1Comparing Volumes from Density
A scientist takes two samples of Earth's crust, each with a mass of 3,240 grams. Sample A has a density of 3.0 grams per cubic centimeter, and sample B has a density of 2.7 grams per cubic centimeter. How much greater is the volume, in cubic centimeters, of sample B than sample A?
Answer Explanation
Find each volume using volume = mass ÷ density. Sample A: 3,240 ÷ 3.0 = 1,080 cm³. Sample B: 3,240 ÷ 2.7 = 1,200 cm³.
The difference is 1,200 − 1,080 = 120 cm³.
2Finding a Unit Rate
The walls of a small apartment were covered using exactly 3 gallons of paint. The paint was spread uniformly over a total area of 960 square feet. What was the rate, in gallons per square foot, at which the paint was used?
- A) 1/960
- B) 1/320
- C) 320
- D) 960
Answer Explanation
Divide the gallons used by the area covered: 3 ÷ 960.
3/960 simplifies to 1/320. The correct answer is B.
3Converting a Squared-Unit Rate
The area of a rectangular region is increasing at a rate of 20 square feet per hour. Which of the following is closest to this rate in square meters per minute? (Use 1 meter = 3.28 feet)
- A) 0.03
- B) 0.10
- C) 1.09
- D) 2.03
Answer Explanation
Convert feet to meters by squaring the conversion factor (since the unit is squared): 1 sq ft = (1/3.28)² sq m ≈ 0.0930 sq m. Then convert hours to minutes by dividing by 60.
20 × 0.0930 ÷ 60 ≈ 0.031, which rounds to 0.03. The correct answer is A.
4Converting a Speed
The speed of a white-throated needletail, a type of bird, in flight was measured to be 49 miles per hour. What was the white-throated needletail's measured speed, in kilometers per hour? (Use 1 mile = 1.6 kilometers)
- A) 30.6
- B) 47.4
- C) 50.6
- D) 78.4
Answer Explanation
Multiply by the conversion factor: 49 × 1.6.
49 × 1.6 = 78.4. The correct answer is D.
5Calculating an Average Speed
An object traveled 30 miles in 2 hours. What was the average speed, in miles per hour, of the object?
Answer Explanation
Divide distance by time: 30 miles ÷ 2 hours.
That gives 15 miles per hour.
6Converting a Squared-Unit Rate
An object's speed is increasing at a rate of 6.80 meters per second squared. What is this rate in miles per minute squared, rounded to the nearest tenth? (Use 1 mile = 1,609 meters)
Answer Explanation
Convert meters to miles (divide by 1,609) and convert per-second-squared to per-minute-squared (multiply by 60² = 3,600, since a rate per larger time units grows larger when the squared unit is bigger).
6.80 × 3,600 ÷ 1,609 ≈ 15.2.
7Converting Volume Units in a Cost Problem
A raised garden is in the shape of a right rectangular prism. Its base has a width of 3 feet and a length of 27 feet, and it will be filled 24 inches high with topsoil. The total cost of the topsoil needed to fill the garden to this height is 198 dollars. What is the unit cost, in dollars, per cubic yard, for the topsoil? (1 yard = 3 feet; 1 foot = 12 inches)
Answer Explanation
Convert the height to feet: 24 inches = 2 feet. Find the volume: 3 × 27 × 2 = 162 cubic feet. Convert to cubic yards: since 1 cubic yard = 3³ = 27 cubic feet, 162 ÷ 27 = 6 cubic yards.
Divide the total cost by the volume: $198 ÷ 6 = 33 dollars per cubic yard.
8Converting a Squared-Unit Area
A certain town has an area of 3.99 square miles. What is the area, in square yards, of this town? (1 mile = 1,760 yards)
- A) 441
- B) 7,022
- C) 776,341
- D) 12,359,424
Answer Explanation
Since the unit is squared, square the conversion factor: 1 square mile = 1,760² = 3,097,600 square yards.
3.99 × 3,097,600 ≈ 12,359,424. The correct answer is D.
9Converting a Squared-Unit Rate
The area of a rectangular region is increasing at a rate of 260 square feet per hour. Which of the following is closest to this rate in square meters per minute? (Use 1 meter = 3.28 feet.)
- A) 0.40
- B) 1.32
- C) 14.21
- D) 24.17
Answer Explanation
Convert feet to meters by squaring the conversion factor: 1 sq ft = (1/3.28)² sq m ≈ 0.0930 sq m. Then divide by 60 to convert hours to minutes.
