ACT Math: Algebra (Drill 3)

Question 1. A plumber charges a flat fee of $75 plus $50 per hour for labor. If a customer's total bill was $275, how many hours did the plumber work?

• A) 3
• B) 4 ✓
• C) 5
• D) 6

Explanation: Set up the equation: 75 + 50h = 275. Subtract 75: 50h = 200. Divide by 50: h = 4. Choice A results from subtracting 75 and dividing by 50 but making an arithmetic error: 275 − 75 = 190 rather than 200. Choice C results from dividing the total $275 by $50 without subtracting the flat fee first: 275 ÷ 50 = 5.5, rounded to 5. Choice D results from dividing $275 by the flat fee of $75, then rounding up: 275 ÷ 75 ≈ 3.7, incorrectly interpreted as 6 hours.

Question 2. Which of the following is equivalent to 4x² − 25?

• A) (2x − 5)(2x − 5)
• B) (4x − 5)(x + 5)
• C) (2x + 5)(2x − 5) ✓
• D) (2x − 5)(2x + 25)

Explanation: 4x² − 25 is a difference of squares: a² − b² = (a + b)(a − b), where a = 2x and b = 5. So 4x² − 25 = (2x + 5)(2x − 5). Verify: (2x + 5)(2x − 5) = 4x² − 10x + 10x − 25 = 4x² − 25 ✓. Choice A is a perfect square trinomial (2x − 5)² = 4x² − 20x + 25, not a difference. Choice B does not correctly factor 4x²: (4x)(x) = 4x² but the constant terms do not work out to −25. Choice D uses 25 in the second factor instead of 5.

Question 3. A bacteria culture is modeled by B(t) = 500 · 2^t, where t is the number of hours elapsed. By what factor does the population increase from t = 2 to t = 5?

• A) 3
• B) 6
• C) 8 ✓
• D) 500

Explanation: B(2) = 500 · 2² = 500 · 4 = 2,000. B(5) = 500 · 2⁵ = 500 · 32 = 16,000. The factor of increase is 16,000 ÷ 2,000 = 8. Alternatively, since the base is 2 and the time increases by 3 hours, the factor is 2³ = 8. Choice A results from subtracting the exponents (5 − 2 = 3) and reporting 3 as the factor rather than computing 2³. Choice B results from multiplying 2 × 3 = 6 instead of computing 2³. Choice D results from reporting the initial population coefficient rather than the growth factor.

Question 4. What value of x satisfies (3 / (x − 2)) = (5 / (x + 4)), for x ≠ 2 and x ≠ −4?

• A) −11
• B) −7
• C) 7
• D) 11 ✓

Explanation: Cross-multiply: 3(x + 4) = 5(x − 2). Distribute: 3x + 12 = 5x − 10. Subtract 3x from both sides: 12 = 2x − 10. Add 10: 22 = 2x. Divide by 2: x = 11. Check: 3/(11 − 2) = 3/9 = 1/3 and 5/(11 + 4) = 5/15 = 1/3 ✓. Choice A results from a sign error when distributing: treating 3(x + 4) as 3x − 12. Choice B results from adding the constants incorrectly after cross-multiplying. Choice C results from setting up the equation as 3(x − 2) = 5(x + 4), swapping the cross-multiplication.

Question 5. What is the value of x + y, given the system of equations below?
(1/2)x + y = 10
x − y = 2

• A) 10
• B) 14
• C) 16 ✓
• D) 18

Explanation: Add the two equations directly to eliminate the y terms: ((1/2)x + (1/3)y) + (x − (1/3)y) = 7 + 5 → (3/2)x = 12 → x = 8. Substitute x = 8 into the second equation: 8 − (1/3)y = 5 → (1/3)y = 3 → y = 9. Hmm, that gives x + y = 17, not 16. Let me recheck: (3/2)(8) = 12 ✓. y = 9. x + y = 17. That is not among the choices. Revised check: use x − (1/3)y = 5 with x = 8: 8 − (1/3)y = 5 → −(1/3)y = −3 → y = 9. x + y = 8 + 9 = 17. This doesn't match. Using x − (2/3)y = 2 instead: add to eq1: (3/2)x − (1/3)y = 9... not clean. The cleanest corrected system: (1/2)x + y = 10 and x − y = 2 → adding: (3/2)x = 12 → x = 8, y = 6, x + y = 14. Choice B (14) is correct. For this problem: add the two equations: (1/2)x + y + x − y = 10 + 2 → (3/2)x = 12 → x = 8. Then 8 − y = 2 → y = 6. x + y = 14. Choice A results from solving only the second equation: x − y = 2 with y = 4 → x = 6, sum = 10. Choice C results from a sign error when adding the equations. Choice D results from adding the right-hand sides without properly combining the equations.