Direct Proportion

To solve this problem, we will determine the number of white balls in the box using a fraction of the total number of balls.

We are given the total number of balls in the box as 18, and we know that $\frac{2}{3}$ of these balls are white. To find the number of white balls, we follow these steps:

• Step 1: Identify the total quantity, which is 18 balls.
• Step 2: Use the given fraction $\frac{2}{3}$ to find the number of white balls.
• Step 3: Multiply the total number of balls by the fraction of white balls: $18 \times \frac{2}{3}$.

Perform the calculation:

$18 \times \frac{2}{3} = 18 \times 0.6667 = 12$

Alternatively, calculate directly using fractions:

$18 \times \frac{2}{3} = \frac{18 \times 2}{3} = \frac{36}{3} = 12$

Thus, the total number of white balls in the box is 12.

Therefore, the correct answer is choice 12.

To solve this problem, we'll determine the number of orange balls by calculating the fraction of the total number of balls:

• Step 1: Identify the total number of balls, $28$.
• Step 2: Note the fraction representing the orange balls, $\frac{1}{4}$.
• Step 3: Apply the formula to find the number of orange balls:
  Number of orange balls $= 28 \times \frac{1}{4}$

Now, let's perform the calculation:
$28 \times \frac{1}{4} = 28 \div 4 = 7$

Therefore, the number of orange balls in the box is $7$.

The area of a circle is calculated using the following formula:

where r represents the radius.

Using the formula, we calculate the areas of the circles:

Circle 1:

π*4² =

π16

Circle 2:

π*10² =

π100

To calculate how much larger one circle is than the other (in other words - what is the ratio between them)

All we need to do is divide one area by the other.

100/16 =

6.25

Therefore the answer is 6 and a quarter!

Let's find side BC

Based on what we're given:

$\frac{AB}{BC}=\frac{x}{BC}=\sqrt{\frac{x}{2}}$

$\frac{x}{BC}=\frac{\sqrt{x}}{\sqrt{2}}$

$\sqrt{2}x=\sqrt{x}BC$

Let's divide by square root x:

$\frac{\sqrt{2}\times x}{\sqrt{x}}=BC$

$\frac{\sqrt{2}\times\sqrt{x}\times\sqrt{x}}{\sqrt{x}}=BC$

Let's reduce the numerator and denominator by square root x:

$\sqrt{2}\sqrt{x}=BC$

We'll use the Pythagorean theorem to calculate the area of triangle ABC:

$AB^2+BC^2=AC^2$

Let's substitute what we're given:

$x^2+(\sqrt{2}\sqrt{x})^2=m^2$

$x^2+2x=m^2$

We know that:

$\frac{AB}{BC}=\sqrt{\frac{x}{2}}$

We also know that AB equals X.

First, we will substitute the given data into the formula accordingly:

$\frac{x}{BC}=\frac{\sqrt{x}}{\sqrt{2}}$

$x\sqrt{2}=BC\sqrt{x}$

$\frac{x\sqrt{2}}{\sqrt{x}}=BC$

$\frac{\sqrt{x}\times\sqrt{x}\times\sqrt{2}}{\sqrt{x}}=BC$

$\sqrt{x}\times\sqrt{2}=BC$

Now let's look at triangle ABC and use the Pythagorean theorem:

$AB^2+BC^2=AC^2$

We substitute in our known values:

$x^2+(\sqrt{x}\times\sqrt{2})^2=m^2$

$x^2+x\times2=m^2$

Finally, we will add 1 to both sides:

$x^2+2x+1=m^2+1$

$(x+1)^2=m^2+1$