Let's solve the questions from the provided image step-by-step.

Question 6

The floor plan of a house is drawn using a scale in which $\frac{1}{2}$ inch corresponds to 1 foot. What is the actual length of a room whose length measures $8 \frac{1}{2}$ inches in the floor plan?

To find the actual length of the room, we can set up a proportion using the given scale:

$\frac{1}{2} \text{ inch} = 1 \text{ foot}$

$8 \frac{1}{2} \text{ inches} = \frac{17}{2} \text{ inches}$

Let $x$ be the actual length in feet:

$\frac{1}{2} \text{ inch} : 1 \text{ foot} = \frac{17}{2} \text{ inches} : x \text{ feet}$

We can solve this proportion by cross-multiplying:

$\frac{1}{2} \times x = 1 \times \frac{17}{2}$

$\frac{x}{2} = \frac{17}{2}$

Multiplying both sides by 2:

$x = 17$

So, the actual length of the room is $17$ feet.

Question 7

Prove the given proportion properties.

(a) If $\frac{a}{b} = \frac{c}{d}$, prove that $\frac{b}{a} = \frac{d}{c}$.

Given:

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

This means:

$a \cdot d = b \cdot c$

Taking the reciprocal of both sides:

$\frac{1}{a \cdot d} = \frac{1}{b \cdot c}$

Rewriting this:

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

Therefore:

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

(b) If $\frac{a}{b} = \frac{c}{d}$, prove that $\frac{a + b}{b} = \frac{c + d}{d}$.

Given:

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

This means:

$a = \frac{b \cdot c}{d}$

Adding $b$ to both sides:

$a + b = \frac{b \cdot c}{d} + b$

Factoring out $b$:

$a + b = b \left( \frac{c}{d} + 1 \right)$

Dividing both sides by $b$:

$\frac{a + b}{b} = \frac{c}{d} + 1$

Rewriting $\frac{c}{d} + 1$:

$\frac{a + b}{b} = \frac{c + d}{d}$

(c) If $\frac{a}{b} = \frac{c}{d}$, prove that $\frac{a - b}{b} = \frac{c - d}{d}$.

Given:

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

This means:

$a = \frac{b \cdot c}{d}$

Subtracting $b$ from both sides:

$a - b = \frac{b \cdot c}{d} - b$

Factoring out $b$:

$a - b = b \left( \frac{c}{d} - 1 \right)$

Dividing both sides by $b$:

$\frac{a - b}{b} = \frac{c}{d} - 1$

Rewriting $\frac{c}{d} - 1$:

$\frac{a - b}{b} = \frac{c - d}{d}$

(d) If $\frac{a}{b} = \frac{c}{d}$, prove that $\frac{a + b}{a - b} = \frac{c + d}{c - d}$.

Given:

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

This means:

$a = \frac{b \cdot c}{d}$

Now we need to prove:

$\frac{a + b}{a - b} = \frac{c + d}{c - d}$

Using $a = \frac{b \cdot c}{d}$, we can write:

$a + b = \frac{b \cdot c}{d} + b = b \left( \frac{c}{d} + 1 \right)$

$a - b = \frac{b \cdot c}{d} - b = b \left( \frac{c}{d} - 1 \right)$

So:

$\frac{a + b}{a - b} = \frac{b \left( \frac{c}{d} + 1 \right)}{b \left( \frac{c}{d} - 1 \right)} = \frac{\frac{c}{d} + 1}{\frac{c}{d} - 1}$

[ = \frac{\frac{c + d}{d}}{\frac{c - d}{d}} = \frac