Comparing Fractions: One of the denominators is the common denominator

To solve this problem, we'll follow these steps:

• Step 1: Simplify the fraction $\frac{4}{12}$.
• Step 2: Compare the simplified fraction with $\frac{1}{3}$.

Now, let's work through each step:
Step 1: Simplifying $\frac{4}{12}$:
The greatest common divisor (GCD) of 4 and 12 is 4. Simplifying the fraction by dividing both numerator and denominator by their GCD, we get:

$\frac{4}{12} = \frac{4 \div 4}{12 \div 4} = \frac{1}{3}$

Step 2: Compare the simplified fraction with $\frac{1}{3}$:
Both fractions are equal since $\frac{4}{12} = \frac{1}{3}$.

Therefore, the solution to the problem is $=$.

To solve this problem, we'll follow these steps:

• Step 1: Find the common denominator for the fractions.
• Step 2: Convert each fraction to this common denominator.
• Step 3: Compare the resulting fractions by examining their numerators.

Now, let's work through each step:
Step 1: Identify the denominators of the fractions. We have $7$ and $21$. The least common multiple (LCM) of $7$ and $21$ is $21$, which we'll use as the common denominator.

Step 2: Convert $\frac{2}{7}$ to a fraction with a denominator of $21$.
We achieve this by multiplying both the numerator and the denominator by $3$:
$\frac{2}{7} = \frac{2 \times 3}{7 \times 3} = \frac{6}{21}$.

The second fraction, $\frac{4}{21}$, already has the denominator $21$.

Step 3: Compare the numerators of the fractions $\frac{6}{21}$ and $\frac{4}{21}$.
We see that $6 > 4$.

Since $6 > 4$, it follows that $\frac{6}{21} > \frac{4}{21}$ and thus $\frac{2}{7} > \frac{4}{21}$.

Therefore, the correct sign to place between $\frac{2}{7}$ and $\frac{4}{21}$ is $\mathbf{>}$.

Hence, the solution to the problem is $>$, which corresponds to choice $2$.

< To solve this problem, we must determine which relational operator (<, >, = ) should be placed between the fractions $\frac{3}{7}$ and $\frac{21}{28}$.

Step 1: Simplify $\frac{21}{28}$.

To simplify $\frac{21}{28}$, find the greatest common divisor (GCD) of 21 and 28. Factors of 21 are 1, 3, 7, 21, and factors of 28 are 1, 2, 4, 7, 14, 28. The GCD is 7.

Divide both the numerator and denominator of $\frac{21}{28}$ by 7:

$\frac{21}{28} = \frac{21 \div 7}{28 \div 7} = \frac{3}{4}$.

Now we compare $\frac{3}{7}$ and $\frac{3}{4}$.

Step 2: Convert both fractions to a common denominator for easy comparison. Use the least common multiple (LCM) of 7 and 4, which is 28.

- Convert $\frac{3}{7}$ to have a denominator of 28:

$\frac{3}{7} = \frac{3 \times 4}{7 \times 4} = \frac{12}{28}$.

- $\frac{3}{4}$ is already simplified and does not need to convert again, as we have considered LCM:

$\frac{3}{4} = \frac{3 \times 7}{4 \times 7} = \frac{21}{28}$.

Step 3: Compare the fractions $\frac{12}{28}$ and $\frac{21}{28}$.

- Since \frac{12}{28} < \frac{21}{28}, therefore, \frac{3}{7} < \frac{21}{28}.

Hence, the missing sign is < .

To solve this problem, we'll follow these steps:

• Step 1: Identify a common denominator for both fractions.
• Step 2: Adjust each fraction to have this same denominator.
• Step 3: Compare the numerators of the converted fractions.

Now, let's work through each step:

Step 1: The denominators of the given fractions are 6 and 12. The least common multiple (LCM) of 6 and 12 is 12.

