Addition of Fractions: Complete the missing numbers

To find the missing fraction $x$ in the equation $\frac{1}{5} + x = \frac{3}{5}$, we can follow these steps:

• Step 1: Start with the given equation: $\frac{1}{5} + x = \frac{3}{5}$.
• Step 2: Subtract $\frac{1}{5}$ from both sides to solve for $x$:

$x = \frac{3}{5} - \frac{1}{5}$

• Step 3: Since the fractions have the same denominator, subtract the numerators directly:

$x = \frac{3 - 1}{5} = \frac{2}{5}$

Thus, the missing fraction is $\frac{2}{5}$.

Therefore, the solution to the problem is $\frac{2}{5}$.

To determine the missing fraction in the equation $\frac{4}{7} + \_\_\_ = \frac{6}{7}$, we follow these steps:

• Step 1: Assume the missing fraction is $\frac{x}{7}$.
• Step 2: Our equation becomes $\frac{4}{7} + \frac{x}{7} = \frac{6}{7}$.
• Step 3: Since the denominators are the same, we equate the numerators: $4 + x = 6$.
• Step 4: Solve for $x$ by subtracting $4$ from both sides: $x = 6 - 4$.
• Step 5: We find $x = 2$.
• Step 6: Substitute $x$ back into the fraction $\frac{x}{7}$, giving us the missing fraction $\frac{2}{7}$.

Thus, the missing fraction is $\frac{2}{7}$.

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

• Step 1: Subtract the original fraction from the resulting fraction.
• Step 2: Simplify the resulting fraction if needed.

Now, let's work through each step:

Step 1: Given the equation $\frac{3}{6} + x = \frac{4}{6}$, we solve for $x$ by subtracting $\frac{3}{6}$ from both sides:
$x = \frac{4}{6} - \frac{3}{6}$.

Step 2: Compute the subtraction:
Since both fractions have the same denominator, subtract the numerators: $4 - 3 = 1$. Thus, the fraction is $\frac{1}{6}$.

Therefore, the missing fraction is $\frac{1}{6}$.

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

• Step 1: Identify the sum and known fraction in the equation $\frac{1}{4} + x = \frac{3}{4}$.
• Step 2: Subtract the known fraction from the total sum to find the missing fraction.
• Step 3: Simplify the result to express the missing fraction clearly.

Now, let's work through each step:
Step 1: We are given that $\frac{1}{4} + x = \frac{3}{4}$. Here, $x$ represents the missing fraction.

Step 2: To isolate $x$, we subtract $\frac{1}{4}$ from both sides of the equation:

$x = \frac{3}{4} - \frac{1}{4}$

Step 3: Perform the subtraction. Since both fractions have the same denominator, subtract the numerators:

$\frac{3}{4} - \frac{1}{4} = \frac{3 - 1}{4} = \frac{2}{4}$

Therefore, the missing fraction is $\frac{2}{4}$, which is one of the given answer choices.

To find the missing fraction in the equation $\frac{6}{10} + \_ = \frac{9}{10}$, we proceed as follows:

• Step 1: Note that both fractions $\frac{6}{10}$ and $\frac{9}{10}$ have the same denominator (10).
• Step 2: We need to find the fraction that, when added to $\frac{6}{10}$, gives $\frac{9}{10}$. This means we need to perform subtraction: $\frac{9}{10} - \frac{6}{10}$.
• Step 3: Conduct the subtraction by subtracting the numerators and keeping the denominator the same: $\frac{9 - 6}{10} = \frac{3}{10}$.

Thus, the missing fraction is $\frac{3}{10}$.

Therefore, the solution to the problem is $\frac{3}{10}$.

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

• Step 1: Identify the numerators of the given fractions. We have $\frac{5}{12}$ and $\frac{7}{12}$.
• Step 2: Since both fractions have the same denominator, subtract the numerator of the given fraction from the resulting fraction's numerator:

$7 - 5 = 2$

Step 3: The missing fraction is $\frac{2}{12}$, as the denominator remains the same.

Therefore, the missing fraction is $\frac{2}{12}$.

To find the missing fraction in the equation $\frac{3}{10} + \_ + \frac{3}{10} = \frac{9}{10}$, we need to determine what must be added to the sums of the known fractions to reach the total sum.

