Solving the Proportion \( \frac{a+b}{b+c} = \frac{c+d}{d+a} \)

The problem gives us a proportion involving four variables \(a, b, c,\) and \(d\):

\( \frac{a+b}{b+c} = \frac{c+d}{d+a} \)

We are also given the condition that \( a \ne c \). We need to find which of the given options correctly describes the relationship between these variables based on the proportion.

Step-by-Step Solution

To solve this proportion, we can use cross-multiplication. This means multiplying the numerator of the left side by the denominator of the right side, and setting it equal to the numerator of the right side multiplied by the denominator of the left side.

Given proportion:

\( \frac{a+b}{b+c} = \frac{c+d}{d+a} \)

Cross-multiply:

\( (a+b)(d+a) = (c+d)(b+c) \)

Expanding Both Sides

Now, we expand both sides of the equation:

Left side expansion:

\( (a+b)(d+a) = a(d+a) + b(d+a) = ad + a^2 + bd + ba \)

Right side expansion:

\( (c+d)(b+c) = c(b+c) + d(b+c) = cb + c^2 + db + dc \)

So the equation becomes:

\( ad + a^2 + bd + ba = cb + c^2 + db + dc \)

Rearranging and Simplifying the Equation

Let's move all terms to one side to simplify. We can subtract \( (cb + c^2 + db + dc) \) from both sides.

\( ad + a^2 + bd + ba - cb - c^2 - db - dc = 0 \)

Notice that the term \( bd \) appears on both sides ( \( bd \) on the left and \( -db \) on the right after moving terms). These terms cancel each other out since \( bd = db \).

\( ad + a^2 + ba - cb - c^2 - dc = 0 \)

Rearrange the terms to group similar parts, maybe related to \(a\) and \(c\):

\( a^2 - c^2 + ad - dc + ba - cb = 0 \)

Factoring the Equation

We can factor each group of terms:

• \( a^2 - c^2 \) is a difference of squares, which factors as \( (a-c)(a+c) \).
• \( ad - dc \) has a common factor \( d \), factoring as \( d(a-c) \).
• \( ba - cb \) (or \( ab - bc \)) has a common factor \( b \), factoring as \( b(a-c) \).

Substitute these factored forms back into the equation:

\( (a-c)(a+c) + d(a-c) + b(a-c) = 0 \)

Now we see that \( (a-c) \) is a common factor in all terms. We can factor it out:

\( (a-c) [ (a+c) + d + b ] = 0 \)

Using the Given Condition

We are given that \( a \ne c \). This means that the term \( (a-c) \) is not equal to zero.

For the entire expression \( (a-c) [ (a+c) + d + b ] \) to be equal to zero, if one factor is not zero, the other factor must be zero.

Since \( (a-c) \ne 0 \), it must be that \( (a+c) + d + b = 0 \).

Rearranging the terms inside the bracket gives us the relationship:

\( a + b + c + d = 0 \)

Checking the Options

Let's compare our derived relationship \( a + b + c + d = 0 \) with the given options:

• Option 1: \( a + b = c + d \) (This is not the same)
• Option 2: \( a + c = b + d \) (This is not the same)
• Option 3: \( a - b - c + d = 0 \) (This is not the same)
• Option 4: \( a + b + c + d = 0 \) (This matches our result)

Therefore, the correct relationship is \( a + b + c + d = 0 \).

Revision Table: Solving Proportions

Additional Information: Algebraic Properties and Conditions

When solving algebraic equations derived from proportions, it's important to be mindful of any conditions given in the problem, like \( a \ne c \) in this case.

• Why \( a \ne c \) is crucial: If \( a \) were equal to \( c \), the term \( (a-c) \) would be zero. Our factored equation \( (a-c)(a+b+c+d) = 0 \) would become \( 0 \times (a+b+c+d) = 0 \). This equation \( 0 = 0 \) is always true, regardless of the values of \( a, b, c, d \) (as long as denominators in the original proportion are non-zero). The condition \( a \ne c \) is what allows us to divide by \( (a-c) \) or conclude that the other factor must be zero.
• Alternative approach (Component and Dividendo - Not suitable here): While properties like Component and Dividendo exist for proportions, they are typically applied to simpler forms like \( \frac{a}{b} = \frac{c}{d} \). The given expression \( \frac{a+b}{b+c} = \frac{c+d}{d+a} \) is more complex and direct algebraic manipulation via cross-multiplication is the most straightforward method.
• Checking for denominator zero: The original proportion has denominators \( b+c \) and \( d+a \). For the proportion to be well-defined, we must have \( b+c \ne 0 \) and \( d+a \ne 0 \). The solution \( a+b+c+d=0 \) doesn't automatically guarantee these are non-zero. For example, if \( a=1, b=0, c=-1, d=0 \), then \( a+b+c+d=0 \) and \( a \ne c \). But \( b+c = 0 + (-1) = -1 \ne 0 \) and \( d+a = 0+1 = 1 \ne 0 \). However, if \( a=1, b=1, c=-1, d=-1 \), then \( a+b+c+d=0 \) and \( a \ne c \). But \( b+c = 1+(-1) = 0 \), which would make the original proportion undefined. The problem assumes the original proportion is valid, implying \(b+c \ne 0\) and \(d+a \ne 0\).