Hard GRE Fractions Questions Practice Questions

Fraction operations account for a significant portion of the Quantitative Reasoning section, often appearing in complex multi-step word problems or data interpretation tasks. Hard GRE fractions questions challenge your ability to manipulate reciprocal relationships, compound ratios, and algebraic fractions under time pressure. Success on these high-level problems requires moving beyond simple arithmetic to understand how fractions behave within larger equations and inequalities.

Concept Explanation

Hard GRE fractions questions are complex mathematical problems that require the integration of fractional arithmetic with algebraic manipulation, ratio reasoning, and number property analysis. To solve these effectively, you must be proficient in finding the Least Common Denominator (LCD) for algebraic expressions, simplifying complex fractions (fractions within fractions), and translating dense word problems into mathematical notation. Key strategies include using the "plugging in numbers" technique with a common multiple of the denominators to eliminate fractions entirely. Additionally, understanding the relationship between fractions, decimals, and percentages is vital, as the GRE Quantitative Reasoning section often blends these formats to test your flexibility. For more foundational practice, you might explore free GRE practice questions before tackling these difficult variations.

Solved Examples

1. Complex Fraction Simplification: Simplify the expression: $$\frac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} - \frac{1}{y}}$$

   1. Find a common denominator for the numerator: $\frac{1}{x} + \frac{1}{y} = \frac{y + x}{xy}$.

   2. Find a common denominator for the denominator: $\frac{1}{x} - \frac{1}{y} = \frac{y - x}{xy}$.

   3. Divide the two resulting fractions by multiplying by the reciprocal: $\frac{y + x}{xy} \times \frac{xy}{y - x}$.

   4. Cancel the $xy$ terms to get the final answer: $\frac{y + x}{y - x}$.

2. Work Rate Problem: Machine A completes a job in $a$ hours and Machine B completes the same job in $b$ hours. If they work together for 2 hours, what fraction of the job remains?

   1. Determine the rate of each machine: Machine A's rate is $\frac{1}{a}$ and Machine B's rate is $\frac{1}{b}$.

   2. Combine their rates: Combined rate = $\frac{1}{a} + \frac{1}{b} = \frac{a + b}{ab}$.

   3. Calculate the work done in 2 hours: 2 \times \left(\frac{a + b}{ab} right) = \frac{2a + 2b}{ab}.

   4. Subtract the work done from the total job (1): $1 - \frac{2a + 2b}{ab} = \frac{ab - 2a - 2b}{ab}$.

3. Remainder Fractions: A traveler spends $\frac{1}{4}$ of his budget on transport and $\frac{2}{3}$ of the remainder on accommodation. If he has $500 left, what was his total budget?

   1. Let $T$ be the total budget. After transport, the remainder is $1 - \frac{1}{4} = \frac{3}{4}T$.

   2. He spends $\frac{2}{3}$ of the remainder on accommodation: $\frac{2}{3} \times \frac{3}{4}T = \frac{1}{2}T$.

   3. The total spent is $\frac{1}{4}T + \frac{1}{2}T = \frac{3}{4}T$.

   4. The amount left is $T - \frac{3}{4}T = \frac{1}{4}T$. Set this equal to $500: $\frac{1}{4}T = 500$.

   5. Solve for $T$: $T = 2,000$.

Practice Questions

1. If $x = \frac{1}{1 + \frac{1}{1 + \frac{1}{2}}}$, what is the value of $x$?

2. In a certain class, $\frac{2}{5}$ of the students are boys. If $\frac{1}{3}$ of the boys and $\frac{1}{2}$ of the girls wear glasses, what fraction of the total students wear glasses?

3. A container is $\frac{1}{3}$ full of water. After adding 10 liters of water, the container is $\frac{3}{4}$ full. What is the total capacity of the container in liters?

4. Compare Quantity A and Quantity B:
   Quantity A: $\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}$
   Quantity B: $\frac{1}{2} \times \frac{1}{3} \times \frac{1}{4} \times \frac{1}{5}$

5. If $a, b, c,$ and $d$ are positive integers such that \frac{a}{b} < \frac{c}{d}, is \frac{a}{b} < \frac{a+c}{b+d} < \frac{c}{d} always true?

