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In mathematics, the slope of a line represents the rate at which the line rises or falls. This rate is expressed as a ratio called the slope formula. The slope formula helps us determine the steepness or incline of the line between two points.The general formula for the slope (\text{slope}) of a line is given as:\[ \text{slope} = \frac{\text{rise}}{\text{run}} \]The 'rise' represents the change in vertical distance between the two points, while the 'run' represents the horizontal distance between those points. This formula works for both positive and negative slopes. Adding the values of 'rise' and 'run' into this formula helps us find how steep a line is.

The idea of 'rise over run' simplifies understanding the slope concept. When calculating the slope of a ski slope, as in this exercise, you measure how much the slope rises or falls vertically (rise) and how far it extends horizontally (run).In our example, the ski slope drops vertically by 25 feet for every 100 feet horizontally. This means the vertical distance (rise) is 25 feet and the horizontal distance (run) is 100 feet. When placing these values in the slope formula, we get:\[\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{25}{100} \]This fraction tells us how steep our slope is, allowing us to understand and compare different slopes easily.

Simplifying fractions is one of the foundational skills in algebra and helps in reducing complex ratios into more understandable forms. When dealing with slope, simplifying the fraction makes it easier to understand and use.In our exercise, we initially have the fraction \(\frac{25}{100}\). To simplify this, we find the greatest common divisor (GCD) of the numerator (25) and the denominator (100). The GCD of 25 and 100 is 25.Thus, we divide both the numerator and the denominator by 25:\[ \frac{25}{100} = \frac{25 \div 25}{100 \div 25} = \frac{1}{4} \]After simplification, our slope value becomes \(\frac{1}{4}\). So, the slope of the ski slope is 1/4, meaning it drops 1 foot for every 4 feet it runs horizontally. This simplified fraction is useful for making quick calculations and easier comparisons.