug in the final mixture should be 0.24(15). So we can set up the equation 0.16x + 0.28(15 - x) = 0.24(15) and solve for x. Simplifying the equation gives 0.16x + 4.2 - 0.28x = 3.6. Combining like terms gives -0.12x + 4.2 = 3.6. Subtracting 4.2 from both sides gives -0.12x = -0.6. Dividing both sides by -0.12 gives x = 5.

Therefore, 5 mL of the 16% drug should be added.

Explanation
To find the amount of additional peanut mixture needed, we can set up a proportion. Since the original mixture contains 40% peanut, that means it contains 0.4 lbs of peanut per 1 lb of mixture. The desired mixture contains 50% peanut, which means it needs to contain 0.5 lbs of peanut per 1 lb of mixture. To find the amount of additional peanut mixture needed, we can set up the proportion: 0.4/1 = x/8. Solving for x, we find x = 3.2 lbs. However, since we already have 8 lbs of mixture, we only need to add 3.2 - 8 = 1.33 lbs of additional peanut mixture to achieve the desired 50% peanut content.

Explanation
The boat travels 5 miles upstream and then back downstream in a total time of 1 hour and 40 minutes. Since the stream current is 4 mph, it means that the boat is traveling against the current for a longer duration than it is traveling with the current. This indicates that the boat's speed in still water must be less than 4 mph. The only option that satisfies this condition is 2 mph, as it allows the boat to travel against the current at a slower speed and with the current at a faster speed, resulting in a total time of 1 hour and 40 minutes.

Explanation
In a geometric progression, each term is obtained by multiplying the previous term by a constant value called the common ratio. To find the first term and the common ratio, we can use the given information. We know that the third term is 32 and the fifth term is 128. From this, we can determine that each term is obtained by multiplying the previous term by the common ratio.

Let's denote the first term as "a" and the common ratio as "r".

The third term can be written as a * r^2 = 32.
The fifth term can be written as a * r^4 = 128.

Dividing the equation for the fifth term by the equation for the third term, we get (a * r^4) / (a * r^2) = 128/32. Simplifying this, we get r^2 = 4.

Taking the square root of both sides, we get r = 2.

Substituting this value of r into the equation for the third term, we get a * (2^2) = 32. Simplifying this, we get 4a = 32, which gives us a = 8.

Therefore, the first term is 8 and the common ratio is 2.

Explanation
Let's assume that Michael's painting time is x minutes. According to the given information, John alone can finish the wall in x + 15 minutes. Together, they can finish the wall in 18 minutes. So, their combined work rate is 1/18 of the wall per minute. From this, we can set up the equation 1/(x + 15) + 1/x = 1/18. By solving this equation, we find that x = 30. Therefore, Michael's painting time is 30 minutes, and John's painting time is 30 + 15 = 45 minutes.

Explanation
Let's assume Jimmy's age as x. According to the given information, Daniel is twice as old as Jimmy, so Daniel's age would be 2x. Terry is one year younger than Daniel, so Terry's age would be 2x - 1. The sum of their ages is 44, so we can set up the equation x + 2x + (2x - 1) = 44. Simplifying this equation, we get 5x - 1 = 44. Solving for x, we find x = 9. Therefore, Daniel's age is 2x = 2 * 9 = 18.

Explanation
Based on the given information, we can deduce that the number of nickels is twice the number of dimes and there is one more quarter than nickels. Let's assume the number of dimes is x. Therefore, the number of nickels is 2x and the number of quarters is 2x + 1. The total value of the coins can be calculated as follows: (0.05 * 2x) + (0.10 * x) + (0.25 * (2x + 1)) = 1.65. Solving this equation, we find that x = 2. Therefore, the number of quarters is 2(2) + 1 = 5.

Explanation
Let's assume the two positive numbers are x and y. We are given that x + y = 35 and xy = 304. To find the smallest number, we need to find the smaller value between x and y. We can solve the equations by substituting one variable in terms of the other. From the first equation, we have y = 35 - x. Substituting this into the second equation, we get x(35 - x) = 304. Simplifying this quadratic equation, we get x^2 - 35x + 304 = 0. Solving this equation, we find x = 16 or x = 19. Since we are looking for the smallest number, the answer is 16.

Explanation
The book is marked 20% off, so the original price of the book can be calculated by dividing the discounted price by 0.8. Let x be the original price of the book. We can set up the equation: x/0.8 = 21.56. Solving for x, we get x = 21.56 * 0.8 = 17.248. The sales tax is 8%, so the final price of the book is x + (x * 0.08) = 17.248 + (17.248 * 0.08) = 17.248 + 1.37984 = 18.62784. Rounding to the nearest cent, the final price is $18.63. Since this is not one of the given answer choices, the question is incomplete or not readable.

Explanation
To calculate the final grade, we need to consider the weightage of each component. The homework grade counts for 15%, the test grades count for 40% (20% each), and the final exam counts for 25%.

To find the score needed on the final exam, we can set up the equation:

0.15(97 + 99 + 100) + 0.4x + 0.25x = 90

Simplifying the equation, we get:

14.55 + 0.4x + 0.25x = 90

Combining like terms:

0.65x = 75.45

Solving for x:

x = 75.45 / 0.65 ≈ 116.08

Since the maximum score on the final exam is 100, the boy would need to score 100 to achieve a final grade of 90. Therefore, the answer of 63.2 is incorrect.

Explanation
The given series is an arithmetic progression with a common difference of 75. To find the sum of the first 50 terms, we can use the formula for the sum of an arithmetic series: Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the series, n is the number of terms, a is the first term, and d is the common difference. Plugging in the values, we get Sn = (50/2)(2(10) + (50-1)(75)) = 25(20 + 49(75)) = 25(20 + 3675) = 25(3695) = 92,375.

Explanation
To find the number of calculators to be produced, we need to calculate the profit per calculator. The profit per calculator is the selling price minus the production cost, which is $11 - $6 = $5.

To earn a total profit of $18000, we divide the total profit by the profit per calculator: $18000 / $5 = 3600.

However, we need to consider the overhead cost as well. The overhead cost is $150000.

To cover the overhead cost, we divide the overhead cost by the profit per calculator: $150000 / $5 = 30000.

So, the total number of calculators to be produced is the sum of the number of calculators needed to cover the profit and the overhead cost: 3600 + 30000 = 33600.

Therefore, the correct answer is 33,600.

Explanation
The sum of a geometric progression (G.P.) can be found using the formula: S = a * (1 - r^n) / (1 - r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms. In this case, the first term is 1, the common ratio is 2, and the number of terms is 20. Plugging these values into the formula, we get: S = 1 * (1 - 2^20) / (1 - 2) = 1 * (1 - 1,048,576) / -1 = 1,048,575. Therefore, the correct answer is 1,048,575.