Comparing Fractions: The common denominator is smaller than the product of the denominators

To solve this problem, we'll compare the fractions: $\frac{1}{4}$ and $\frac{5}{6}$.

• Step 1: Determine the least common denominator (LCD) of the fractions' denominators, 4 and 6. The LCM of 4 and 6 is 12.
• Step 2: Convert $\frac{1}{4}$ to an equivalent fraction with a denominator of 12:

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

• Step 3: Convert $\frac{5}{6}$ to an equivalent fraction with a denominator of 12:

$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$

• Step 4: Compare the numerators: $\frac{3}{12}$ compared to $\frac{10}{12}$.

Since $3$ is less than $10$, we conclude that $\frac{1}{4}$ is less than $\frac{5}{6}$.

Thus, the missing sign is < .

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

• Step 1: Identify the denominators of the fractions $\frac{3}{8}$ and $\frac{1}{4}$.
• Step 2: Determine the least common denominator of 8 and 4.
• Step 3: Convert each fraction to have this common denominator.
• Step 4: Compare the fractions by examining their numerators.

Let's begin by determining the least common denominator (LCD) of the fractions:

The denominators are 8 and 4. The least common multiple (LCM) of these two numbers is 8.

Convert each fraction to an equivalent fraction with the common denominator of 8:

• $\frac{3}{8}$ already has the denominator 8, so it remains $\frac{3}{8}$.
• For $\frac{1}{4}$, convert by multiplying the numerator and denominator by 2:
$\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$.

Now, compare these equivalent fractions: $\frac{3}{8}$ and $\frac{2}{8}$.

Since $3 > 2$, it follows that $\frac{3}{8} > \frac{2}{8}$.

Therefore, the correct sign for the missing space is $>$.

Thus, $\frac{3}{8} > \frac{1}{4}$.

To solve this problem, we'll take the following steps:

• Step 1: Simplify the fraction $\frac{2}{5}$.
• Step 2: Simplify the fraction $\frac{6}{15}$.
• Step 3: Compare the simplified forms.

Let's go through these steps:

Step 1: $\frac{2}{5}$ is already in its simplest form because 2 and 5 have no common divisors other than 1.

Step 2: Simplify the fraction $\frac{6}{15}$:
- The greatest common divisor (GCD) of 6 and 15 is 3.
- Divide both the numerator and the denominator by their GCD to simplify:
$\frac{6}{15} = \frac{6 \div 3}{15 \div 3} = \frac{2}{5}$

Step 3: Compare the simplified forms:
Both $\frac{2}{5}$ and $\frac{6}{15}$ simplify to $\frac{2}{5}$. Thus, they are equivalent.

Therefore, the correct mathematical sign between the fractions $\frac{2}{5}$ and $\frac{6}{15}$ is $=$.

So, the missing sign is $\mathbf{=}$.

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

• Step 1: Identify the given fractions $\frac{4}{6}$ and $\frac{3}{4}$.
• Step 2: Calculate the least common denominator (LCD) of 6 and 4.
• Step 3: Convert each fraction to have the common denominator and compare numerators.

Now, let's work through these steps:

Step 1: The given fractions are $\frac{4}{6}$ and $\frac{3}{4}$.

Step 2: Find the least common denominator of 6 and 4. The prime factorization of 6 is $2 \times 3$ and of 4 is $2^2$. The LCD is $2^2 \times 3 = 12$.

Step 3: Convert each fraction to this common denominator.

• Convert $\frac{4}{6}$ to have a denominator of 12: Multiply both numerator and denominator by 2 to get $\frac{4 \times 2}{6 \times 2} = \frac{8}{12}$.
• Convert $\frac{3}{4}$ to have a denominator of 12: Multiply both numerator and denominator by 3 to get $\frac{3 \times 3}{4 \times 3} = \frac{9}{12}$.

Compare the numerators: 8 and 9. Since 8 is less than 9, we find that $\frac{4}{6} < \frac{3}{4}$.

