Solving Proportions - SHSAT Math

1. A recipe that serves 6 people calls for $\frac{3}{4}$ cup of sugar. How many cups of sugar are needed to serve 16 people using this recipe?
   Answer: $2$ cups of sugar needed Set up proportion: $\frac{\frac{3}{4} \text{ cup}}{6 \text{ people}} = \frac{x \text{ cups}}{16 \text{ people}}$. Cross multiply: $\frac{3}{4} \times 16 = 6x$, so $12 = 6x$, giving $x = 2$ cups. Choice A results from using 12 people instead of 16. Choice B comes from incorrectly calculating $\frac{3}{4} \times \frac{16}{6} = \frac{3 \times 16}{4 \times 6} = \frac{48}{24} = 2$ but making an arithmetic error. Choice D assumes the recipe needs to be scaled by $\frac{16}{6} = \frac{8}{3}$ and calculates $\frac{3}{4} \times \frac{8}{3} = 2$ but adds $\frac{1}{4}$ incorrectly.
2. Solve for $x$ in the proportion $\dfrac{7}{x}=\dfrac{14}{20}$.
   Answer: 10 When you encounter a proportion like this, you're dealing with two equivalent ratios that you can solve using cross-multiplication. A proportion states that two fractions are equal, so $\frac{7}{x} = \frac{14}{20}$. To solve this, cross-multiply by multiplying the numerator of each fraction by the denominator of the other fraction: $7 \times 20 = 14 \times x$. This gives you $140 = 14x$. Now divide both sides by 14 to isolate $x$: $x = \frac{140}{14} = 10$. You can verify this by substituting back: $\frac{7}{10} = \frac{14}{20}$. Simplifying the right side: $\frac{14}{20} = \frac{7}{10}$ ✓ Looking at the wrong answers: Choice B (8) would give you $\frac{7}{8} = \frac{14}{20}$, but $\frac{7}{8} = 0.875$ while $\frac{14}{20} = 0.7$, so these aren't equal. Choice C (4) would make $\frac{7}{4} = \frac{14}{20}$, but $\frac{7}{4} = 1.75$ while $\frac{14}{20} = 0.7$. Choice D (2) gives $\frac{7}{2} = \frac{14}{20}$, but $\frac{7}{2} = 3.5$ while $\frac{14}{20} = 0.7$. The answer is A (10). Strategy tip: Always cross-multiply when solving proportions—it's the most reliable method. After finding your answer, quickly substitute it back into the original proportion to check that both sides are equal. This catches calculation errors and builds confidence.
3. A car travels 180 miles in 3 hours at a constant speed. At this same rate, how many hours will it take to travel an additional 420 miles?
   Answer: 7 hours exactly First find the speed: $\frac{180 \text{ miles}}{3 \text{ hours}} = 60$ mph. Then use proportion for additional distance: $\frac{60 \text{ miles}}{1 \text{ hour}} = \frac{420 \text{ miles}}{x \text{ hours}}$. Cross multiply: $60x = 420$, so $x = 7$ hours. Choice A incorrectly uses total distance 600 miles. Choice C results from using 450 miles instead of 420. Choice D comes from miscalculating the initial speed as 52.5 mph.
4. A printer can print 240 pages in 8 minutes. At this constant rate, how long will it take to print a document that has 420 pages?
   Answer: 14 minutes exactly Set up proportion: $\frac{240 \text{ pages}}{8 \text{ minutes}} = \frac{420 \text{ pages}}{x \text{ minutes}}$. Cross multiply: $240x = 420 \times 8 = 3360$, so $x = \frac{3360}{240} = 14$ minutes exactly. Choice B results from miscalculating $\frac{3360}{240}$ as 14.5 minutes. Choice C comes from using 450 pages instead of 420. Choice D incorrectly calculates the printing rate as 28 pages per minute instead of 30.
5. A machine produces widgets at a constant rate. In 45 minutes, it produces 180 widgets. At this rate, how many widgets will it produce in 2 hours and 15 minutes?
