Hard GRE Proportion Questions Practice Questions

Proportions on the GRE Quantitative Reasoning section represent the equality of two ratios, often appearing in complex word problems involving rates, mixtures, and geometric scaling. While basic proportions require simple cross-multiplication, hard GRE proportion questions challenge test-takers by introducing multiple variables, indirect relationships, and multi-step conversions. Succeeding on these problems requires a deep understanding of how quantities change relative to one another in both linear and non-linear environments.

Concept Explanation

A proportion is a mathematical statement asserting that two ratios are equal, typically expressed in the form $\frac{a}{b} = \frac{c}{d}$. In the context of the GRE, these relationships can be direct, where one variable increases as the other increases, or inverse, where one variable increases as the other decreases. Harder problems often involve "compound proportions," where a single quantity depends on two or more other variables—for example, the amount of work done depends on both the number of workers and the time spent working. According to Khan Academy's foundational math resources, the key to solving these is identifying the constant of proportionality. On the GRE, you must also be wary of units; a proportion might compare liters to minutes in one part of the question and milliliters to seconds in another. To prepare for these nuances, students often utilize an adaptive GRE practice test to simulate the increasing difficulty of the exam. Understanding the difference between a part-to-part ratio and a part-to-whole ratio is also vital for avoiding traps in data interpretation and word problems.

Solved Examples

1. Example 1: Compound Work Rates
   If 8 machines can produce 1,200 widgets in 6 hours, how many widgets can 10 machines of the same efficiency produce in 4 hours?
   1. Identify the rate per machine per hour: $\text{Rate} = \frac{1,200}{8 \times 6} = \frac{1,200}{48} = 25$ widgets per machine per hour.
   2. Apply the rate to the new conditions: $\text{Total Widgets} = 10 \text{ machines} \times 4 \text{ hours} \times 25 \text{ widgets/hr}$.
   3. Calculate the final result: $40 \times 25 = 1,000$ widgets.
2. Example 2: Inverse Proportions
   A group of 12 volunteers can complete a project in 15 days. If the project must be finished in 9 days, how many additional volunteers are needed, assuming all volunteers work at the same rate?
   1. Recognize this as an inverse proportion: $( \text{Volunteers})_1 \times ( \text{Days})_1 = ( \text{Volunteers})_2 \times ( \text{Days})_2$.
   2. Set up the equation: $12 \times 15 = x \times 9$.
   3. Solve for $x$: $180 = 9x \rightarrow x = 20$.
   4. Find the "additional" count: $20 - 12 = 8$ additional volunteers.
3. Example 3: Geometric Proportions
   The surface area of two spheres is in the ratio 4:9. What is the ratio of their volumes?
   1. Recall that surface area scales with the square of the radius: $\frac{SA_1}{SA_2} = (\frac{r_1}{r_2})^2$.
   2. Solve for the radius ratio: $\frac{4}{9} = (\frac{r_1}{r_2})^2 \rightarrow \frac{r_1}{r_2} = \frac{2}{3}$.
   3. Recall that volume scales with the cube of the radius: $\frac{V_1}{V_2} = (\frac{r_1}{r_2})^3$.
   4. Calculate the volume ratio: $(\frac{2}{3})^3 = \frac{8}{27}$. The ratio is 8:27.

Practice Questions

1. A solution is 20% acid and 80% water. If 5 liters of pure water are added to 20 liters of this solution, what is the new ratio of acid to water?
2. In a certain factory, the number of units produced is directly proportional to the square root of the number of workers. If 16 workers produce 200 units, how many units will 100 workers produce?
3. Three pumps, working together at their respective constant rates, can fill a tank in 4 hours. Pump A works twice as fast as Pump B, and Pump B works three times as fast as Pump C. How many hours would it take Pump C alone to fill the tank?

4. The ratio of the number of men to the number of women in a committee is 5:3. If 6 men leave and 6 women join, the ratio becomes 1:1. How many people were originally in the committee?
5. A map uses a scale where 0.5 inches represents 12 miles. If two cities are 3.5 inches apart on the map, and a car travels between them at an average speed of 60 miles per hour, how many minutes does the trip take?
6. If $y$ is inversely proportional to the cube of $x$, and $y = 2$ when $x = 2$, what is the value of $y$ when $x = 4$?
7. A mixture contains sand, cement, and gravel in the ratio 3:2:5 by weight. If the mixture is replaced by a new one where the weight of sand is doubled and the weight of gravel is halved, what is the new ratio of cement to the total weight?
8. Machine X produces $w$ units in $h$ hours. Machine Y produces the same number of units in $0.75h$ hours. If both machines work together, what fraction of the total work is performed by Machine Y?

