Solving an Equation by Multiplication/ Division: Using fractions

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

• Identify the given fraction equation.
• Multiply both sides of the equation by the reciprocal of the coefficient of $x$.
• Simplify to isolate $x$.

Now, let's work through these steps:
Step 1: The problem gives us the equation $\frac{1}{3} x = \frac{1}{9}$.
Step 2: We multiply both sides by 3 to eliminate the fraction on the left side:

$3 \times \frac{1}{3} x = 3 \times \frac{1}{9}$

Step 3: Simplifying both sides results in:

$x = \frac{3}{9}$

Further simplification of $\frac{3}{9}$ yields:

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

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

We begin by multiplying the simple fraction by y:

$\frac{-1}{5}\times y=-25$

We then reduce both terms by $-\frac{1}{5}$

$y=\frac{-25}{-\frac{1}{5}}$

Finally we multiply the fraction by negative 5:

$y=-25\times(-5)=125$

To solve the equation $3\frac{1}{2} \cdot y = 21$, we'll follow these steps:

• Convert the mixed number to an improper fraction.
• Divide both sides of the equation by the coefficient of $y$.

Let's analyze these steps in detail:

Step 1: Convert the mixed number to an improper fraction.
The coefficient of $y$ is $3\frac{1}{2}$. Converting to an improper fraction, we have:

$3\frac{1}{2} = \frac{7}{2}$

Step 2: Divide both sides of the equation by $\frac{7}{2}$.
The equation becomes:

$\frac{7}{2} \cdot y = 21$

To isolate $y$, divide both sides by $\frac{7}{2}$:

$y = 21 \div \frac{7}{2}$

Dividing by a fraction is equivalent to multiplying by its reciprocal, so:

$y = 21 \cdot \frac{2}{7}$

Carrying out the multiplication, we calculate:

$y = \frac{21 \cdot 2}{7} = \frac{42}{7}$

Dividing the numerator by the denominator gives us:

$y = 6$

Thus, the solution to the equation is $y = 6$.

To solve the equation $3b = \frac{7}{6}$ for the variable $b$, we will perform the following steps:

• Step 1: Identify the equation. The given equation is $3b = \frac{7}{6}$.
• Step 2: Isolate the variable. Divide both sides by 3 to solve for $b$.

When we divide both sides of the equation by 3, we obtain:

Step 3: Simplify the expression. Dividing a fraction by an integer is equivalent to multiplying the denominator of the fraction by that integer:

The denominator becomes:

Thus, the solution to the equation is $b = \frac{7}{18}$.

This matches the correct answer choice among the given options.

Therefore, the value of $b$ is $b = \frac{7}{18}$.

To solve the equation $\frac{1}{5}x - 4 = 6$, we will follow these steps:

• Step 1: Add 4 to both sides of the equation to eliminate the subtraction and isolate the fractional term.
• Step 2: Multiply both sides by 5 to clear the fraction and solve for $x$.

Let's apply these steps to solve the equation:

Step 1: Add 4 to both sides:
$\frac{1}{5}x - 4 + 4 = 6 + 4$
This simplifies to:
$\frac{1}{5}x = 10$

Step 2: Multiply both sides by 5 to solve for $x$:
$5 \times \frac{1}{5}x = 10 \times 5$
This simplifies to:
$x = 50$

Therefore, the solution to the equation is $x = 50$.

To solve the equation $\frac{2}{8}x - 3 = 7$, we'll follow these steps:

• Step 1: Simplify the fraction. The coefficient $\frac{2}{8}$ simplifies to $\frac{1}{4}$.
• Step 2: Eliminate the constant term by adding 3 to both sides of the equation.
• Step 3: Solve for $x$ by removing the coefficient of $x$ using division.

Let's solve the equation step-by-step:

Step 1: Simplify the equation:
The equation $\frac{2}{8}x - 3 = 7$ simplifies to $\frac{1}{4}x - 3 = 7$.