Therefore, the correct sign to fill in is $<$, and the correct answer is:

$\frac{4}{6} < \frac{3}{4}$.

To solve this problem, we'll use the cross-multiplication method to compare the fractions $\frac{1}{4}$ and $\frac{5}{12}$.

• Step 1: Cross-multiply the fractions.
  We multiply the numerator of the first fraction by the denominator of the second, and the numerator of the second fraction by the denominator of the first:
• $1 \times 12 = 12$
• $5 \times 4 = 20$
• Step 2: Compare the results from cross-multiplication.
  Now, compare the results: $12$ and $20$.
  Since $12 < 20$, it follows that $\frac{1}{4} < \frac{5}{12}$.

Therefore, the correct sign to fill in the blank is $<$.

The solution to the problem is $\frac{1}{4} < \frac{5}{12}$.

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

• Step 1: Identify the least common multiple (LCM) of the denominators 12 and 8.
• Step 2: Convert each fraction to have this common denominator.
• Step 3: Compare the converted fractions by examining their numerators.

Let's proceed with the solution:
Step 1: The LCM of 12 and 8 needs to be found. The factorization of 12 is $2^2 \times 3$ and for 8 is $2^3$. The LCM is $2^3 \times 3 = 24$. Therefore, the common denominator is 24.

Step 2: Convert each fraction to the new denominator:

• Convert $\frac{5}{12}$: The equivalent fraction is found by multiplying both the numerator and the denominator by a number that equals the common denominator when the original denominator is multiplied by it. $24 ÷ 12 = 2$. Thus, multiply both the numerator and denominator by 2: $\frac{5 \times 2}{12 \times 2} = \frac{10}{24}$.
• Convert $\frac{7}{8}$: Similarly, multiply by $24 ÷ 8 = 3$: $\frac{7 \times 3}{8 \times 3} = \frac{21}{24}$.

Step 3: Compare the two equivalent fractions $\frac{10}{24}$ and $\frac{21}{24}$. Comparing the numerators while the denominators are the same: 10 < 21.

Therefore, $\frac{5}{12}$ is less than $\frac{7}{8}$.

Thus, the correct sign to fill in is $<$.

To solve this problem, we'll compare the two fractions, $\frac{1}{4}$ and $\frac{5}{16}$, by converting them to have a common denominator:

• Step 1: Identify the denominators of the two fractions: 4 and 16.
• Step 2: Find the least common denominator (LCD). The smallest number both 4 and 16 can divide into is 16.
• Step 3: Convert $\frac{1}{4}$ to have a denominator of 16: Multiply both the numerator and the denominator of $\frac{1}{4}$ by 4 to obtain $\frac{4}{16}$.
• Step 4: Now we compare the two fractions: $\frac{4}{16}$ and $\frac{5}{16}$.
• Step 5: Since both fractions have the same denominator, compare the numerators: 4 and 5.

Based on the comparison, 4 is less than 5, which means $\frac{4}{16}$ is less than $\frac{5}{16}$.

Therefore, the correct sign to fill in the blank is $<$.

To solve this problem, let's proceed as follows:

Step 1: Simplify the fractions.
The first fraction is $\frac{1}{7}$, which is already in its simplest form.
The second fraction is $\frac{4}{14}$, which can be simplified by dividing both the numerator and denominator by their greatest common divisor, 2:
$\frac{4 \div 2}{14 \div 2} = \frac{2}{7}$.

Step 2: Compare the simplified fractions.
Now, compare $\frac{1}{7}$ and $\frac{2}{7}$.
Since both fractions have the same denominator (7), we compare the numerators directly:
Since $1 < 2$, it follows that $\frac{1}{7} < \frac{2}{7}$.

Therefore, the missing sign in $\frac{1}{7} ☐ \frac{4}{14}$ is $\lt$.

The correct answer to this problem is $\lt$.