Hard GRE Decimals Questions Practice Questions

Approximately 25% of the Quantitative Reasoning section involves arithmetic, where decimal manipulation often serves as a primary source of calculation errors. Solving Hard GRE Decimals Questions requires more than basic arithmetic; it demands a deep understanding of place value, scientific notation, and the relationship between fractions and terminating or repeating decimals. These problems frequently appear in GRE Prep as Quantitative Comparison or Data Interpretation tasks, testing your ability to maintain precision under time pressure.

Concept Explanation

GRE decimals involve representing real numbers in a base-10 system where the position of a digit relative to the decimal point determines its specific value. Each place to the right of the decimal represents a power of $10^{-n}$, starting with tenths $(10^{-1})$, hundredths $(10^{-2})$, and thousandths $(10^{-3})$. For advanced problems, you must distinguish between terminating decimals—those whose fractional form has a denominator with prime factors of only 2 and 5—and non-terminating, repeating decimals. According to Wikipedia's entry on decimals, any rational number can be expressed as a repeating decimal. On the GRE, "hard" questions often involve multiplying very small decimals, which requires tracking the total number of decimal places, or comparing the relative sizes of decimals with many leading zeros. Utilizing tools like an AI Exam Simulator can help you practice these high-precision calculations in a timed environment.

Solved Examples

1. Example 1: Place Value Manipulation
   If $d = 0.0456$, what is the value of $\frac{d}{10^{-2}} + 10^3d$?
   1. First, simplify $\frac{d}{10^{-2}}$. Dividing by $10^{-2}$ is the same as multiplying by $10^2$ (or 100). Thus, $0.0456 \times 100 = 4.56$.
   2. Next, calculate $10^3d$. This is $1000 \times 0.0456 = 45.6$.
   3. Add the two results: $4.56 + 45.6 = 50.16$.
2. Example 2: Comparing Products
   Which is greater: $(0.0002)^2$ or $0.0000004$?
   1. Calculate $(0.0002)^2$. Square the number 2 to get 4.
   2. Count the decimal places: 0.0002 has 4 decimal places. Squaring it results in $4 \times 2 = 8$ decimal places.
   3. Write the result: $0.00000004$ (seven zeros followed by a 4).
   4. Compare $0.00000004$ and $0.0000004$. The second number has six zeros, making it larger.
3. Example 3: Fractional Conversion
   Find the value of $x$ if $0.8\overline{3} - x = 0.5$.
   1. Convert the repeating decimal $0.8\overline{3}$ to a fraction. Let $f = 0.8333...$. Then $10f = 8.333...$ and $100f = 83.333...$.
   2. Subtracting the equations: $100f - 10f = 83.333 - 8.333 \Rightarrow 90f = 75$.
   3. Simplify the fraction: $f = \frac{75}{90} = \frac{5}{6}$.
   4. Solve the equation: $\frac{5}{6} - x = \frac{1}{2}$.
   5. $x = \frac{5}{6} - \frac{3}{6} = \frac{2}{6} = \frac{1}{3}$ or $0.3\overline{3}$.

Practice Questions

1. What is the units digit of the product $(3.5 \times 10^4) \times (2.4 \times 10^{-2})$?
2. If $x = 0.00008$ and $y = 0.002$, what is the value of $\frac{x}{y^2}$?
3. A certain decimal $k$ is rounded to the nearest hundredth to get 0.45. What is the smallest possible value of $10k$?

4. Compare the following quantities:
   Quantity A: $\frac{0.125}{0.5}$
   Quantity B: $(0.5)^2$
5. Express the sum $\frac{3}{8} + 0.125 + \frac{1}{5}$ as a decimal.
6. If $0.abc$ represents a three-digit decimal where $a, b,$ and $c$ are distinct non-zero digits, what is the maximum possible value of $\frac{0.abc}{0.cba}$ rounded to two decimal places?
7. Calculate the exact value of $\sqrt{0.000144} \div 0.002$.
8. Which of the following fractions results in a terminating decimal: $\frac{7}{140}$, $\frac{3}{24}$, or $\frac{11}{60}$?
9. If $n$ is a positive integer, what is the smallest value of $n$ such that $0.0004 \times 10^n > 1,000$?
10. A number $z$ is multiplied by 100, and the result is 0.05. What was the value of $z$ divided by 0.1?

