Comparing Fractions and Finding the True Statement

To determine which statement is true, we need to compare the fractions given in each option. When comparing fractions with different denominators, it is often helpful to find a common denominator or convert the fractions to decimals.

Let's analyze each option:

Analyzing Option 1: $\frac{29}{6} > \frac{43}{12}$

To compare $\frac{29}{6}$ and $\frac{43}{12}$, we can find a common denominator. The least common multiple (LCM) of 6 and 12 is 12.

Convert $\frac{29}{6}$ to an equivalent fraction with a denominator of 12:

$\frac{29}{6} = \frac{29 \times 2}{6 \times 2} = \frac{58}{12}$

Now we compare $\frac{58}{12}$ and $\frac{43}{12}$. Since the denominators are the same, we compare the numerators:

$58$ vs $43$

Since $58 > 43$, it means $\frac{58}{12} > \frac{43}{12}$.

Therefore, $\frac{29}{6} > \frac{43}{12}$ is a true statement.

Analyzing Option 2: $\frac{29}{6} = \frac{43}{12}$

From the analysis of Option 1, we know that $\frac{29}{6}$ is equivalent to $\frac{58}{12}$.

The statement is $\frac{58}{12} = \frac{43}{12}$.

Since the numerators $58$ and $43$ are not equal, the fractions are not equal.

Therefore, $\frac{29}{6} = \frac{43}{12}$ is a false statement.

Analyzing Option 3: $29 < \frac{43}{12}$

This statement compares a whole number, 29, with a fraction, $\frac{43}{12}$.

Let's convert 29 into a fraction with a denominator of 12:

$29 = \frac{29 \times 12}{12} = \frac{348}{12}$

The statement is $\frac{348}{12} < \frac{43}{12}$.

Comparing the numerators, we have $348$ vs $43$.

Since $348 > 43$, the inequality $\frac{348}{12} < \frac{43}{12}$ is false.

Alternatively, we can convert $\frac{43}{12}$ to a mixed number or decimal:

$\frac{43}{12} = 3 \frac{7}{12}$ or $43 \div 12 \approx 3.58$

The statement becomes $29 < 3.58$, which is clearly false.

Therefore, $29 < \frac{43}{12}$ is a false statement.

Analyzing Option 4: $\frac{29}{6} = \frac{53}{12}$

From the analysis of Option 1, we know that $\frac{29}{6}$ is equivalent to $\frac{58}{12}$.

The statement is $\frac{58}{12} = \frac{53}{12}$.

Since the numerators $58$ and $53$ are not equal, the fractions are not equal.

Therefore, $\frac{29}{6} = \frac{53}{12}$ is a false statement.

Summary of Comparisons

Based on the analysis, only the statement in Option 1 is true.

Revision Table: Understanding Fraction Comparisons

Comparing fractions is a fundamental skill. Here's a quick review:

• To compare fractions with the same denominator, compare the numerators. The fraction with the larger numerator is greater. Example: $\frac{5}{7} > \frac{3}{7}$ because $5 > 3$.
• To compare fractions with different denominators, find a common denominator (usually the LCM of the denominators). Convert each fraction to an equivalent fraction with the common denominator, then compare the numerators. Example: Compare $\frac{1}{3}$ and $\frac{2}{5}$. LCM of 3 and 5 is 15. $\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}$, $\frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15}$. Since $5 < 6$, $\frac{5}{15} < \frac{6}{15}$, so $\frac{1}{3} < \frac{2}{5}$.
• Alternatively, you can convert fractions to decimals by dividing the numerator by the denominator and comparing the decimal values. Example: $\frac{1}{3} \approx 0.333$, $\frac{2}{5} = 0.4$. Since $0.333 < 0.4$, $\frac{1}{3} < \frac{2}{5}$.

Additional Information: Types of Numbers and Comparisons

The numbers in this question are rational numbers (fractions). Comparing numbers is a basic operation in mathematics used to determine the relative size or order of two numbers. The symbols used for comparison are:

• $=$ : equal to
• $>$ : greater than
• $<$ : less than
• $\ge$ : greater than or equal to
• $\le$ : less than or equal to
• $\ne$ : not equal to

Understanding how to compare different types of numbers, including fractions, decimals, and whole numbers, is crucial for solving many mathematical problems.