Step 2: Convert $\frac{4}{6}$ to have a denominator of 12:

$\frac{4}{6} = \frac{4 \times 2}{6 \times 2} = \frac{8}{12}$

The fraction $\frac{5}{12}$ already has the denominator 12, so it remains:

$\frac{5}{12} = \frac{5}{12}$

Step 3: Compare the numerators of the fractions $\frac{8}{12}$ and $\frac{5}{12}$.

We have $8 > 5$.

Therefore, $\frac{8}{12} > \frac{5}{12}$, which implies $\frac{4}{6} > \frac{5}{12}$.

Thus, the correct comparison sign is $>$.

To solve this problem, we need to compare the fractions $\frac{1}{3}$ and $\frac{6}{9}$.

We'll follow these steps:

• Step 1: Simplify the fraction $\frac{6}{9}$.
• Step 2: Compare it to $\frac{1}{3}$.

Step 1:
The fraction $\frac{6}{9}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3.
Thus, $\frac{6}{9} = \frac{6 \div 3}{9 \div 3} = \frac{2}{3}$.

Step 2:
Now, we compare $\frac{1}{3}$ with the simplified form $\frac{2}{3}$.
Since $\frac{1}{3}$ has a smaller numerator than $\frac{2}{3}$ while both have the same denominator, it is evident that $\frac{1}{3}$ is less than $\frac{2}{3}$.

Therefore, the correct sign to place between $\frac{1}{3}$ and $\frac{6}{9}$ is $<$.

Final Solution: The correct answer is $\frac{1}{3} < \frac{6}{9}$.

To solve this problem, let's carefully follow these steps:

• Step 1: Simplify the fraction $\frac{2}{24}$.
• Step 2: Compare the simplified fraction with $\frac{1}{12}$.

Step 1: We start by simplifying $\frac{2}{24}$.
We find the greatest common divisor of 2 and 24, which is 2. Thus, we divide both the numerator and denominator by 2:

$\frac{2 \div 2}{24 \div 2} = \frac{1}{12}$

Step 2: Now that we have simplified $\frac{2}{24}$ to $\frac{1}{12}$, we can compare it with the original fraction $\frac{1}{12}$.

Both fractions are now $\frac{1}{12}$. Therefore, they are equal.

This means the correct sign to insert is the equality sign $(=)$.

Hence, the solution to the problem is $\boxed{=}$.

To solve this problem, we will follow these steps:

• Step 1: Convert $\frac{2}{5}$ to an equivalent fraction with a denominator of 10.
• Step 2: Compare the numerators of the two fractions with the common denominator.

Now, let's work through each step:

Step 1: Convert $\frac{2}{5}$ to an equivalent fraction with denominator 10.

To convert $\frac{2}{5}$ to have a denominator of 10, we need to determine what number, when multiplied with 5, gives us 10. It is 2. Hence, we multiply both the numerator and denominator of $\frac{2}{5}$ by 2:

$\frac{2}{5} \times \frac{2}{2} = \frac{4}{10}$

Now, $\frac{2}{5}$ is equivalent to $\frac{4}{10}$.

Step 2: Compare the fractions $\frac{4}{10}$ and $\frac{7}{10}$.

Both fractions now have the same denominator, 10. We can compare them directly by looking at their numerators:

$4 \quad \text{and} \quad 7$

Since 4 is less than 7, we have:

$\frac{4}{10} < \frac{7}{10}$

Therefore, $\frac{2}{5} < \frac{7}{10}$.

The correct mathematical sign that completes the statement is $\lt$, so:

$\frac{2}{5} < \frac{7}{10}$

To solve this problem, let's follow these steps:

• Step 1: Identify a common denominator. Since 8 is a multiple of 4, the least common denominator is 8.
• Step 2: Convert $\frac{3}{4}$ into eighths. Multiply numerator and denominator by 2 to get $\frac{6}{8}$.
• Step 3: Compare $\frac{2}{8}$ and $\frac{6}{8}$. Because 2 is less than 6, we can conclude that $\frac{2}{8}$ is less than $\frac{6}{8}$.

Therefore, the missing symbol is < .