First, calculate the cost of buying 6 individual pens: 6 * $1.50 = $9.00. The cost of the pack is $7.20. The total savings is $9.00 - $7.20 = $1.80. The question asks for the savings per pen. To find this, divide the total savings by the number of pens in the pack: $1.80 / 6 = $0.30. So, you save $0.30 per pen.

Set up a proportion to solve this problem. Let x be the actual distance in miles. The ratio of inches to miles is constant: 2 inches / 5 miles = 4.5 inches / x miles. Cross-multiply: 2x = 5 * 4.5. 2x = 22.5. Divide by 2: x = 11.25. The actual distance is 11.25 miles.

Let the number of cows be 5x and the number of sheep be 6x. The number of brown cows is (1/3) of the total cows, so (1/3) * 5x = (5/3)x. The number of brown sheep is (1/2) of the total sheep, so (1/2) * 6x = 3x. The ratio of brown cows to brown sheep is (5/3)x : 3x. The 'x' cancels out, leaving the ratio (5/3) : 3. To eliminate the fraction, multiply both sides of the ratio by 3: (5/3)3 : 33, which simplifies to 5:9.

The number of rotations of two connected gears is inversely proportional to their number of teeth. The ratio of the number of teeth is 48:36, which simplifies to 4:3. Therefore, the ratio of their rotations (larger gear to smaller gear) will be the inverse, 3:4. Let R_L be the rotations of the large gear and R_S be the rotations of the small gear. R_L / R_S = 3 / 4. We are given R_L = 15. So, 15 / R_S = 3 / 4. Cross-multiply: 3 * R_S = 15 * 4 = 60. R_S = 60 / 3 = 20. The smaller gear makes 20 rotations.

First, calculate the number of gallons of gas used on the trip. Gallons used = Total miles / miles per gallon = 350 miles / 28 miles/gallon = 12.5 gallons. The tank started with 15 gallons. The amount of gas remaining is 15 - 12.5 = 2.5 gallons. The question asks for the fraction of the tank that is full. This is the remaining gas divided by the tank's capacity: 2.5 gallons / 15 gallons. To simplify this fraction, you can write it as 25/150. Divide both the numerator and the denominator by 25: 25/25 = 1 and 150/25 = 6. The fraction is 1/6.

First, convert the total leakage to milliliters: 12 liters * 1000 mL/liter = 12,000 mL. Next, find the total time in hours for this leakage: 12,000 mL / 200 mL/hour = 60 hours. Finally, convert the hours to days. There are 24 hours in a day. 60 hours / 24 hours/day = 2.5 days. Since the question asks for the number of full days it will take, the leak will be completed during the 3rd day. So, it will take 3 full days to ensure at least 12 liters have leaked.

When you encounter problems comparing volumes and surface areas of similar shapes, remember that these measurements scale differently. Volume scales with the cube of the linear dimension, while surface area scales with the square.

Since the volume ratio of Cube A to Cube B is 64:27, you can find the ratio of their edge lengths. If the edge of Cube A is $s_A$ and the edge of Cube B is $s_B$, then:

$$\frac{s_A^3}{s_B^3} = \frac{64}{27}$$

Taking the cube root of both sides: $$\frac{s_A}{s_B} = \frac{4}{3}$$

Now you can find the surface area ratio. Since each cube has 6 faces and surface area is $6s^2$:

$$\frac{\text{Surface Area of A}}{\text{Surface Area of B}} = \frac{6s_A^2}{6s_B^2} = \frac{s_A^2}{s_B^2} = \left(\frac{4}{3}\right)^2 = \frac{16}{9}$$

Choice A (4:3) gives you the ratio of edge lengths, not surface areas. Choice B (8:9) appears to mix up the relationship and gets the wrong values entirely. Choice C (8:3) incorrectly applies the relationship between volume and surface area scaling. Choice D (16:9) correctly squares the edge length ratio to get the surface area ratio.

The key insight is recognizing the scaling relationships: when linear dimensions change by a factor of $k$, areas change by $k^2$ and volumes by $k^3$. Always work backward from the given ratio to find the linear scale factor first.

First, find the rate in hours per room. The rate is 5 hours / 2 rooms = 2.5 hours per room. To paint 3 rooms, it will take 3 rooms * 2.5 hours/room = 7.5 hours. The question asks for the time in minutes. Convert hours to minutes: 7.5 hours * 60 minutes/hour = 450 minutes.

The ratio of red to blue is 3:4. The total number of parts is 3 + 4 = 7. The total amount of paint needed is 21 gallons. The amount of red paint needed is (3/7) * 21 = 9 gallons. The amount of blue paint needed is (4/7) * 21 = 12 gallons. Now calculate the cost. Cost of red paint = 9 gallons * $20/gallon = $180. Cost of blue paint = 12 gallons * $15/gallon = $180. The total cost is $180 + $180 = $360.

First, find the original number of security guards. Let S be the number of guards. The ratio is S / 1260 = 3 / 140. To solve for S, multiply both sides by 1260: S = (3 * 1260) / 140. S = (3 * 9) = 27. So there were originally 27 guards. Then, 4 additional guards are hired, making the new total 27 + 4 = 31 guards. The number of fans remains 1,260. The new ratio is 31 guards to 1,260 fans, or 31:1260. This ratio cannot be simplified.