Answers & Explanations

1. Answer: 4:21. In 20 liters of solution, acid = $0.20 \times 20 = 4$ liters and water = 16 liters. Adding 5 liters of water makes the new water total 21 liters. The acid remains 4 liters. The ratio is 4:21.
2. Answer: 500 units. The relationship is $U = k\sqrt{W}$. Using the first set: $200 = k\sqrt{16} \rightarrow 200 = 4k \rightarrow k = 50$. For 100 workers: $U = 50\sqrt{100} = 50 \times 10 = 500$.
3. Answer: 40 hours. Let Pump C's rate be $r$. Then B = $3r$ and A = $2(3r) = 6r$. Combined rate = $r + 3r + 6r = 10r$. They fill the tank in 4 hours, so the total work is $4 \times 10r = 40r$. Pump C alone takes $\frac{40r}{r} = 40$ hours.
4. Answer: 48. Let men = $5x$ and women = $3x$. After the change: $\frac{5x-6}{3x+6} = \frac{1}{1}$. Cross-multiplying gives $5x - 6 = 3x + 6 \rightarrow 2x = 12 \rightarrow x = 6$. Original total = $8x = 8(6) = 48$.
5. Answer: 84 minutes. Distance = $(\frac{3.5}{0.5}) \times 12 = 7 \times 12 = 84$ miles. Time = $\frac{84 \text{ miles}}{60 \text{ mph}} = 1.4$ hours. Convert to minutes: $1.4 \times 60 = 84$ minutes.
6. Answer: 0.25 (or 1/4). The relationship is $y = \frac{k}{x^3}$. So $2 = \frac{k}{2^3} \rightarrow k = 16$. When $x = 4$, $y = \frac{16}{4^3} = \frac{16}{64} = 0.25$.
7. Answer: 2/10.5 (or 4/21). Original: Sand 3, Cement 2, Gravel 5 (Total 10). New: Sand 6, Cement 2, Gravel 2.5 (Total 10.5). Ratio of cement to total = $\frac{2}{10.5} = \frac{4}{21}$.
8. Answer: 4/7. Rates are $\frac{w}{h}$ for X and $\frac{w}{0.75h} = \frac{4w}{3h}$ for Y. Combined rate = $\frac{3w}{3h} + \frac{4w}{3h} = \frac{7w}{3h}$. Fraction by Y = $\frac{ \text{Rate}_Y}{ \text{Total Rate}} = \frac{4/3}{7/3} = \frac{4}{7}$.

For more targeted practice, you might find GRE practice questions with explanations helpful for identifying common traps in these complex ratios. Additionally, you can generate custom sets using an AI Question Generator to focus specifically on your weak areas.

1. If the ratio of A to B is 4:5 and the ratio of B to C is 2:3, what is the ratio of A to C?

• A8:15
• B4:3
• C6:8
• D2:5

Frequently Asked Questions

What is the difference between direct and inverse proportions on the GRE?

Direct proportion means as one value increases, the other increases at a constant rate (y = kx), whereas inverse proportion means as one value increases, the other decreases (y = k/x).

How do I handle ratios with three or more terms?

Treat them as a total number of parts; for a ratio of 2:3:5, there are 10 total parts, and each quantity represents a fraction (2/10, 3/10, 5/10) of the whole.

Can proportions involve squared or cubed units?

Yes, especially in geometry where lengths are proportional to $x$, areas are proportional to $x^2$, and volumes are proportional to $x^3$.

What is a "constant of proportionality"?

It is the fixed value (often denoted as 'k') that relates two proportional quantities, found by dividing or multiplying the known values of the variables.

How do I solve work-rate problems involving proportions?

Use the formula Work = Rate × Time, and remember that rates are additive when multiple entities work together, while the time taken is inversely proportional to the total rate.

Why are proportions considered "hard" on the GRE?

They are difficult because the test often combines proportions with other concepts like percentages, geometry, or complex algebraic expressions that require multiple steps to isolate the variable.