Step 2: Eliminate the constant term:
Add 3 to both sides to isolate the term involving $x$:

$\frac{1}{4}x - 3 + 3 = 7 + 3$

This simplifies to:

$\frac{1}{4}x = 10$

Step 3: Solve for $x$:
Multiply both sides by the reciprocal of $\frac{1}{4}$ to solve for $x$:

$4 \cdot \frac{1}{4}x = 4 \cdot 10$

This simplifies to:

$x = 40$

Therefore, the solution to the equation is $x = 40$.

To solve the equation $\frac{3x}{4} = 16$, we will eliminate the fraction by multiplying both sides by 4.

• Step 1: Multiply both sides by 4:
  $\left(\frac{3x}{4}\right) \times 4 = 16 \times 4$
• Step 2: Simplify:
  $3x = 64$
• Step 3: Solve for $x$ by dividing both sides by 3:
  $x = \frac{64}{3}$
• Step 4: Simplify the fraction to a mixed number:
  $x = 21\frac{1}{3}$

Therefore, the solution to the equation $\frac{3x}{4} = 16$ is $x = 21\frac{1}{3}$.

To solve the equation $\frac{x}{4} + 2x - 18 = 0$, we proceed as follows:

• Step 1: Eliminate the fraction by multiplying the entire equation by 4:
  $(4) \Big(\frac{x}{4} + 2x - 18\Big) = (4)(0)$
• Step 2: Distribute and simplify:
  $x + 8x - 72 = 0$
• Step 3: Combine like terms:
  $9x - 72 = 0$
• Step 4: Isolate $9x$ by adding 72 to both sides:
  $9x = 72$
• Step 5: Solve for $x$ by dividing both sides by 9:
  $x = \frac{72}{9}$
• Step 6: Simplify the division:
  $x = 8$

Thus, the solution to the problem is $x = 8$.

First, we cross multiply:

$8\times(x+4)=3\times7$

We multiply the right section and expand the parenthesis, multiplying each of the terms by 8:

$8x+32=21$

We rearrange the equation remembering change the plus and minus signs accordingly:

$8x=21-32$
Solve the subtraction exercise on the right side and divide by 8:

$8x=-11$

$\frac{8x}{8}=-\frac{11}{8}$

Convert the simple fraction into a mixed fraction:

$x=-1\frac{3}{8}$

First, we will arrange the equation so that we have variables on one side and numbers on the other side.

Therefore, we will move $\frac{5}{6}$ to the other side, and we will get

$\frac{1}{3}x=-\frac{1}{6}-\frac{5}{6}$

Note that the two fractions on the right side share the same denominator, so you can subtract them:

$\frac{1}{3}x=-\frac{6}{6}$

Observe the minus sign on the right side!

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

Now, we will try to get rid of the denominator, we will do this by multiplying the entire exercise by the denominator (that is, all terms on both sides of the equation):

$1x=-3$

$x=-3$

Let's proceed with solving the equation step by step:

1. Start with the equation $\frac{2}{3}x + \frac{1}{4} = \frac{3}{4}$.

2. Subtract $\frac{1}{4}$ from both sides to remove the constant term on the left:
   $\frac{2}{3}x + \frac{1}{4} - \frac{1}{4} = \frac{3}{4} - \frac{1}{4}$.

3. This simplifies to: $\frac{2}{3}x = \frac{3}{4} - \frac{1}{4}$.

4. Perform the subtraction on the right-hand side:
   $\frac{2}{3}x = \frac{2}{4} = \frac{1}{2}$.

5. Now solve for $x$ by dividing both sides of the equation by $\frac{2}{3}$:
   $x = \frac{1}{2} \div \frac{2}{3}$.

6. Dividing by a fraction is the same as multiplying by its reciprocal:
   $x = \frac{1}{2} \times \frac{3}{2}$.

7. Simplify the multiplication:
   $x = \frac{3}{4}$.

Therefore, the value of the parameter $x$ is $\frac{3}{4}$.

Let's solve the equation $\frac{2}{3}x - \frac{4}{6} = \frac{1}{3}$.

Step 1: Simplify the fractions.

• The term $\frac{4}{6}$ is equivalent to $\frac{2}{3}$ after simplification.

Now, the equation can be rewritten as:

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

Step 2: Add $\frac{2}{3}$ to both sides to isolate the term with $x$.