6. A tank is filled by Pipe A in 4 hours and emptied by Pipe B in 6 hours. If both pipes are opened simultaneously when the tank is empty, what fraction of the tank will be full after 3 hours?

7. If $n$ is a positive integer, which of the following must be the greatest?
   (A) $\frac{n+1}{n}$ (B) $\frac{n+2}{n+1}$ (C) $\frac{n+3}{n+2}$ (D) $\frac{n}{n+1}$

8. A sum of money is divided among three people: A, B, and C. A receives $\frac{1}{3}$ of the total, B receives $\frac{2}{5}$ of the total, and C receives the remaining $120. How much did B receive?

Answers & Explanations

1. Answer: 3/5. Start from the bottom: $1 + \frac{1}{2} = \frac{3}{2}$. Then $\frac{1}{3/2} = \frac{2}{3}$. Then $1 + \frac{2}{3} = \frac{5}{3}$. Finally, $\frac{1}{5/3} = \frac{3}{5}$.

2. Answer: 13/30. Assume 30 students (LCM of 5, 3, 2). Boys = $\frac{2}{5} \times 30 = 12$; Girls = 18. Boys with glasses = $\frac{1}{3} \times 12 = 4$. Girls with glasses = $\frac{1}{2} \times 18 = 9$. Total with glasses = $4 + 9 = 13$. Fraction = $\frac{13}{30}$.

3. Answer: 24 liters. Let $C$ be capacity. $\frac{3}{4}C - \frac{1}{3}C = 10$. Find LCD (12): $\frac{9}{12}C - \frac{4}{12}C = 10 \Rightarrow \frac{5}{12}C = 10$. Solving for $C$, $C = 10 \times \frac{12}{5} = 24$.

4. Answer: Quantity A is greater. A sum of fractions greater than 0 will always be larger than their product when the fractions are less than 1 (since multiplying by a fraction less than 1 decreases the value).

5. Answer: Yes. This is a property of mediants. Adding the numerators and denominators of two positive fractions always results in a value between the two original fractions.

6. Answer: 1/4. Net rate = $\frac{1}{4} - \frac{1}{6} = \frac{3-2}{12} = \frac{1}{12}$ per hour. In 3 hours: $3 \times \frac{1}{12} = \frac{1}{4}$.

7. Answer: (A). Each fraction (A, B, C) is of the form $1 + \frac{1}{x}$. As the denominator $x$ increases, the value of the fraction decreases. Since n < n+1 < n+2, $\frac{n+1}{n}$ is the largest.

8. Answer: $180. Total fraction for A and B = $\frac{1}{3} + \frac{2}{5} = \frac{5+6}{15} = \frac{11}{15}$. C's share = $1 - \frac{11}{15} = \frac{4}{15}$. If $\frac{4}{15}T = 120$, then $T = 450$. B's share = $\frac{2}{5} \times 450 = 180$.

1. If the value of a fraction is increased by adding the same positive integer to both the numerator and the denominator, what must be true about the original fraction?

• AThe original fraction was equal to 1
• BThe original fraction was greater than 1
• CThe original fraction was less than 1
• DThe original fraction was negative

Frequently Asked Questions

How do I compare two large fractions without a calculator?

The most efficient method is cross-multiplication: to compare $\frac{a}{b}$ and $\frac{c}{d}$, compare the products $ad$ and $bc$. If ad > bc, then \frac{a}{b} > \frac{c}{d}.

What is the best way to handle "fraction of a remainder" word problems?

Always define the remainder clearly after each step before applying the next fraction. Alternatively, multiply the remaining fractions together (e.g., if you spend $1/3$, you have $2/3$ left) to find the final remaining portion of the whole.

Can I use the GRE on-screen calculator for fraction questions?

The GRE Prep resources suggest using the calculator sparingly for fractions because it only handles decimals. It is often faster and more accurate to keep numbers in fraction form to allow for easy cancellation.

What happens to a fraction when the denominator increases?

If the numerator stays constant and positive, increasing the denominator will always decrease the total value of the fraction. This is a common logic check for GRE practice questions with explanations.

How do I simplify a fraction where the numerator or denominator contains a square root?

You should rationalize the denominator by multiplying both the top and bottom by the conjugate or the square root itself. This is a common requirement in adaptive GRE practice tests that feature geometry or advanced algebra.