The ratio of concentrate to water is 2:7. The total number of parts in the mixture is 2 + 7 = 9 parts. The fraction of the solution that is water is 7/9. To find the amount of water in 36 liters of solution, multiply the total volume by this fraction: (7/9) * 36 liters = 7 * 4 = 28 liters. The quantity in Column A is 28 liters, and the quantity in Column B is 28 liters. Therefore, the two quantities are equal.

First, calculate the height of the larger version by multiplying the replica's height by the scale factor: 5 inches * 2.5 = 12.5 inches. The question asks for the difference in height, so subtract the original height from the new height: 12.5 inches - 5 inches = 7.5 inches.

First, find the linear scaling factor, which is the ratio of the new side length to the original side length: 18 cm / 6 cm = 3. For Column B, the perimeter scales by the same factor as the side lengths. The ratio of the perimeters is 3. For Column A, the area scales by the square of the linear scaling factor. The ratio of the areas is $3^2$ = 9. Since 9 is greater than 3, the quantity in Column A is greater.

First, find the scaling factor for the recipe. The baker is making 42 muffins instead of 12, so the scaling factor is 42 / 12 = 3.5. Now, calculate the new amounts of flour and milk. New flour amount: 1.5 cups * 3.5 = 5.25 cups. New milk amount: 0.75 cups * 3.5 = 2.625 cups. Finally, find the difference: 5.25 - 2.625 = 2.625 cups. Alternatively, find the original difference (1.5 - 0.75 = 0.75 cups) and multiply it by the scaling factor (0.75 * 3.5 = 2.625 cups).

When you encounter percentage increase problems, you're working with the concept that a percentage increase means adding that percentage of the original value to the original value itself.

To find the new length, you need to calculate what 40% of 50 inches equals, then add it to the original 50 inches. First, convert 40% to a decimal: 40% = 0.40. Then multiply: $$0.40 \times 50 = 20$$ inches. The new length is $$50 + 20 = 70$$ inches.

Alternatively, you can use the shortcut method: a 40% increase means the new value is 140% of the original, or $$1.40 \times 50 = 70$$ inches. Since Column A equals 70 inches and Column B is also 70 inches, the quantities are equal.

Looking at why the other answers are wrong: Choice (A) suggests Column A is greater than 70, which would mean you either calculated the percentage increase incorrectly or added an extra step. Choice (B) suggests 70 is greater than the new length, which would happen if you mistakenly calculated a percentage decrease or used the wrong base number. Choice (C) implies insufficient information, but you have everything needed: the original length and the percentage increase.

Remember that "increased by X%" means the new value equals the original value times (1 + X%), where X is the percentage as a decimal. This formula works for any percentage increase problem and helps you avoid calculation errors.

For Column A, first find the price per foot of the rope: $6 / 4 feet = $1.50 per foot. Then, calculate the cost of 10 feet: 10 feet * $1.50/foot = $15.00. For Column B, calculate the cost directly: 3 feet * $5.50/foot = $16.50. Comparing the two quantities, $15.00 is less than $16.50. Therefore, the quantity in Column B is greater.

First, find the original area: 4 miles * 3 miles = 12 square miles. The original density is 240 rabbits / 12 sq mi = 20 rabbits/sq mi. Now, find the new dimensions and area. New length = 4 * 2 = 8 miles. New width = 3 * 2 = 6 miles. New area = 8 * 6 = 48 square miles. The new rabbit population is 240 * 2 = 480 rabbits. The new population density is 480 rabbits / 48 square miles = 10 rabbits per square mile.

First, find the car's fuel efficiency in miles per gallon (mpg). Efficiency = 168 miles / 6 gallons = 28 mpg. Next, to find the number of gallons needed for the new trip, divide the trip distance by the car's efficiency: 406 miles / 28 mpg = 14.5 gallons.

First, find the unit rate of the scale. If 2 inches = 70 miles, then 1 inch = 70 / 2 = 35 miles. Next, multiply this unit rate by the map distance to find the actual distance: 5 inches * 35 miles/inch = 175 miles.

The linear scaling factor from the small box to the large box is the ratio of their side lengths: 12 inches / 4 inches = 3. Volume scales by the cube of the linear scaling factor. Therefore, the ratio of the volumes is $3^3$ = 3 * 3 * 3 = 27. The volume of the larger box is 27 times the volume of the smaller box. Alternatively, one could calculate the volumes: Small Volume = $4^3$ = 64 cubic inches. Large Volume = $12^3$ = 1728 cubic inches. The ratio is 1728 / 64 = 27.