Step 1: Calculate the sum of known fractions. We have:

• $\frac{3}{10} + \frac{3}{10}$

Add the numerators together, since both fractions have the same denominator:

• $3 + 3 = 6$

Thus, the combined fraction is:

• $\frac{6}{10}$

Step 2: To find the missing fraction, subtract the sum of known fractions from the total:

• $\frac{9}{10} - \frac{6}{10} = \frac{3}{10}$

This shows that the missing fraction is indeed $\frac{3}{10}$.

Therefore, the solution to the problem is $\frac{3}{10}$.

To solve this problem, follow these steps:

• Step 1: Add the known fractions: We first add the fractions given on the left-hand side. So, $\frac{2}{12} + \frac{3}{12} = \frac{5}{12}$.
• Step 2: Subtract this sum from the total: We know the sum of all the fractions is $\frac{9}{12}$. To find the missing fraction, we need to calculate $\frac{9}{12} - \frac{5}{12}$.
• Step 3: Perform the subtraction: Since the fractions have the same denominator, subtract numerators: $\frac{9 - 5}{12} = \frac{4}{12}$.
• Step 4: Verify your solution: Verify by adding all fractions including the missing one: $\frac{2}{12} + \frac{4}{12} + \frac{3}{12} = \frac{9}{12}$, which confirms our answer.

Therefore, the missing fraction is $\frac{4}{12}$.

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

• Step 1: Compute the sum of the given fractions $\frac{2}{9}$ and $\frac{5}{9}$.
• Step 2: Subtract the result from the total $\frac{8}{9}$ to find the missing fraction.

First, let's compute the sum of the known fractions:

$\frac{2}{9} + \frac{5}{9} = \frac{2 + 5}{9} = \frac{7}{9}$

Next, subtract this result from the total sum:

$\frac{8}{9} - \frac{7}{9} = \frac{8 - 7}{9} = \frac{1}{9}$

Thus, the missing fraction that completes the equation is $\frac{1}{9}$.

Therefore, the solution to the problem is $\frac{1}{9}$.

The goal is to find the missing fraction in the equation $\frac{5}{15} + \_ + \frac{3}{15} = \frac{13}{15}$.

To find the missing fraction, observe the following:
- Start by focusing only on the numerators because they have the same denominator.

We write the equation for the numerators:

• $5 + x + 3 = 13$

Combine the known terms on the left side, $5 + 3$, which results in 8:

• $8 + x = 13$

Now, solve for $x$ by subtracting 8 from both sides:

• $x = 13 - 8$
• $x = 5$

The missing fraction is $\frac{x}{15} = \frac{5}{15}$.

Therefore, the missing fraction that completes the equation is $\frac{5}{15}$.

To solve this problem, we will focus on simplifying the fraction on the left side and comparing it to the fraction on the right side:

• Step 1: Simplify the fraction $\frac{7}{14}$. Since both 7 and 14 have a common factor of 7, we can divide both the numerator and denominator by 7 to get $\frac{1}{2}$.
• Step 2: We now have the equation $\frac{1}{2} + ? = \frac{1}{2}$.
• Step 3: To determine what needs to be added to $\frac{1}{2}$ to result in $\frac{1}{2}$, we can see that nothing needs to be added. Therefore, the unknown term $?$ should be 0.

This makes sense because adding zero to any number does not change its value.

Therefore, the solution to the problem is $0$.

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

• Convert the fractions to have a common denominator.
• Subtract to find the missing value.
• Simplify the resulting fraction.

Step 1: Start with the equation $\frac{1}{10} + ? = \frac{3}{4}$.
Step 2: Rewrite it to find the missing term: $? = \frac{3}{4} - \frac{1}{10}$.

Step 3: To subtract, find a common denominator. The least common multiple of 4 and 10 is 20:

• Convert $\frac{3}{4}$ to have a denominator of 20: $\frac{3}{4} = \frac{15}{20}$.
• Convert $\frac{1}{10}$ to have a denominator of 20: $\frac{1}{10} = \frac{2}{20}$.

