Problem 71 - Elementary Algebra

Question

If the ratio of rise to run is to be \(\frac{3}{5}\) for some stairs, and the measure of the rise is 19 centimeters, find the measure of the run to the nearest centimeter.

Step-by-Step Solution

Answer

The run is 32 centimeters.

Step 1: Understand the Ratio Concept

The ratio of the rise to the run is given as \( \frac{3}{5} \). This tells us that for every 3 units of rise, there should be 5 units of run. Our task is to find the length of the run when the rise is 19 cm.

Step 2: Set Up the Proportion Equation

If the ratio is \( \frac{3}{5} \), then we can set up the proportion equation: \( \frac{3}{5} = \frac{19}{x} \), where \( x \) is the length of the run we need to find.

Step 3: Solve the Equation for x

We solve the proportion \( \frac{3}{5} = \frac{19}{x} \) by cross-multiplying, which gives \( 3x = 95 \). We then divide both sides by 3 to find \( x = \frac{95}{3} \).

Step 4: Calculate the Run

Calculate \( \frac{95}{3} \) to find that \( x \approx 31.67 \). Since we need the run to the nearest centimeter, we round this to the nearest whole number.

Step 5: Round to the Nearest Whole Number

Round \( x = 31.67 \) to the nearest whole number, giving us \( x = 32 \) cm.

Key Concepts

Rise and Run

The terms "rise" and "run" are crucial when discussing slopes, such as in staircases or hills. In this context, the "rise" refers to the vertical height between two steps, whereas "run" pertains to the horizontal distance between those steps.

Think of rise and run as creating a right triangle, where the rise is one side, the run is another, and the slope or incline is the hypotenuse. This relationship is often expressed as a ratio, which forms the basis of understanding how steep something is.

• "Rise" measures how much you ascend vertically.
• "Run" measures how far you move horizontally.
• The greater the rise compared to the run, the steeper the slope.

In design and construction, having an appropriate balance between rise and run ensures both functionality and safety.

Proportion Equation

Proportion equations involve two ratios that are set equal to each other. In staircases, for example, we aim to keep the ratio of rise to run constant.

When we construct a proportion equation, such as \( \frac{3}{5} = \frac{19}{x} \), it allows us to find unknown lengths while maintaining the ratio consistent.

• A proportion equation helps to solve for unknown variables by keeping the ratios equivalent.
• It is important in real-world problems to maintain balance and equality.

These equations are beneficial because they simplify complex relationships by using a simple linking ratio that is understandable and easy to apply.

Cross-Multiplication

Cross-multiplication is a handy technique often used to solve proportion equations. It allows us to easily clear fractions by rearranging the equation into a solvable linear format.

For instance, to solve \( \frac{3}{5} = \frac{19}{x} \), we use cross-multiplication to get the equation:

• Multiply diagonally: \(3 \times x\) and \(5 \times 19\), leading to \(3x = 95\).
• Then, solve the linear equation by isolating the variable: divide both sides by 3 to find \(x = \frac{95}{3}\).

This process streamlines finding the unknown run value when the rise and its ratio are known, making it a practical tool for tackling various mathematical problems.