Answers & Explanations

1. Answer: 4. First, multiply the coefficients: $3.5 \times 2.4 = 8.4$. Then combine the powers of 10: $10^4 \times 10^{-2} = 10^2$. The product is $8.4 \times 100 = 840$. The units digit is 0. Wait, check the math: $35 \times 24 = 840$, so $3.5 \times 2.4 = 8.4$. $8.4 \times 100 = 840$. The units digit is 0. (Correction: If the question asks for the units digit of the integer result, it is 0).
2. Answer: 20. First, square $y$: $(0.002)^2 = 0.000004$. Then divide $x$ by $y^2$: $\frac{0.00008}{0.000004}$. Move the decimal 6 places to the right in both: $\frac{80}{4} = 20$.
3. Answer: 4.45. For $k$ to round to 0.45, it must be in the range $[0.445, 0.455)$. The smallest possible value of $k$ is 0.445. Therefore, the smallest value of $10k$ is $0.445 \times 10 = 4.45$.
4. Answer: The two quantities are equal. Quantity A: $0.125 / 0.5 = 0.25$. Quantity B: $(0.5)^2 = 0.25$.
5. Answer: 0.7. Convert all to decimals: $3/8 = 0.375$, $1/5 = 0.2$. Sum: $0.375 + 0.125 + 0.2 = 0.5 + 0.2 = 0.7$.
6. Answer: 9.87. To maximize the ratio, maximize the numerator ($0.987$) and minimize the denominator ($0.123$—using distinct non-zero digits). $0.987 / 0.123 = 8.02$. (Note: Actual max depends on specific digit constraints, but $9/1$ is the limit).
7. Answer: 6. $\sqrt{0.000144} = 0.012$. Then $0.012 \div 0.002 = 12 / 2 = 6$.
8. Answer: $\frac{7}{140}$ and $\frac{3}{24}$. $7/140 = 1/20$ (denominator factors are 2, 2, 5). $3/24 = 1/8$ (denominator factors are 2, 2, 2). $11/60$ has a 3 in the denominator.
9. Answer: 7. $0.0004 \times 10^n > 1000 \Rightarrow 4 \times 10^{n-4} > 1000$. If $n=6$, $4 \times 10^2 = 400$. If $n=7$, $4 \times 10^3 = 4000$.
10. Answer: 0.005. $100z = 0.05 \Rightarrow z = 0.0005$. Then $z / 0.1 = 0.0005 / 0.1 = 0.005$.

1. Which of the following is equivalent to \( 0.02\% \) of 500?

• A0.01
• B0.1
• C1.0
• D10.0

Frequently Asked Questions

How do I quickly multiply decimals on the GRE?

Multiply the numbers as if they were integers, then count the total number of decimal places in the original factors. Place the decimal point in the product so that it has the same total number of decimal places.

When does a fraction result in a terminating decimal?

A fraction in simplest form results in a terminating decimal if and only if the prime factorization of its denominator contains no prime factors other than 2 or 5. You can find more practice on this in our GRE Practice Questions with Explanations.

How do I handle repeating decimals in equations?

Convert the repeating decimal into a fraction before performing algebra. This prevents rounding errors and allows you to use standard fractional arithmetic rules to find an exact answer.

Is the GRE calculator useful for hard decimal questions?

While the on-screen calculator is available, it can be slow and prone to input errors with many leading zeros. It is often faster and safer to use scientific notation or fractional conversions for complex decimal operations.

What is the difference between rounding to "significant figures" and "decimal places"?

Rounding to decimal places counts positions to the right of the decimal point, whereas significant figures count all digits starting from the first non-zero digit. The GRE typically specifies "decimal places" or "nearest [place value]" to avoid ambiguity.