Step 4: Subtract $\frac{2}{20}$ from $\frac{15}{20}$:
$\frac{15}{20} - \frac{2}{20} = \frac{13}{20}$.

Step 5: Verify with the provided choices. The correct answer choice is the fraction $\frac{13}{20}$, which matches choice 4.

Therefore, the solution to the problem is $\frac{13}{20}$.

To find the missing fraction in the equation $\frac{2}{9} + ? = \frac{2}{3}$, we will perform the following steps:

• Step 1: Convert the fraction $\frac{2}{3}$ to have the same denominator as $\frac{2}{9}$.
• Step 2: Subtract $\frac{2}{9}$ from the new fraction.

Let's execute these steps:
Step 1: Convert $\frac{2}{3}$ into a fraction with a denominator of 9. To do this, multiply both the numerator and the denominator of $\frac{2}{3}$ by 3 to obtain an equivalent fraction:
$\frac{2}{3} \times \frac{3}{3} = \frac{6}{9}$

Step 2: Subtract $\frac{2}{9}$ from $\frac{6}{9}$:
$\frac{6}{9} - \frac{2}{9} = \frac{4}{9}$

Thus, the fraction that we need to add to $\frac{2}{9}$ to get $\frac{2}{3}$ is $\frac{4}{9}$.

The correct answer to the problem is $\frac{4}{9}$.

To solve this problem, we aim to find $x$ in the equation $x + \frac{3}{4} = \frac{4}{5}$.

Step 1: Isolate $x$ by subtracting $\frac{3}{4}$ from both sides.

$x = \frac{4}{5} - \frac{3}{4}$

Step 2: Find a common denominator for the fractions $\frac{4}{5}$ and $\frac{3}{4}$. The least common denominator of 5 and 4 is 20.

Convert $\frac{4}{5}$ to a fraction with a denominator of 20:

$\frac{4}{5} = \frac{4 \times 4}{5 \times 4} = \frac{16}{20}$

Convert $\frac{3}{4}$ to a fraction with a denominator of 20:

$\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}$

Step 3: Subtract the two fractions:

$x = \frac{16}{20} - \frac{15}{20} = \frac{16 - 15}{20} = \frac{1}{20}$

Therefore, the missing fraction $x$ is $\frac{1}{20}$.

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

• Step 1: Find a common denominator for the fractions involved.
• Step 2: Subtract $\frac{1}{6}$ from $\frac{2}{3}$ to isolate $?$.
• Step 3: Simplify the resulting fraction to find $?$.

Let's work through each step:
Step 1: The denominators are $6$ and $3$. The common denominator can be $6$ since it is the least common multiple of both numbers.

Step 2: Convert $\frac{2}{3}$ to an equivalent fraction with a denominator of $6$. Multiply the numerator and the denominator by $2$ to get:

$\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}$

Now the equation becomes:

$? + \frac{1}{6} = \frac{4}{6}$

Subtract $\frac{1}{6}$ from $\frac{4}{6}$:

$? = \frac{4}{6} - \frac{1}{6} = \frac{4 - 1}{6} = \frac{3}{6}$

Step 3: Simplify $\frac{3}{6}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $3$:

$\frac{3}{6} = \frac{3 \div 3}{6 \div 3} = \frac{1}{2}$

Thus, the value of the missing fraction $?$ is $\frac{1}{2}$.

The correct answer choice is $\frac{1}{2}$.

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

• Step 1: Identify the common denominator for $\frac{1}{3}$ and $\frac{2}{5}$.
• Step 2: Rewrite each fraction using this common denominator.
• Step 3: Perform the fraction subtraction.
• Step 4: Simplify the resulting fraction, if needed.

Now, let's work through each step:

Step 1: Find the common denominator.

The denominators of the fractions are 3 and 5. The least common multiple of 3 and 5 is 15.

Step 2: Rewrite each fraction with the common denominator of 15.

$\frac{1}{3} = \frac{5}{15}$, and $\frac{2}{5} = \frac{6}{15}$.

Step 3: Subtract $\frac{5}{15}$ from $\frac{6}{15}$.

$\frac{6}{15} - \frac{5}{15} = \frac{1}{15}$.

Step 4: Simplification.

The fraction $\frac{1}{15}$ is already in its simplest form.

