Comparing Fractions: Finding a Common Denominator by Multiplying the Denominators

To solve this problem, we'll compare the fractions $\frac{1}{3}$ and $\frac{3}{10}$ by finding a common denominator.

Let's follow these steps:

• Step 1: Identify the denominators of the given fractions, which are 3 and 10.
• Step 2: Calculate the least common denominator (LCD) of 3 and 10. Since 3 and 10 are co-prime, their product, 30, is the LCD.
• Step 3: Adjust the fractions to have a common denominator of 30:
• Convert $\frac{1}{3}$ to a fraction with denominator 30 by multiplying the numerator and the denominator by 10:
$\frac{1}{3} = \frac{1 \times 10}{3 \times 10} = \frac{10}{30}$
• Convert $\frac{3}{10}$ to a fraction with denominator 30 by multiplying the numerator and the denominator by 3:
$\frac{3}{10} = \frac{3 \times 3}{10 \times 3} = \frac{9}{30}$
• Step 4: Compare the numerators of the fractions: 10 (from $\frac{10}{30}$) and 9 (from $\frac{9}{30}$).
• Since 10 is greater than 9, $\frac{10}{30}$ is greater than $\frac{9}{30}$.
• Thus, $\frac{1}{3}$ is greater than $\frac{3}{10}$.

Therefore, the correct comparison sign is $>$.

To compare the fractions $\frac{1}{5}$ and $\frac{3}{4}$, we'll use the following steps:

• Step 1: Find a common denominator for the fractions.
• Step 2: Convert each fraction to an equivalent fraction with the common denominator.
• Step 3: Compare the numerators of the equivalent fractions.

Step 1: The denominators of the two fractions are 5 and 4. To find a common denominator, we multiply these together, getting $5 \times 4 = 20$. So, our common denominator is 20.

Step 2: Convert each fraction to have the denominator of 20.

• $\frac{1}{5}$: Multiply the numerator and denominator by 4 to get $\frac{1 \times 4}{5 \times 4} = \frac{4}{20}$.
• $\frac{3}{4}$: Multiply the numerator and denominator by 5 to get $\frac{3 \times 5}{4 \times 5} = \frac{15}{20}$.

Step 3: Now compare the numerators of the equivalent fractions:

$\frac{4}{20}$ compared to $\frac{15}{20}$ (both having the denominator of 20):

$4 < 15$ implies $\frac{4}{20} < \frac{15}{20}$.

Therefore, $\frac{1}{5} < \frac{3}{4}$.

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

Therefore, the missing sign in the expression $\frac{1}{5} ☐ \frac{3}{4}$ is $<$.

To solve this problem, we will use cross-multiplication to compare the two fractions $\frac{2}{3}$ and $\frac{6}{4}$.

• Step 1: Calculate the cross products. Multiply the numerator of the first fraction by the denominator of the second fraction, and the denominator of the first fraction by the numerator of the second fraction.
• Step 2: Compare the two products to determine which fraction is larger.

Now, let's work through the steps:

Step 1: Cross-multiply the two fractions:
$\text{First Product} = 2 \times 4 = 8$
$\text{Second Product} = 3 \times 6 = 18$

Step 2: Compare the resulting products:
Since $8 < 18$, it follows that $\frac{2}{3} < \frac{6}{4}$.

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

To determine the correct inequality sign to compare $\frac{3}{4}$ and $\frac{1}{9}$, we'll use the technique of cross-multiplication. This method involves multiplying the numerator of each fraction by the denominator of the other fraction. By calculating and comparing these products, we can determine the relative size of the fractions.

Let's perform the cross-multiplication:

• Multiply the numerator of the first fraction by the denominator of the second fraction:

• Multiply the denominator of the first fraction by the numerator of the second fraction:

Now we compare these two results. Since $27$ is greater than $4$, we conclude that:

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

To solve this problem, we will follow these steps:

• Step 1: Identify the fractions we need to compare, which are $\frac{3}{4}$ and $\frac{2}{6}$.
• Step 2: Simplify $\frac{2}{6}$ to its simplest form.
• Step 3: Find a common denominator for the two fractions.
• Step 4: Convert each fraction to have the common denominator.
• Step 5: Compare the resulting numerators to determine the relationship.

Now, let's work through each step:

Step 1: The fractions we have are $\frac{3}{4}$ and $\frac{2}{6}$.

Step 2: Simplify $\frac{2}{6}$. The greatest common factor of 2 and 6 is 2, so $\frac{2}{6} = \frac{1}{3}$.

Step 3: Find a common denominator for $\frac{3}{4}$ and $\frac{1}{3}$. The least common multiple of 4 and 3 is 12.

Step 4: Convert each fraction to have the common denominator:

$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$

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

Step 5: Compare the numerators of the converted fractions:

Now, compare $\frac{9}{12}$ and $\frac{4}{12}$.

Since $9 > 4$, it follows that $\frac{9}{12} > \frac{4}{12}$.

Therefore, $\frac{3}{4} > \frac{1}{3}$, and hence $\frac{3}{4} > \frac{2}{6}$.

The correct comparison sign is $>$.

