Checking Correctness of Comparison Statements

We are asked to determine which of the given comparison statements involving fractions and decimals are correct. To do this, we need to convert the fractions into their decimal equivalents and then compare them with the given decimal values.

Analyzing Statement I: $(\frac{3}{11}) > 0.3$

First, let's convert the fraction $\frac{3}{11}$ into a decimal. We do this by dividing the numerator (3) by the denominator (11).

Calculation for $\frac{3}{11}$:

$\frac{3}{11} = 3 \div 11$

Performing the division:

$$ \begin{array}{r} 0.2727\dots \\ 11\overline{)3.0000} \\ -22\downarrow \\ \hline 80 \\ -77\downarrow \\ \hline 30 \\ -22\downarrow \\ \hline 80 \\ \dots \end{array} $$

The decimal equivalent of $\frac{3}{11}$ is approximately $0.2727...$.

Now, we compare $0.2727...$ with $0.3$.

The statement is $0.2727... > 0.3$.

Comparing the decimal values:

• $0.2727...$
• $0.3000...$

Since the first decimal digit of $0.2727...$ is 2, and the first decimal digit of $0.3$ is 3, we see that $0.2727...$ is less than $0.3$.

So, the statement $(\frac{3}{11}) > 0.3$ is incorrect.

Analyzing Statement II: $(\frac{7}{8}) > 0.86$

Next, let's convert the fraction $\frac{7}{8}$ into a decimal. We do this by dividing the numerator (7) by the denominator (8).

Calculation for $\frac{7}{8}$:

$\frac{7}{8} = 7 \div 8$

Performing the division:

$$ \begin{array}{r} 0.875 \\ 8\overline{)7.000} \\ -64\downarrow \\ \hline 60 \\ -56\downarrow \\ \hline 40 \\ -40 \\ \hline 0 \end{array} $$

The decimal equivalent of $\frac{7}{8}$ is $0.875$.

Now, we compare $0.875$ with $0.86$.

The statement is $0.875 > 0.86$.

Comparing the decimal values:

• $0.875$
• $0.860$

Comparing digit by digit from left to right after the decimal point:

• The first decimal digit is 8 for both.
• The second decimal digit for $0.875$ is 7, and for $0.860$ is 6.

Since $7 > 6$, we conclude that $0.875$ is greater than $0.86$.

So, the statement $(\frac{7}{8}) > 0.86$ is correct.

Conclusion on Statement Correctness

Based on our analysis:

• Statement I: $(\frac{3}{11}) > 0.3$ is incorrect.
• Statement II: $(\frac{7}{8}) > 0.86$ is correct.

Therefore, only Statement II is correct.

Revision Table: Fraction to Decimal Conversions

Additional Information: Comparing Fractions and Decimals

Comparing fractions and decimals is a common task in mathematics. The easiest way to compare a fraction and a decimal is often to convert the fraction into its decimal form. Division is used for this conversion. Once both numbers are in decimal form, comparison becomes straightforward, much like comparing whole numbers, but considering the place value after the decimal point.

When comparing decimals:

1. Compare the whole number parts first (the digits before the decimal point).
2. If the whole number parts are equal, compare the digits after the decimal point from left to right.
3. The decimal with the larger digit at the first differing place is the greater number.
4. If one decimal has more digits after the decimal point, you can add trailing zeros to the other decimal to make the comparison clearer (e.g., $0.3 = 0.300$).

Understanding fraction-to-decimal conversion is fundamental for solving various mathematical problems involving different number formats.