Therefore, the solution to the equation is $\frac{1}{15}$.

To solve the problem $\frac{1}{2} + x = \frac{2}{5}$, we need to find $x$. We will achieve this by following these steps:

• Calculate a common denominator for the fractions $\frac{1}{2}$ and $\frac{2}{5}$.
• Subtract $\frac{1}{2}$ from $\frac{2}{5}$ once both fractions have equivalent denominators.
• Determine the simplified result for $x$.

Here's the solution:

Step 1: The denominators for $\frac{1}{2}$ and $\frac{2}{5}$ are 2 and 5. The least common denominator is 10.

Step 2: Convert each fraction to have the denominator of 10.

Convert $\frac{1}{2}$ to $\frac{5}{10}$ because $\frac{1 \cdot 5}{2 \cdot 5} = \frac{5}{10}$.

Convert $\frac{2}{5}$ to $\frac{4}{10}$ because $\frac{2 \cdot 2}{5 \cdot 2} = \frac{4}{10}$.

Step 3: Subtract the converted $\frac{1}{2}$ from $\frac{2}{5}$.

This gives us $\frac{4}{10} - \frac{5}{10} = -\frac{1}{10}$.

Therefore, the value of $x$ that satisfies the equation is $-\frac{1}{10}$.

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

• Step 1: Rearrange the equation $\frac{1}{6} + x = \frac{1}{2}$ to solve for $x$.
• Step 2: Subtract $\frac{1}{6}$ from both sides of the equation, so $x = \frac{1}{2} - \frac{1}{6}$.
• Step 3: Using a common denominator for subtraction.

Now, let's work through each step:
Step 1: Rearrange the equation: $x = \frac{1}{2} - \frac{1}{6}$.
Step 2: To subtract fractions, we need a common denominator. The least common denominator of 2 and 6 is 6.
Step 3: Rewrite each fraction with the common denominator:
$\frac{1}{2} = \frac{3}{6}$ And $\frac{1}{6}$ is already with the denominator 6.
Step 4: Subtract the fractions: $x = \frac{3}{6} - \frac{1}{6} = \frac{2}{6}$.
Step 5: Simplify $\frac{2}{6}$ to $\frac{1}{3}$.

Therefore, the solution to the problem is $x = \frac{1}{3}$.

To find the missing number in the equation $? + \frac{2}{8} = \frac{3}{4}$, we will follow these steps:

• Step 1: Simplify the fraction $\frac{2}{8}$ to $\frac{1}{4}$.
• Step 2: Recognize that we need to find a fraction that, when added to $\frac{1}{4}$, results in $\frac{3}{4}$.
• Step 3: Set up the equation $? + \frac{1}{4} = \frac{3}{4}$.
• Step 4: Solve for the missing fraction by subtracting $\frac{1}{4}$ from $\frac{3}{4}$:
  $? = \frac{3}{4} - \frac{1}{4} = \frac{2}{4}$.
• Step 5: Simplify the resulting fraction: $\frac{2}{4} = \frac{1}{2}$.

Therefore, the missing number in the equation is $\frac{2}{4}$.

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

• Step 1: Simplify the fraction $\frac{2}{8}$.
• Step 2: Determine a common denominator for $\frac{1}{4}$ and $\frac{3}{5}$.
• Step 3: Subtract $\frac{1}{4}$ from $\frac{3}{5}$ with this common denominator.
• Step 4: Simplify the result if necessary.

Now, let's work through each step:

Step 1: Simplify $\frac{2}{8}$ to $\frac{1}{4}$ since both 2 and 8 share a common factor of 2.
Step 2: Calculate the least common denominator (LCD) for $\frac{1}{4}$ and $\frac{3}{5}$.
The LCD of 4 and 5 is 20.
Rewrite $\frac{1}{4}$ as $\frac{5}{20}$ and $\frac{3}{5}$ as $\frac{12}{20}$.

Step 3: Subtract the fractions: $\frac{12}{20} - \frac{5}{20} = \frac{7}{20}$.

Therefore, the missing fraction needed to satisfy the original equation is $\frac{7}{20}$.

Thus, the solution to the problem is $\frac{7}{20}$.