Multiplication of Fractions: In combination with other operations

To solve this mathematical problem, follow these steps:

• Step 1: Multiply the fractions $\frac{3}{4}$ and $\frac{3}{4}$.
• Step 2: Subtract $\frac{1}{4}$ from the product obtained in Step 1.

Let's execute each step in detail:

Step 1: Calculate the product of $\frac{3}{4}$ and $\frac{3}{4}$.

To multiply two fractions, multiply their numerators and their denominators separately:

$\frac{3}{4} \times \frac{3}{4} = \frac{3 \times 3}{4 \times 4} = \frac{9}{16}$

Step 2: Subtract $\frac{1}{4}$ from $\frac{9}{16}$.

Before we subtract $\frac{1}{4}$ from $\frac{9}{16}$, we need a common denominator. The common denominator for these fractions is 16:

$\frac{1}{4} = \frac{1 \times 4}{4 \times 4} = \frac{4}{16}$

Now subtract $\frac{4}{16}$ from $\frac{9}{16}$:

$\frac{9}{16} - \frac{4}{16} = \frac{9 - 4}{16} = \frac{5}{16}$

Therefore, the solution to the given problem is $\frac{5}{16}$.

To solve $\frac{1}{2} \times \frac{1}{2} + \frac{3}{4}$, follow these steps:

• Step 1: Multiply $\frac{1}{2} \times \frac{1}{2}$ by using the multiplication of fractions formula:
  $\frac{1}{2} \times \frac{1}{2} = \frac{1 \times 1}{2 \times 2} = \frac{1}{4}$.
• Step 2: Add $\frac{1}{4}$ to $\frac{3}{4}$.
  Since $\frac{1}{4}$ and $\frac{3}{4}$ already have the same denominator, the addition can be done directly:
  $\frac{1}{4} + \frac{3}{4} = \frac{1 + 3}{4} = \frac{4}{4} = 1$.

Therefore, the correct solution to the expression is $1$.

To solve the problem $\frac{3}{5} \times \frac{2}{3} + \frac{2}{5}$, we proceed with the following steps:

• Step 1: Multiply the fractions $\frac{3}{5}$ and $\frac{2}{3}$.

The multiplication yields:

$\frac{3}{5} \times \frac{2}{3} = \frac{3 \times 2}{5 \times 3} = \frac{6}{15}$

• Step 2: Simplify the product $\frac{6}{15}$.

Both 6 and 15 share a common factor of 3:

$\frac{6}{15} = \frac{6 \div 3}{15 \div 3} = \frac{2}{5}$

• Step 3: Add $\frac{2}{5}$ to the simplified result $\frac{2}{5}$.

Since the fractions $\frac{2}{5}$ and $\frac{2}{5}$ have the same denominator, add the numerators while keeping the denominator:

$\frac{2}{5} + \frac{2}{5} = \frac{2+2}{5} = \frac{4}{5}$

Therefore, the solution to the problem is $\frac{4}{5}$.

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

• Step 1: Multiply the fractions. Calculate $\frac{2}{3} \times \frac{1}{3}$.
• Step 2: Add the product to another fraction. Add the result to $\frac{2}{9}$.

Now, let's work through the calculations:

Step 1: Multiply $\frac{2}{3}$ by $\frac{1}{3}$.

The formula for multiplying fractions is:

$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$.

Substitute the values:

$\frac{2}{3} \times \frac{1}{3} = \frac{2 \times 1}{3 \times 3} = \frac{2}{9}$.

Step 2: Add $\frac{2}{9}$ to the product.

We found in Step 1 that $\frac{2}{3} \times \frac{1}{3} = \frac{2}{9}$.

Now add $\frac{2}{9} + \frac{2}{9} = \frac{2 + 2}{9} = \frac{4}{9}$.

Therefore, the solution to the expression is $\frac{4}{9}$.

To solve the problem $\frac{3}{4} \times \frac{1}{2} + \frac{5}{8}$, we'll follow these steps:

• Step 1: Multiply the fractions $\frac{3}{4} \times \frac{1}{2}$.
• Step 2: Add the result to $\frac{5}{8}$.

Now, let's work through the steps:

Step 1: Compute the product of the first two fractions:
$\frac{3}{4} \times \frac{1}{2} = \frac{3 \times 1}{4 \times 2} = \frac{3}{8}$

Step 2: Add the resulting fraction to $\frac{5}{8}$ by finding a common denominator:

The fractions $\frac{3}{8}$ and $\frac{5}{8}$ already have the same denominator, so we can simply add them:
$\frac{3}{8} + \frac{5}{8} = \frac{3 + 5}{8} = \frac{8}{8} = 1$

Therefore, the solution to the problem is $1$.

To solve the expression $\frac{4}{4} \times \frac{1}{2} + \frac{3}{8}$, follow these steps:

• Step 1: Simplify $\frac{4}{4}$. Since $\frac{4}{4} = 1$, the expression becomes $1 \times \frac{1}{2} + \frac{3}{8}$.
• Step 2: Perform the multiplication.
  Calculate $1 \times \frac{1}{2} = \frac{1}{2}$.
• Step 3: Add $\frac{1}{2}$ to $\frac{3}{8}$.
  First, convert $\frac{1}{2}$ to an equivalent fraction with a denominator of 8: $\frac{1}{2} = \frac{4}{8}$.
• Step 4: Now, add the fractions: $\frac{4}{8} + \frac{3}{8} = \frac{7}{8}$.