$\frac{2}{3}x = \frac{1}{3} + \frac{2}{3}$

Simplify the right side:

$\frac{2}{3}x = \frac{3}{3}$

$\frac{3}{3} = 1$

So the equation becomes:

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

Step 3: Solve for $x$ by multiplying both sides by the reciprocal of $\frac{2}{3}$.

Multiply both sides by $\frac{3}{2}$:

$x = 1 \times \frac{3}{2}$

Thus, the solution is:

$x = \frac{3}{2}$

The solution to the problem is $x = \frac{3}{2}$.

To find the value of $x$ in the given equation, we will perform the following steps:

• Step 1: Start with the equation given: $3x - \frac{1}{9} = \frac{8}{9}$.
• Step 2: To eliminate the constant $-\frac{1}{9}$ on the left, add $\frac{1}{9}$ to both sides:

$3x - \frac{1}{9} + \frac{1}{9} = \frac{8}{9} + \frac{1}{9}$

This simplifies to:

$3x = \frac{8}{9} + \frac{1}{9}$

Combine the fractions on the right side:

$\frac{8}{9} + \frac{1}{9} = \frac{9}{9} = 1$

So, now we have:

$3x = 1$

• Step 3: Divide both sides by 3 to solve for $x$:

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

Thus, the solution to the equation is:

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

To solve for $x$ in the equation $\frac{7}{8}x = \frac{2}{5}$, we will follow these steps:

• Multiply both sides of the equation by the reciprocal of $\frac{7}{8}$, which is $\frac{8}{7}$.
• Simplify the resulting expression to find the value of $x$.

Let's work through these steps:

First, multiply both sides by $\frac{8}{7}$ to isolate $x$ on the left side.

$\frac{8}{7} \times \frac{7}{8}x = \frac{8}{7} \times \frac{2}{5}$

This simplifies to:

$x = \frac{8}{7} \times \frac{2}{5}$

Now, perform the multiplication of the fractions:

$x = \frac{8 \times 2}{7 \times 5} = \frac{16}{35}$

Thus, the value of $x$ is $\frac{16}{35}$.

To solve the equation $12y + 4y + 5 - 3 = 2y$, we'll follow these steps:

• **Step 1**: Simplify the left side of the equation by combining like terms: $12y + 4y = 16y$.
• **Step 2**: Replace that in the equation: $16y + 5 - 3 = 2y$. Simplify further to $16y + 2 = 2y$.
• **Step 3**: Isolate $y$ by getting all the terms involving $y$ on one side. Subtract $2y$ from both sides, yielding: $16y - 2y = -2$.
• **Step 4**: This simplifies to $14y = -2$.
• **Step 5**: Divide each side by 14 to solve for $y$: $y = \frac{-2}{14}$.
• **Step 6**: Simplify the fraction: $y = -\frac{1}{7}$.

Therefore, the solution to the problem is $y = -\frac{1}{7}$.

To solve the equation $17.5 - 18x - 5.5x = 19.2 + 14\frac{1}{2} - 5x$, follow these steps:

Step 1: Combine like terms on both sides of the equation.

• On the left side, combine $-18x - 5.5x$, which simplifies to $-23.5x$.
• On the right side, simplify $19.2 + 14.5 - 5x$. The fraction $14\frac{1}{2}$ is converted to decimal form as $14.5$, giving $19.2 + 14.5 = 33.7$.

Step 2: Rewrite the equation with the simplified terms:

$17.5 - 23.5x = 33.7 - 5x$.

Step 3: Get all terms involving $x$ on one side of the equation and constant terms on the other.

• Add $5x$ to both sides to move all $x$ related terms to the left:
• $17.5 - 23.5x + 5x = 33.7$

This further simplifies to $17.5 - 18.5x = 33.7$.

Step 4: Isolate the term with $x$ by subtracting $17.5$ from both sides:

$-18.5x = 33.7 - 17.5$.

The right side evaluates to $16.2$.

Thus, we have $-18.5x = 16.2$.

Step 5: Solve for $x$ by dividing both sides by $-18.5$:

$x = \frac{16.2}{-18.5} \approx -0.8757$.