260 × 0.0930 ÷ 60 ≈ 0.40. The correct answer is A.
10Converting a Depth Measurement
A remotely operated vehicle (ROV) is used for underwater ocean research. When the ROV is at a depth of 41 fathoms, what is the ROV's depth in feet? (1 fathom = 6 feet)
Answer Explanation
Multiply by the conversion factor: 41 × 6.
That gives 246 feet.
11Converting a Squared-Unit Rate
An object's speed is increasing at a rate of 3.9 meters per second squared. What is this rate, in miles per minute squared, rounded to the nearest tenth? (Use 1 mile = 1,609 meters)
- A) 0.1
- B) 8.7
- C) 104.6
- D) 412.6
Answer Explanation
Convert meters to miles (divide by 1,609) and convert per-second-squared to per-minute-squared (multiply by 3,600).
3.9 × 3,600 ÷ 1,609 ≈ 8.7. The correct answer is B.
12Converting Volume Units in a Cost Problem
A raised garden is in the shape of a right rectangular prism. Its base has a width of 3 feet and a length of 27 feet, and it will be filled 24 inches high with topsoil. The total cost of the topsoil needed to fill the garden to this height is 234 dollars. What is the unit cost, in dollars per cubic yard, for the topsoil? (1 yard = 3 feet; 1 foot = 12 inches)
Answer Explanation
As before, the garden's volume is 3 × 27 × 2 (feet, with 24 inches converted to 2 feet) = 162 cubic feet, which converts to 162 ÷ 27 = 6 cubic yards.
Divide the total cost by the volume: $234 ÷ 6 = 39 dollars per cubic yard.
13Converting a Squared-Unit Rate
An object's speed is increasing at the rate of 12.1 meters per second squared. What is this rate, in miles per minute squared, rounded to the nearest tenth? (Use 1 mile = 1,609 meters.)
- A) 0.5
- B) 27.1
- C) 133
- D) 324.5
Answer Explanation
Convert meters to miles (divide by 1,609) and convert per-second-squared to per-minute-squared (multiply by 3,600).
12.1 × 3,600 ÷ 1,609 ≈ 27.1. The correct answer is B.
14Converting a Squared-Unit Rate
An object's speed is increasing at a rate of 5.70 meters per second squared. What is this rate in miles per minute squared, rounded to the nearest tenth? (Use 1 mile = 1,609 meters.)
Answer Explanation
Convert meters to miles (divide by 1,609) and convert per-second-squared to per-minute-squared (multiply by 3,600).
5.70 × 3,600 ÷ 1,609 ≈ 12.8.
15Converting a Linear Measurement
How many centimeters are equivalent to 35 meters? (1 meter = 100 centimeters)
Answer Explanation
Multiply by the conversion factor: 35 × 100.
That gives 3500 centimeters.
16Converting a Squared-Unit Rate
An object's speed is increasing at a rate of 1.1 meters per second squared. What is this rate in miles per minutes squared, round to the nearest tenth? (Use 1 mile = 1,609 meters.)
- A) 0
- B) 29.5
- C) 2.5
- D) 1462.7
Answer Explanation
Convert meters to miles (divide by 1,609) and convert per-second-squared to per-minute-squared (multiply by 3,600).
1.1 × 3,600 ÷ 1,609 ≈ 2.5. The correct answer is C.
17Converting a Squared-Unit Rate
The area of a rectangular region is increasing at a rate of 280 square feet per hour. Which of the following is closest to this rate in square meters per minute? (Use 1 meter = 3.28 feet.)
- A) 0.43
- B) 1.42
- C) 15.31
- D) 26.03
Answer Explanation
Convert feet to meters by squaring the conversion factor: 1 sq ft = (1/3.28)² sq m ≈ 0.0930 sq m. Then divide by 60 to convert hours to minutes.
280 × 0.0930 ÷ 60 ≈ 0.43. The correct answer is A.
18Converting a Speed
An object moves at a speed of 3/50 feet per second. What is this speed, in yards per second? (3 feet = 1 yard)
Answer Explanation
Divide by the conversion factor, since converting from a smaller unit (feet) to a larger one (yards) reduces the numeric value: (3/50) ÷ 3.
(3/50) ÷ 3 = 3/150 = 1/50. The correct answer is D.