Thus, the final result is $\frac{7}{8}$.

To solve the given problem, we will follow these steps:

• Step 1: Perform the multiplication of the fractions.
• Step 2: Simplify the result, if applicable.
• Step 3: Add the simplified fractional result to the given fraction, ensuring the denominators align properly.
• Step 4: Simplify the final result, if necessary.

Let's go through each step:

Step 1: Multiply the fractions $\frac{2}{3} \times \frac{2}{3} = \frac{2 \times 2}{3 \times 3} = \frac{4}{9}$.

Step 2: The result from step 1 is $\frac{4}{9}$, which cannot be further simplified.

Step 3: Add the result from Step 2 to $\frac{4}{9}$ given in the problem:
We have two fractions $\frac{4}{9}$ and $\frac{4}{9}$, and since they already have a common denominator, we add them directly:
$\frac{4}{9} + \frac{4}{9} = \frac{4 + 4}{9} = \frac{8}{9}$.

Step 4: The fraction $\frac{8}{9}$ is already in its simplest form.

Therefore, the solution to the problem is $\frac{8}{9}$.

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

• Step 1: Multiply the given fractions.
• Step 2: Simplify if necessary.
• Step 3: Perform the addition of resulting fractions.

Now, let's work through each step:

Step 1: Multiply the fractions:
$\frac{1}{4} \times \frac{1}{2} = \frac{1 \times 1}{4 \times 2} = \frac{1}{8}$

Step 2: We find that the result is already simplified.

Step 3: Add $\frac{1}{8}$ to $\frac{3}{8}$:
$\frac{1}{8} + \frac{3}{8} = \frac{1 + 3}{8} = \frac{4}{8} = \frac{1}{2}$

The fractions have the same denominator, allowing for direct addition.

Therefore, the solution to the problem is $\frac{1}{2}$.

To solve this problem, we'll approach it in the following steps:

Step 1: Perform the Multiplication
The expression begins with multiplying two fractions: $\frac{1}{4} \times \frac{4}{5}$. Using the formula for multiplying fractions, we get:
$\frac{1 \times 4}{4 \times 5} = \frac{4}{20}$
Simplifying $\frac{4}{20}$ by dividing both numerator and denominator by 4 gives:
$\frac{1}{5}$

Step 2: Add the Result to the Second Fraction
Now, we need to add $\frac{1}{5}$ to $\frac{11}{20}$. To do this, we first find a common denominator.
The least common denominator between 5 and 20 is 20. Convert $\frac{1}{5}$ to twentieths:
$\frac{1}{5} = \frac{4}{20}$
Now add $\frac{4}{20}$ to $\frac{11}{20}$:
$\frac{4}{20} + \frac{11}{20} = \frac{15}{20}$

Step 3: Simplify the Final Result
Simplify $\frac{15}{20}$ by dividing the numerator and the denominator by 5:
$\frac{15 \div 5}{20 \div 5} = \frac{3}{4}$

Therefore, the solution to the problem is $\frac{3}{4}$. This matches choice 1, which is $\frac{3}{4}$.

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

• Step 1: Multiply the first two fractions.
• Step 2: Add the result to the third fraction.
• Step 3: Simplify the final result.

Now, let's work through each step:
Step 1: Multiply $\frac{4}{5}$ by $\frac{1}{2}$. According to the multiplication rule for fractions, we have:
$\frac{4}{5} \times \frac{1}{2} = \frac{4 \times 1}{5 \times 2} = \frac{4}{10}$ Step 2: We need to add $\frac{4}{10}$ to $\frac{3}{10}$. Since these fractions have the same denominator, we can add them directly:
$\frac{4}{10} + \frac{3}{10} = \frac{4 + 3}{10} = \frac{7}{10}$ Step 3: The sum $\frac{7}{10}$ is already in simplest form.

Therefore, the solution to the problem is $\frac{7}{10}$, which matches choice $\text{(3)}$.

To solve the given expression, follow these steps:

First, multiply the fractions $\frac{3}{5}$ and $\frac{1}{2}$:

$\frac{3}{5} \times \frac{1}{2} = \frac{3 \times 1}{5 \times 2} = \frac{3}{10}$

Now, add $\frac{3}{10}$ to the result of the multiplication:

Since the fractions $\frac{3}{10}$ and $\frac{3}{10}$ have the same denominator, we can simply add their numerators:

$\frac{3}{10} + \frac{3}{10} = \frac{3 + 3}{10} = \frac{6}{10}$

Simplify $\frac{6}{10}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

$\frac{6}{10} = \frac{6 \div 2}{10 \div 2} = \frac{3}{5}$

Therefore, the solution to the problem is $\frac{3}{5}$.

According to the order of operations, we will first solve the expression in parentheses.

Note that since the denominators are not common, we will look for a number that is both divisible by 2 and 3. That is 6.

We will multiply one-third by 2 and one-half by 3, now we will get the expression:

$\frac{1}{4}\times(\frac{2+3}{6})=$

Let's solve the numerator of the fraction:

$\frac{1}{4}\times\frac{5}{6}=$

We will combine the fractions into a multiplication expression:

$\frac{1\times5}{4\times6}=\frac{5}{24}$