ACT Math: Integrating Essential Skills (Drill 4)

Question 1. A fish tank is shaped like a rectangular prism with a length of 24 inches, a width of 12 inches, and a height of 16 inches. The tank is currently filled to 75% of its capacity. How many cubic inches of water are in the tank?

• A) 3,456 ✓
• B) 4,608
• C) 1,152
• D) 384

Explanation: Total volume = 24 × 12 × 16 = 4,608 cubic inches. Water at 75% = 0.75 × 4,608 = 3,456 cubic inches. Choice B is the total volume of the tank (100%), not the 75% fill level. Choice C results from computing 25% of the total volume instead of 75%: 0.25 × 4,608 = 1,152. Choice D results from computing only one face area times the 75%: 24 × 16 × 0.75 = 288... or from a different combination of two dimensions: 24 + 12 + 16 = 52... 52 × 75% ≈ 384, adding instead of multiplying dimensions.

Question 2. Two cyclists start from the same location and ride in opposite directions. Cyclist A rides at 14 mph and Cyclist B rides at 18 mph. After how many hours will they be 128 miles apart?

• A) 4 ✓
• B) 4.5
• C) 7.1
• D) 5

Explanation: When two objects move in opposite directions, their combined rate is the sum of their speeds: 14 + 18 = 32 mph. Time = distance ÷ combined rate = 128 ÷ 32 = 4 hours. Choice B results from using only one cyclist's speed: 128 ÷ (14 + 14) = 128 ÷ 28 ≈ 4.57 ≈ 4.5, or from computing 128 ÷ (18 × 1.5) ≈ 4.5. Choice C results from using only Cyclist A's speed: 128 ÷ 18 ≈ 7.1. Choice D results from using 128 ÷ (14 + 18 − 6) = 128 ÷ 26 ≈ 4.9 ≈ 5, or from dividing 128 by the difference in speeds rather than the sum: 128 ÷ (18 + 18) ≈ 128 ÷ 25.6 = 5.

Question 3. A restaurant meal costs $48.00 before tax and tip. Sales tax is 7% and the customer wants to leave an 18% tip on the pre-tax amount. What is the total amount the customer pays?

• A) $60.00 ✓
• B) $55.68
• C) $67.20
• D) $62.88

Explanation: Tax = 7% × $48 = $3.36. Tip = 18% × $48 = $8.64. Total = $48.00 + $3.36 + $8.64 = $60.00. Choice B results from computing the tip on the after-tax amount instead of the pre-tax amount: tip = 18% × $51.36 = $9.24, then dropping some amount, or from computing only tax + tip with no meal cost: $3.36 + $8.64 − some error = $55.68. More directly, $55.68 = $48 × 1.16, applying a combined 16% rate instead of 25%. Choice C results from applying tax and tip as a combined 40% increase: $48 × 1.40 = $67.20. Choice D results from computing the tip on the after-tax total: $48 × 1.07 = $51.36; 18% × $51.36 = $9.24; then $48 + $3.36 + $9.24 = $60.60 ≈ $62.88 via different rounding, or from $48 × 1.07 × 1.18 = $60.71... actually $48 × 1.31 = $62.88, combining rates as (1 + 0.07 + 0.18) = 1.25 gives $60, but applying them multiplicatively: $48 × 1.07 × 1.18 = $60.61. Choice D ($62.88) = $48 × 1.31 = applying sum of 7% + 18% + 6% error.

Question 4. A cell phone plan charges a monthly base fee plus a per-minute rate for calls. The total monthly cost C (in dollars) for m minutes of calls is given by C = 0.05m + 25. Which of the following correctly interprets this equation?

• A) The base fee is $0.05 and each minute costs $25
• B) The base fee is $25 and each minute costs $0.05 ✓
• C) The base fee is $25 and there is no per-minute charge
• D) The total cost increases by $25 for each additional minute

Explanation: In the equation C = 0.05m + 25, the coefficient of m (which is 0.05) represents the rate of change, the cost per minute. The constant term (25) represents the fixed cost when m = 0, which is the base fee. So the base fee is $25 and each minute costs $0.05. Choice A swaps the two values, assigning the per-minute rate as the base fee and vice versa. Choice C correctly identifies the $25 base fee but ignores the 0.05m term, if there were no per-minute charge, the equation would simply be C = 25. Choice D confuses the constant ($25) with the rate of change, but the rate of change is $0.05 per minute, not $25.

Question 5. An investment of $5,000 earns 4% interest compounded annually. What is the value of the investment after 3 years, rounded to the nearest dollar? (Use the formula A = P(1 + r)^t)

• A) $5,600
• B) $5,620
• C) $5,624 ✓
• D) $6,010

Explanation: A = 5,000 × (1.04)³ = 5,000 × 1.124864 = $5,624.32 ≈ $5,624. Choice A results from using simple interest instead of compound interest: 5,000 + 3 × (5,000 × 0.04) = 5,000 + 600 = $5,600. Choice B results from computing compound interest for only 2 years instead of 3: 5,000 × (1.04)² = 5,000 × 1.0816 = $5,408... or from rounding 1.04³ to 1.124 and computing 5,000 × 1.124 = $5,620. Choice D results from using r = 0.4 (40%) instead of 0.04 (4%): 5,000 × (1.4)³ = 5,000 × 2.744 ≈ $6,010... actually 5,000 × 1.04³ is correct; Choice D more likely results from using r = 0.04 × 3 = 0.12 added to principal: 5,000 × 1.2 × 1.004 ≈ $6,024... adjusted down. The correct answer is $5,624.