   Answer: 540 widgets produced This is a rate problem that tests your ability to work with proportional relationships and unit conversions. When you see "constant rate" problems, you're looking for widgets per unit time. First, find the machine's rate. In 45 minutes, it produces 180 widgets, so the rate is $\frac{180 \text{ widgets}}{45 \text{ minutes}} = 4 \text{ widgets per minute}$. Next, convert the target time to minutes: 2 hours and 15 minutes = 120 + 15 = 135 minutes. Finally, multiply the rate by the time: $4 \text{ widgets/minute} \times 135 \text{ minutes} = 540 \text{ widgets}$. This confirms answer D is correct. Let's examine why the other answers are wrong. Answer A (520 widgets) likely comes from incorrectly calculating the rate or making an arithmetic error in the final multiplication. Answer B (580 widgets) might result from adding instead of properly scaling the proportion, or from a conversion mistake. Answer C (560 widgets) could come from using an incorrect rate like 4.15 widgets per minute instead of exactly 4. The key trap in rate problems is often unit conversion errors or setting up the wrong proportion. Always double-check that your units cancel properly and that you've converted time correctly. Strategy tip: For constant rate problems, always find the unit rate first (amount per one unit of time), then multiply by your target time. This systematic approach prevents most calculation errors and makes the math more manageable.
6. A recipe calls for 3 cups of flour to make 24 cookies. Maria wants to make 40 cookies but only has 4.5 cups of flour available. How many additional cups of flour does she need?
   Answer: 0.5 cups First, set up a proportion to find flour needed for 40 cookies: $\frac{3 \text{ cups}}{24 \text{ cookies}} = \frac{x \text{ cups}}{40 \text{ cookies}}$. Cross multiply: $3 \times 40 = 24x$, so $120 = 24x$, giving $x = 5$ cups needed. Since Maria has 4.5 cups, she needs $5 - 4.5 = 0.5$ additional cups. Choice B incorrectly calculates $4.5 - 3.5 = 1$. Choice C uses $6 - 4.5 = 1.5$ from incorrect proportion setup. Choice D assumes she needs 6.5 total cups.
7. In a scale model, 2.5 inches represents 15 feet of actual length. If the actual building is 78 feet tall, what is the height of the model building in inches?
   Answer: 13.0 inches Set up the proportion: $\frac{2.5 \text{ inches}}{15 \text{ feet}} = \frac{x \text{ inches}}{78 \text{ feet}}$. Cross multiply: $2.5 \times 78 = 15x$, so $195 = 15x$, giving $x = 13$ inches. Choice A results from using 60 feet instead of 78 feet. Choice C comes from incorrectly calculating $\frac{2.5 \times 81}{15}$. Choice D results from rounding $\frac{195}{14}$ instead of dividing by 15.
8. Two gears are connected so that when the first gear makes 8 complete rotations, the second gear makes 12 complete rotations. If the first gear makes 20 rotations, how many rotations does the second gear make?
   Answer: 30 rotations exactly This is a proportional reasoning problem involving gear ratios. When gears are connected, their rotation rates maintain a constant ratio - if one gear speeds up or slows down, the other changes proportionally. First, establish the ratio between the gears. When the first gear makes 8 rotations, the second makes 12 rotations. This gives us the ratio $8:12$, which simplifies to $2:3$. This means for every 2 rotations of the first gear, the second gear makes 3 rotations. To find how many rotations the second gear makes when the first makes 20 rotations, set up a proportion: $\frac{8}{12} = \frac{20}{x}$. Cross multiply: $8x = 12 \times 20 = 240$. Solving for x: $x = \frac{240}{8} = 30$ rotations. You can verify this using the simplified ratio: since $20 \div 2 = 10$, the first gear completes 10 cycles of its "2-rotation units." The second gear must complete 10 cycles of its "3-rotation units," giving $10 \times 3 = 30$ rotations. Choice A (28) might result from incorrectly adding 8 + 20 for some misguided reasoning. Choice B (36) could come from setting up the proportion backwards as $\frac{12}{8} = \frac{30}{20}$ and getting confused. Choice C (32) might result from adding 12 + 20, perhaps thinking you add the "extra" rotations to the original 12. Study tip: In gear ratio problems, always establish the basic ratio first, then scale it up proportionally. Double-check by ensuring your answer maintains the same ratio as the given relationship.

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