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### 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

Learn how to solve proportion and scale problems, including a detailed step-by-step solution for converting measurements using scales. Suitable for high school students studying mathematics.

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