Let the measure of the first angle be \(x\). Then the second angle is \(2x\) and the third angle is \(x + 20\). The sum of the angles in a triangle is 180 degrees. So, \(x + 2x + (x + 20) = 180\). Combine like terms: \(4x + 20 = 180\). Subtract 20 from both sides: \(4x = 160\). Divide by 4: \(x = 40\). The angles are \(x=40\) degrees, \(2x = 2(40) = 80\) degrees, and \(x + 20 = 40 + 20 = 60\) degrees. The largest of these is 80 degrees.

Let \(s\) be the number of sales employees and \(p\) be the number of production employees. The total number of employees is 45. There are 8 in administration, so \(s + p + 8 = 45\). This means \(s + p = 37\). We are also told that \(s = 2p + 1\). Substitute this into the previous equation: \((2p + 1) + p = 37\). Combine like terms: \(3p + 1 = 37\). Subtract 1: \(3p = 36\). Divide by 3: \(p = 12\). There are 12 employees in production. To find the number of sales employees, use \(s = 2p + 1\): \(s = 2(12) + 1 = 24 + 1 = 25\). There are 25 employees in the sales department.

First, distribute on both sides of the equation: \(6x - 8 = 7x - 7\). Next, subtract \(6x\) from both sides to get \(-8 = x - 7\). Finally, add 7 to both sides to isolate \(x\): \(-8 + 7 = x\), which simplifies to \(x = -1\).

Let \(t\) be the number of ride tickets Marco bought. The total cost is the sum of the entrance fee and the cost of the tickets. The equation is \(2t + 8 = 34\). First, subtract the $8.00 entrance fee from the total amount spent: \(2t = 34 - 8\), which is \(2t = 26\). Then, divide the remaining amount by the cost per ticket to find the number of tickets: \(t = 26 / 2 = 13\). Marco bought 13 tickets.

Let \(C\) be the total capacity of the tank. The initial amount of water is \(\frac{1}{4}C\). After adding 10 gallons, the amount is \(\frac{1}{4}C + 10\), which is equal to \(\frac{2}{3}C\). The equation is \(\frac{1}{4}C + 10 = \frac{2}{3}C\). To solve for \(C\), subtract \(\frac{1}{4}C\) from both sides: \(10 = \frac{2}{3}C - \frac{1}{4}C\). Find a common denominator (12): \(10 = \frac{8}{12}C - \frac{3}{12}C\), which simplifies to \(10 = \frac{5}{12}C\). Multiply by \(\frac{12}{5}\) to isolate \(C\): \(C = 10 \times \frac{12}{5} = \frac{120}{5} = 24\). The capacity is 24 gallons.

To find the value of \(k\), substitute \(x = 5\) into the equation. The equation becomes \(3(5 - 2) + k = 4(5) - 1\). Simplify both sides: \(3(3) + k = 20 - 1\), which is \(9 + k = 19\). To solve for \(k\), subtract 9 from both sides: \(k = 19 - 9 = 10\).

This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 12x + 5 = 53, where the unknown variable is isolated by subtracting 5 from both sides to get 12x = 48 and then dividing by 12 to find x = 4. The correct answer, choice A, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice B is incorrect due to a common arithmetic error where students forget to subtract and divide 53 by something to get x = 48. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.

This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 3x = 150, where the unknown variable is isolated by dividing both sides by 3 to find x = 50. The correct answer, choice C, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice A is incorrect due to a common arithmetic error where students multiply instead of dividing, such as 3 * 150 = 450. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.

This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 8x = 48, where the unknown variable is isolated by dividing both sides by 8 to find x = 6. The correct answer, choice A, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice B is incorrect due to a common arithmetic error where students multiply instead of dividing, such as 8 * 5 = 40. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.

This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 4x = 24, where the unknown variable is isolated by dividing both sides by 4 to find x = 6. The correct answer, choice B, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice A is incorrect due to a common arithmetic error where students multiply instead of dividing, such as 4 * 7 = 28. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.