For similar figures, the ratio of areas equals the square of the ratio of corresponding sides. The side ratio is 3:5, so the area ratio is 3²:5² = 9:25. If the smaller triangle has area 36, then 36/larger area = 9/25. Cross multiply: 9 × larger area = 36 × 25 = 900, so larger area = 100. Choice A uses the linear ratio 3:5. Choice B incorrectly calculates. Choice D doubles the correct answer.

Let the initial number of boys be 5x and girls be 7x. After 6 girls join: boys = 5x, girls = 7x + 6. The new ratio 5x:(7x + 6) = 5:8. Cross multiply: 8(5x) = 5(7x + 6), so 40x = 35x + 30, giving 5x = 30. Therefore, there are 30 boys. Choice B assumes x = 5. Choice C assumes x = 7. Choice D assumes x = 4.

In the ratio 3:2:1, chocolate chips represent 1 part. With 1.2 pounds of chocolate chips, each part equals 1.2 pounds. Nuts represent 3 parts, so nuts needed = 3 × 1.2 = 3.6 pounds. Choice A uses 2 parts instead of 3. Choice C incorrectly multiplies 1.2 by 4. Choice D uses 1.5 parts.

Let current ages be 2x, 3x, and 5x. In 4 years: (2x+4), (3x+4), (5x+4). The ratio becomes 3:4:6, so (2x+4):(3x+4):(5x+4) = 3:4:6. From the first two terms: (2x+4)/(3x+4) = 3/4. Cross multiply: 4(2x+4) = 3(3x+4), so 8x+16 = 9x+12, giving x = 4. The youngest is 2x = 8 years old. Choice B uses x directly. Choice C uses 2.5x. Choice D uses x incorrectly.

When you see gears with "inversely proportional" speeds, you're dealing with a fundamental mechanical relationship: larger gears turn slower, smaller gears turn faster. This inverse relationship means that as one quantity increases, the other decreases proportionally.

Set up the inverse proportion using the formula: $$r_1 \times \text{speed}_1 = r_2 \times \text{speed}_2$$, where the product of radius and rotational speed remains constant for connected gears.

Given the 6 cm gear rotating at 150 rpm, and needing the speed of the 4 cm gear:

$$6 \times 150 = 4 \times \text{speed}_2$$

$$900 = 4 \times \text{speed}_2$$

$$\text{speed}_2 = 225 \text{ rpm}$$

The smaller gear rotates at 225 rpm, confirming answer D.

Looking at the wrong answers: A (200 rpm) likely comes from incorrectly using direct proportion, thinking smaller radius means proportionally slower speed. B (240 rpm) might result from calculation errors or misapplying the ratio. C (180 rpm) could come from confusing which gear is which or making arithmetic mistakes in the inverse relationship.

Remember this key pattern: for inverse proportions, multiply the given values to find the constant, then divide by the new variable to find your answer. On gear problems specifically, always check that your answer makes intuitive sense—smaller gears should spin faster than larger ones when connected.

This is a rate problem that tests your ability to set up and solve proportions. When you see a question giving you one rate and asking for another quantity at the same rate, think proportion or unit rate.

First, find the car's fuel efficiency (miles per gallon): $$180 \text{ miles} ÷ 6 \text{ gallons} = 30 \text{ miles per gallon}$$. Now you can find how many gallons are needed for 420 miles: $$420 \text{ miles} ÷ 30 \text{ miles per gallon} = 14 \text{ gallons}$$.

Alternatively, you can set up a proportion: $$\frac{180 \text{ miles}}{6 \text{ gallons}} = \frac{420 \text{ miles}}{x \text{ gallons}}$$. Cross-multiply: $$180x = 420 × 6 = 2520$$, so $$x = 2520 ÷ 180 = 14$$ gallons.

Choice A (12 gallons) might result from incorrectly calculating the unit rate as 15 miles per gallon instead of 30, then dividing 420 by 35. Choice B (16 gallons) could come from setting up the proportion incorrectly or making arithmetic errors in cross-multiplication. Choice C (15 gallons) might result from dividing 420 by 28 instead of 30, suggesting a calculation error in finding the unit rate.

For rate problems, always identify what stays constant (the rate itself) and what changes (the quantities). Calculate the unit rate first—it makes the math cleaner and helps you catch errors. Double-check by multiplying your answer by the unit rate to see if you get back to 420 miles.