This question asks for the height of a larger rectangular poster that is similar to a smaller one, given the smaller poster's dimensions of 18 inches wide and 12 inches tall, and the larger poster's width of 30 inches. Since the posters are similar, the ratio of their corresponding sides is constant, so set up the proportion of widths to heights as 18/12 = 30/h, or equivalently, the scale factor from smaller to larger is 30/18. Simplify the scale factor to 5/3, then multiply the smaller height by this factor: h = 12 * (5/3) = 20 inches. Alternatively, cross-multiply in the proportion 18/12 = 30/h to get 18h = 360, so h = 20 inches. A common error is mistakenly using the heights for the ratio instead of matching corresponding sides, which could lead to incorrect scaling. Another mistake might be adding dimensions instead of proportioning them, ignoring similarity. When dealing with similar figures, always confirm units are consistent and identify corresponding sides before setting up the proportion.

This question asks for the grams of salt in 260 grams of a solution that is 35% salt by mass. The percentage means salt/solution = 35/100 = 0.35, so set up the proportion 0.35 = s/260, where s is salt in grams. Solve by multiplying: s = 0.35 * 260 = 91 grams. Alternatively, convert percentage to decimal and scale directly, ensuring units are in grams throughout. A key error is using volume instead of mass or misplacing the decimal in percentage conversion. Another mistake could be subtracting instead of multiplying. In percentage problems, confirm units match (here, mass) and double-check decimal placement for accuracy.

This question asks for the length of a model car in meters, given a 1:18 scale (model to actual) and an actual car length of 4.5 meters. The scale means model/actual = 1/18, so set up the proportion 1/18 = m/4.5, where m is the model length. Solve by cross-multiplying: 18m = 4.5, so m = 4.5 / 18 = 0.25 meters. Alternatively, directly divide the actual length by 18: 4.5 / 18 = 0.25 meters, ensuring units are in meters. A key error is reversing the scale to 18:1, which would incorrectly enlarge the model. Another mistake could be multiplying instead of dividing. In scale models, confirm the ratio direction and unit consistency to get accurate dimensions.

This question asks for the liters of red paint to mix with 6 liters of blue paint to maintain a 5:8 red-to-blue ratio. The ratio means red/blue = 5/8, so set up the proportion 5/8 = r/6, where r is red paint in liters. Solve by cross-multiplying: 8r = 30, so r = 3.75 liters. Alternatively, scale the ratio by multiplying 8 by the factor to reach 6 (which is 6/8 = 0.75), then apply to red: 5 * 0.75 = 3.75 liters, keeping units in liters. A common error is inverting the ratio, like using 8:5, leading to more red than intended. Another mistake might be adding instead of proportioning. For mixing ratios, write the proportion clearly and check units to ensure the mixture stays consistent.

This question asks for the length of the leg in triangle B that corresponds to the 12 cm leg in similar triangle A, where triangle A has legs 9 cm and 12 cm, and triangle B's corresponding leg to 9 cm is 15 cm. Since the triangles are similar, the ratio of corresponding sides is constant, so set up the proportion 9/15 = 12/x, or use the scale factor 15/9 = 5/3. Multiply the smaller leg by the scale factor: x = 12 * (5/3) = 20 cm. Alternatively, cross-multiply in the proportion: 9x = 180, so x = 20 cm, matching units in cm. A key error is mismatching corresponding sides, such as using 12/15 directly. Another mistake could be adding ratios instead of multiplying. In similar figures, identify correspondences carefully and verify units to avoid setup errors.

This question asks for the number of cellos in a school orchestra with a total of 45 musicians, where the ratio of violins to cellos to flutes is 7:3:5. The total parts in the ratio are 7 + 3 + 5 = 15, so set up the proportion for cellos as 3/15 of the total musicians. Calculate one part by dividing 45 by 15, yielding 3 musicians per part, then multiply by 3 for cellos: 3 * 3 = 9. Alternatively, directly compute (3/15) * 45 = 9 cellos, ensuring the total adds up correctly. A common error is forgetting to sum the ratio parts, leading to incorrect fractions like 3/7. Another mistake might be distributing unevenly without proportioning. For multi-part ratios, always sum parts first and check if the total matches by adding individual counts.

This question asks for the actual distance in miles between two towns, given a map scale of 1 inch = 12 miles and a map distance of 3.75 inches. Set up the proportion based on the scale: 1 inch / 12 miles = 3.75 inches / d miles, where d is the actual distance. Solve by cross-multiplying: 1 * d = 3.75 * 12, so d = 45 miles. Alternatively, scale directly by multiplying the map distance by the miles per inch: 3.75 * 12 = 45 miles, keeping units consistent. A common error is inverting the scale, like using 12 inches per mile, which would yield a much smaller distance. Another mistake might be forgetting to multiply and just adding values. In scale problems, always verify the units in the proportion setup to ensure accurate conversion.

Because time is the same, speeds are proportional to distances: 15/6 times 8 is 20 mph. The other choices come from inverting the ratio, adding distances to the speed, or averaging incorrectly.

Let boys = 5k and girls = 7k; solving $(5k - 4)/(7k + 6) = 2/3$ gives k = 24, so total originally = 12k = 288. The distractors result from misapplying the ratio or halving totals.

Convert 9 miles ≈ 14.4 km; times are 7.2/6 + 7.2/12 = 1.8 hours = 108 minutes. 96 uses a simple average speed, 135 uses miles without converting, and 72 uses the faster speed for the whole trip.