Solving an Equation by Multiplication/ Division: Using ratios for calculation

To solve this problem, we will find a common equation to account for all students:

• Activity: Playing catch, Fraction: $\frac{1}{5}x$
• Activity: Playing soccer, Fraction: 20% of $x$ which is $\frac{1}{5}x$
• Activity: Watching a movie, Number: $15$ students

The total number of students involved is $x$. Thus, the setup for the equation is:

$\frac{1}{5}x + \frac{1}{5}x + 15 = x$

Simplify and solve for $x$:

$\frac{2}{5}x + 15 = x$

Subtract $\frac{2}{5}x$ from both sides:

$15 = x - \frac{2}{5}x$

$15 = \frac{3}{5}x$

To isolate $x$, multiply both sides by $\frac{5}{3}$:

$x = 15 \times \frac{5}{3}$

$x = 25$

Therefore, the total number of students is 25 students.

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

• Step 1: Define variables based on the problem.
• Step 2: Set up an equation using the given total and ratio.
• Step 3: Solve the equation to find the number of girls.
• Step 4: Calculate the number of boys using the ratio.

Now, let's work through each step:
Step 1: Let $g$ be the number of girls. According to the problem, there are 3 times as many boys as girls, so the number of boys is $3g$.
Step 2: Since the total number of students is 28, we set up the equation:
$\ g + 3g = 28$
Step 3: Combine like terms to simplify the equation:
$\ 4g = 28$
To solve for $g$, divide both sides by 4:
$\ g = \frac{28}{4} = 7$
Step 4: Calculate the number of boys as $3g$:
$\ 3g = 3 \times 7 = 21$

Therefore, the solution to the problem is 21 boys in the class.

To solve this problem, we shall adhere to the following steps:

• Step 1: Utilize the area formula for triangles.
• Step 2: Simplify the equation to find the variable $x$.
• Step 3: Verify the result against the multiple-choice options.

Now, let us execute these steps:

Step 1: Start by applying the triangle area formula $A = \frac{1}{2} \times \text{base} \times \text{height}$.
The given area is $10 \, \text{cm}^2$, the base is $x$, and the height is $5x$. Thus, the formula becomes:

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

Step 2: Simplify the equation:
$10 = \frac{1}{2} \times 5x^2$ $10 = \frac{5}{2}x^2$

Multiply both sides by $2$ to eliminate the fraction:

$20 = 5x^2$

Divide both sides by $5$:

$4 = x^2$

Take the square root of both sides:

$x = 2$

So, the value of $x$ is $\boxed{2}$.

Step 3: Upon reviewing the given multiple-choice options, the answer $x = 2$ corresponds to one of the listed choices, ensuring our calculations align with the expected solution.

Therefore, the solution to the problem is $x = 2$.