The ratio of students wearing sneakers to not wearing sneakers is 9:7, with 63 students wearing sneakers. Set up the proportion: 9/7 = 63/x, where x is students not wearing sneakers. Cross-multiply: 9x = 441, so x = 49 students not wearing sneakers. A common error is using 63 as the total instead of just those wearing sneakers.

This is a ratios and unit conversion question requiring a two-step process. Choice B (24) is correct — first convert quarts to ounces: 5 quarts × 32 oz/quart = 160 oz of water. Then set up the proportion: 3 salt/20 water = x salt/160 water → x = (3 × 160)/20 = 480/20 = 24 oz of salt. Choice A (12) skips the unit conversion: 3/20 = x/5 → x = 0.75... or uses only half of 160: 3/20 × 80 = 12. Choice C (48) doubles the correct answer — perhaps computing 3/10 = x/160 (using 10 instead of 20). Choice D (75) treats 5 quarts as 5 × 5 = 25 units of something: 3/20 × 25 × ... or 5 × 15 = 75. Pro tip: Any problem mixing units requires conversion before setting up proportions. Convert 5 quarts to 160 ounces first, THEN apply the ratio. The ratio is salt:water = 3:20, so for every 20 oz of water, use 3 oz of salt — scale that relationship up to 160 oz.

The ratio of oil to vinegar is 2:5, and the chef uses 20 tablespoons of vinegar. Set up the proportion: 2/5 = x/20, where x is tablespoons of oil. Cross-multiply: 5x = 40, so x = 8 tablespoons of oil. A common error is reversing the ratio setup, which would give 50 tablespoons.

The ratio of cats to dogs is 4:5, and there are 30 dogs. Set up the proportion: 4/5 = x/30, where x is the number of cats. Cross-multiply: 5x = 120, so x = 24 cats. A common mistake is using the ratio backwards (5:4) which would incorrectly give 37.5 cats.

To find an equivalent ratio to 6:9, we need to simplify by finding the greatest common factor. The GCF of 6 and 9 is 3. Dividing both parts by 3: 6÷3 : 9÷3 = 2:3. We can verify: 2×3 = 6 and 3×3 = 9, confirming the equivalence. Option C (12:15) is also equivalent but not in simplest form.

With a ratio of cats to dogs of 4:3, we set up the proportion 4/3 = x/21 where x is the number of cats. Cross-multiplying gives us 3x = 4 × 21, so 3x = 84. Dividing both sides by 3, we get x = 28 cats. We can verify: 28:21 simplifies to 4:3 when divided by 7. A common mistake would be reversing the ratio or incorrectly setting up the proportion.

With a scale of 1:50, the model is 50 times smaller than the actual object. If the model measures 2 meters, the actual length is 2 × 50 = 100 meters. We can verify this by checking that 100/2 = 50, which matches our scale factor. A common error would be dividing instead of multiplying, which would give an unreasonably small actual measurement.

To solve $5/x = 10/15$, we first simplify the right side: $10/15 = 2/3$. So our equation becomes $5/x = 2/3$. Cross-multiplying gives us $2x = 5 \times 3$, so $2x = 15$. Dividing both sides by 2, we get $x = 7.5$. We can verify: $5/7.5 = 2/3$, which matches $10/15 = 2/3$. A common error would be not simplifying first or making arithmetic mistakes during cross-multiplication.

The scale factor is the ratio of corresponding dimensions from the smaller to larger rectangle. Comparing the lengths: 16/4 = 4, and comparing the widths: 32/8 = 4. Both ratios give us a scale factor of 4, confirming the rectangles are similar. This means each dimension of the larger rectangle is 4 times the corresponding dimension of the smaller one. A common mistake would be taking the reciprocal, giving 1/4 instead of 4.

With an apples to oranges ratio of 2:3 and 12 oranges, we set up the proportion 2/3 = x/12 where x is the number of apples. Cross-multiplying gives us 3x = 2 × 12, so 3x = 24. Dividing both sides by 3, we get x = 8 apples. We can verify: 8:12 simplifies to 2:3 when divided by 4. A common mistake would be reversing the ratio or incorrectly setting up the proportion.