Multiply. Write your answer in lowest terms. \( \frac{125-5 z}{49} \cdot \frac{245}{7 z-175} \)

To solve the expression \( \frac{125-5 z}{49} \cdot \frac{245}{7 z-175} \), we start by simplifying each fraction. First, notice that \( 125 - 5z \) can be factored out as \( 5(25 - z) \), and \( 7z - 175 \) can be factored by taking out 7, yielding \( 7(z - 25) \). So, we rewrite the expression: \[ \frac{5(25 - z)}{49} \cdot \frac{245}{7(z - 25)} \] Next, we factor \( 245 \) as \( 35 \cdot 7 \) which allows us to cancel out the \( 7 \) in the denominator with the \( 7 \) in \( 245 \): \[ = \frac{5(25 - z) \cdot 35}{49 \cdot (z - 25)} \] Now, substituting \( z - 25 \) as \( -(25 - z) \): \[ = \frac{5 \cdot 35}{49 \cdot -1} \quad (because \, 25 - z \, is \, equivalent \, to \, -(z - 25)) \] This simplifies to: \[ -\frac{175}{49} \] Now we can reduce \( \frac{175}{49} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 7: \[ -\frac{175 \div 7}{49 \div 7} = -\frac{25}{7} \] Thus, the answer in lowest terms is: \[ -\frac{25}{7} \] Engaging historical nugget: Did you know that fractions date back to ancient civilizations? The Egyptians used a system based on unit fractions—fractions with a numerator of 1—over 4,000 years ago! Their method laid the groundwork for our current fraction representation. And for a sprinkle of real-world application: Fractions are essential in cooking and baking! When adjusting a recipe, you may need to multiply fractions to find out how much of each ingredient to use. Knowing how to simplify those fractions ensures your cookies come out just right!