Rounding $-0.8757$ to two decimal places gives $x = -0.87$.

Therefore, the solution to the equation is $x = -0.87$.

This corresponds to option 2 in the given choices.

To solve the equation $22x - \frac{1}{2} + 16\frac{1}{2} = 14.5x - 12$, we will follow these steps:
1. Combine like terms on both sides of the equation.
2. Isolate the variable $x$.
3. Solve for $x$.

Let's start by simplifying each side:

• Simplify the left-hand side: $22x - \frac{1}{2} + 16\frac{1}{2}$.

The term $16\frac{1}{2}$ is equivalent to $16.5$, so the left-hand side becomes:
$22x - 0.5 + 16.5 = 22x + 16$.

• Now simplify the right-hand side: $14.5x - 12$.

The right-hand side remains as $14.5x - 12$.

Now, let's collect like terms. Move the term involving $x$ from the right-hand side to the left:

• Subtract $14.5x$ from both sides:
  $22x + 16 - 14.5x = -12$

This simplifies to:
$7.5x + 16 = -12$.

Next, isolate the constant term. Subtract 16 from both sides:

• $7.5x + 16 - 16 = -12 - 16$

This simplifies to:
$7.5x = -28$.

Finally, solve for $x$ by dividing both sides by 7.5:

• $x = \frac{-28}{7.5}$

Calculating the fraction gives approximately:
$x \approx -3.73$.

Therefore, the solution to the problem is $x = -3.73$.

To solve the equation $\frac{1}{6}x - \frac{1}{3} = \frac{1}{3}$, we will take the following steps:

• Step 1: Eliminate fractions by multiplying the entire equation by the least common multiple of the denominators $6$.
• Step 2: Simplify the resulting equation.
• Step 3: Isolate the variable $x$.

Let's proceed with the solution:

Step 1: Multiply the entire equation by $6$ to clear fractions:
$6 \left(\frac{1}{6}x - \frac{1}{3}\right) = 6 \times \frac{1}{3}$

Step 2: Simplify:
$x - 2 = 2$

Step 3: Solve for $x$ by adding $2$ to both sides:
$x = 2 + 2$

Therefore, $x = 4$.

To solve the equation $\frac{2}{5}x = \frac{3}{8}$, we need to isolate $x$. We can achieve this by multiplying both sides by the reciprocal of $\frac{2}{5}$.

Step 1: Multiply both sides by $\frac{5}{2}$, which is the reciprocal of $\frac{2}{5}$:

$\frac{5}{2} \times \frac{2}{5}x = \frac{5}{2} \times \frac{3}{8}$

Step 2: Simplify the left side. The $\frac{5}{2}$ and $\frac{2}{5}$ cancel each other out:

$x = \frac{5 \times 3}{2 \times 8}$

Step 3: Simplify the right side by multiplying the numerators and denominators:

$x = \frac{15}{16}$

Therefore, the solution to the equation is $\boxed{\frac{15}{16}}$, which matches choice 3.

To solve the equation $\frac{4}{9} + \frac{3}{5}x = \frac{4}{3}$, we will follow these steps:

• Step 1: Subtract $\frac{4}{9}$ from both sides to isolate the term involving $x$.
• Step 2: Divide by the coefficient of $x$ to solve for $x$.

Step 1: Subtract $\frac{4}{9}$ from both sides:

$\frac{3}{5}x = \frac{4}{3} - \frac{4}{9}$

To subtract these fractions, find a common denominator. The least common denominator for 3 and 9 is 9. Rewrite $\frac{4}{3}$ as $\frac{12}{9}$ (since $4 \times 3 = 12$), resulting in:

$\frac{3}{5}x = \frac{12}{9} - \frac{4}{9} = \frac{8}{9}$

Step 2: Divide both sides by $\frac{3}{5}$ to solve for $x$:

$x = \frac{8}{9} \div \frac{3}{5} = \frac{8}{9} \times \frac{5}{3}$

Multiply the fractions. The result is:

$x = \frac{8 \times 5}{9 \times 3} = \frac{40}{27}$

Thus, the solution to the equation is $x = \frac{40}{27}$.