To solve this problem, we'll convert both fractions to have a common denominator and compare:

• Step 1: Identify the denominators $(15$ and $3)$ of the fractions $\frac{2}{15}$ and $\frac{1}{3}$.
• Step 2: Find the common denominator. The least common multiple (LCM) of $15$ and $3$ is $15$.
• Step 3: Convert each fraction to have the common denominator $15$.
• For $\frac{2}{15}$, the fraction is already over $15$, so it stays $\frac{2}{15}$.
• For $\frac{1}{3}$, multiply both the numerator and the denominator by $5$ to get an equivalent fraction: $\frac{1 \times 5}{3 \times 5} = \frac{5}{15}$.
• Step 4: Compare the equivalent fractions $\frac{2}{15}$ and $\frac{5}{15}$.
• Step 5: Since $2 \lt 5$, it follows that $\frac{2}{15} \lt \frac{5}{15}$.

Therefore, the correct sign to fill in the missing slot is $\lt$.

Consequently, the completed expression is $\frac{2}{15} \lt \frac{1}{3}$.

Therefore, the solution to the problem is $\lt$.

To solve this problem, we'll compare the fractions $\frac{2}{9}$ and $\frac{3}{14}$ by finding a common denominator:

• Step 1: Find the common denominator. The denominators are $9$ and $14$. The least common multiple of $9$ and $14$ is $126$.
• Step 2: Convert each fraction to an equivalent fraction with a denominator of $126$.
  For $\frac{2}{9}$: $\frac{2}{9} = \frac{2 \times 14}{9 \times 14} = \frac{28}{126}$.
  For $\frac{3}{14}$: $\frac{3}{14} = \frac{3 \times 9}{14 \times 9} = \frac{27}{126}$.
• Step 3: Compare the numerators of the equivalent fractions. $28 > 27$, so $\frac{28}{126} > \frac{27}{126}$.

Therefore, the fraction $\frac{2}{9}$ is greater than $\frac{3}{14}$, and the correct inequality sign is $>$.

Hence, the answer is $\frac{2}{9} > \frac{3}{14}$.

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

• Step 1: Determine a common denominator for the fractions.
• Step 2: Convert each fraction to have the common denominator.
• Step 3: Compare the numerators to find the correct inequality.

Now, let's work through each step:
Step 1: Determine a common denominator for $\frac{3}{7}$ and $\frac{1}{6}$. The common denominator is the product of the denominators $7 \times 6 = 42$.
Step 2: Convert each fraction to have this common denominator:
- Convert $\frac{3}{7}$: Multiply the numerator and denominator by 6: $\frac{3 \times 6}{7 \times 6} = \frac{18}{42}$.
- Convert $\frac{1}{6}$: Multiply the numerator and denominator by 7: $\frac{1 \times 7}{6 \times 7} = \frac{7}{42}$.
Step 3: Compare the numerators: $18$ and $7$.
Since $18 > 7$, we have $\frac{18}{42} > \frac{7}{42}$. Thus, $\frac{3}{7} > \frac{1}{6}$.

Therefore, the correct inequality sign is $>$.

To solve this problem, we will compare the fractions $\frac{3}{7}$ and $\frac{1}{8}$ by converting them to have a common denominator.

Step 1: Find the least common multiple (LCM) of the denominators 7 and 8. Since 7 and 8 are coprime (have no common factors other than 1), the LCM is simply the product of the two numbers:

$\text{LCM}(7, 8) = 7 \times 8 = 56$

Step 2: Convert each fraction to an equivalent fraction with the common denominator of 56.

• Convert $\frac{3}{7}$:

• $\frac{3}{7} = \frac{3 \times 8}{7 \times 8} = \frac{24}{56}$

• Convert $\frac{1}{8}$:

• $\frac{1}{8} = \frac{1 \times 7}{8 \times 7} = \frac{7}{56}$

Step 3: Compare the new numerators:

$\frac{24}{56} \quad \text{and} \quad \frac{7}{56}$

Since 24 > 7, we conclude that \frac{24}{56} > \frac{7}{56}.

Therefore, the original inequality we are solving is:

\frac{3}{7} > \frac{1}{8}

Thus, the correct sign to fill in the blank is \bm{>}.

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

• Step 1: Calculate the least common denominator of the fractions
• Step 2: Convert both fractions to have this common denominator
• Step 3: Compare the numerators of the fractions with the common denominator
• Step 4: Select the appropriate inequality sign

Now, let's work through each step:

Step 1: The denominators of the given fractions are 8 and 9. The least common multiple (LCM) of these numbers is 72. Therefore, the least common denominator (LCD) is 72.

Step 2: Convert each fraction to an equivalent fraction with a denominator of 72.

• For $\frac{3}{8}$: Multiply the numerator and the denominator by 9 (since $\frac{72}{8} = 9$ ). This gives $\frac{3 \times 9}{8 \times 9} = \frac{27}{72}$.
• For $\frac{1}{9}$: Multiply the numerator and the denominator by 8 (since $\frac{72}{9} = 8$ ). This gives $\frac{1 \times 8}{9 \times 8} = \frac{8}{72}$.

Step 3: Now compare the numerators of $\frac{27}{72}$ and $\frac{8}{72}$.

Since $27$ is greater than $8$, it follows that $\frac{27}{72}$ is greater than $\frac{8}{72}$.

Step 4: Therefore, the correct inequality sign to fill the blank is $>$.

Thus, we conclude that $\frac{3}{8} > \frac{1}{9}$.

The correct answer is $\gt$.