<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-cke-white-space-first="1" data-cke-white-space-last="1"> <mfrac> <mrow> <mfrac> <mrow> <mn>y</mn> </mrow> <mrow> <mn>b</mn> </mrow> </mfrac> <mo>−</mo> <mfrac> <mrow> <mn>z</mn> </mrow> <mrow> <mn>c</mn> </mrow> </mfrac> </mrow> <mrow> <mspace width="5px"></mspace> <mspace width="5px"></mspace> <mfrac> <mrow> <mn>y</mn> </mrow> <mrow> <mn>b</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>z</mn> </mrow> <mrow> <mn>c</mn> </mrow> </mfrac> <mspace width="5px"></mspace> <mspace width="5px"></mspace> </mrow> </mfrac> <mo>=</mo> </mrow></math> simplify

To simplify the expression:

\[
\frac{\frac{y}{b} - \frac{z}{c}}{\frac{y}{b} + \frac{z}{c}}
\]

we will simplify both the numerator and the denominator separately before dividing them.

**Step 1: Simplify the Numerator**

The numerator is:

\[
\frac{y}{b} - \frac{z}{c}
\]

To subtract these fractions, we need a common denominator, which is \( bc \).

\[
\frac{y}{b} = \frac{y \cdot c}{b \cdot c} = \frac{yc}{bc}
\]

\[
\frac{z}{c} = \frac{z \cdot b}{c \cdot b} = \frac{zb}{bc}
\]

Now, subtract the fractions:

\[
\frac{yc}{bc} - \frac{zb}{bc} = \frac{yc - zb}{bc}
\]

**Step 2: Simplify the Denominator**

The denominator is:

\[
\frac{y}{b} + \frac{z}{c}
\]

Again, using the common denominator \( bc \):

\[
\frac{y}{b} = \frac{y \cdot c}{b \cdot c} = \frac{yc}{bc}
\]

\[
\frac{z}{c} = \frac{z \cdot b}{c \cdot b} = \frac{zb}{bc}
\]

Now, add the fractions:

\[
\frac{yc}{bc} + \frac{zb}{bc} = \frac{yc + zb}{bc}
\]

**Step 3: Divide the Simplified Numerator by the Simplified Denominator**

Now, divide the simplified numerator by the simplified denominator:

\[
\frac{\frac{yc - zb}{bc}}{\frac{yc + zb}{bc}} = \frac{yc - zb}{bc} \times \frac{bc}{yc + zb}
\]

Simplify by canceling out the \( bc \):

\[
\frac{yc - zb}{yc + zb}
\]

Thus, the simplified expression is:

\[
\frac{yc - zb}{yc + zb}
\]

Would you like more details on any part of this process, or do you have any specific questions?

Here are some related questions to explore:

1. How do you handle addition and subtraction of algebraic fractions?
2. How can common factors in fractions help simplify expressions?
3. What techniques can be used to simplify complex rational expressions?
4. How does factoring help in simplifying algebraic fractions?
5. What are the rules for dividing fractions?

**Tip:** When simplifying expressions involving fractions, always factor common terms where possible to make the process easier!

Learn how to simplify the complex fractional expression (y/b - z/c) / (y/b + z/c) step-by-step. This problem covers algebraic concepts including fraction addition, subtraction, and the use of common denominators. Suitable for high school students in Grades 9-12.

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