Problem 63 - Algebra for College Students

Question

If the ratio of rise to run is to be \(\frac{2}{3}\) for some steps, and the run is 28 centimeters, find the rise to the nearest centimeter.

Step-by-Step Solution

Answer

The rise is 19 cm.

Step 1: Understanding Ratios

The problem asks us to use a ratio of rise to run, which is \(\frac{2}{3}\). This means for every 2 units of rise, there are 3 units of run. The run is given as 28 cm.

Step 2: Set Up the Proportion

We'll use the given ratio \(\frac{2}{3}\) to find the rise. Set up the proportion: \(\frac{2}{3} = \frac{\text{rise}}{28}\).

Step 3: Solve for the Rise

To find the rise, cross-multiply: \[2 \times 28 = 3 \times \text{rise} \]This simplifies to 56 = 3 × rise.

Step 4: Complete the Calculations

Now solve for the rise: \[\text{rise} = \frac{56}{3} \approx 18.67\]Round 18.67 to the nearest whole number, which is 19.

Key Concepts

Understanding and Solving Proportions

A proportion is an equation that states two ratios are equal. In the given exercise, the proportion was set up using the ratio of rise to run, which was given as \(\frac{2}{3}\). The idea is to maintain this constant relationship even when we change the values numerically. Here, we were tasked with finding the rise when the run is 28 cm, using this ratio. The proportion equation therefore becomes \(\frac{2}{3} = \frac{\text{rise}}{28}\).

To solve the proportion, we use a method called cross-multiplication. Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the products equal. In this case, we multiply 2 (rise's numerator) by 28 (run's denominator) and 3 (run's numerator) by the rise, leading to \[2 \times 28 = 3 \times \text{rise}\].

This simplifies to the equation \(56 = 3 \times \text{rise}\). Solving this gives us the initial answer for the rise which is 18.67 cm.

Rounding Decimals in Proportion Problems

Once you've calculated the exact rise of 18.67 cm, you need to round it to make it practical. Rounding means adjusting the number to the nearest whole number or specified decimal place, depending on the problem requirements. In this case, the rise was rounded to the nearest centimeter.

When rounding, look at the first digit after the decimal (the tenths place). If it is 5 or more, you round the whole number up. If it's less than 5, you keep the number as it is. Here, 18.67 has a 6 in the tenths place, so we round the number up to 19.

It's important to remember that rounding is not merely about cutting off numbers; it involves a thoughtful decision based on the context and what level of precision is needed. In practical scenarios like construction, using whole numbers is often more pragmatic than using decimal fractions.

Mathematical Problem Solving Using Proportions

Mathematical problem solving is all about applying strategies and logical thinking to find solutions. When working with proportions, the key is to maintain the balance of ratios while solving the equation.

• First, understand the problem: Read carefully and determine what is being asked.
• Next, gather information: Identify known values and how they relate to each other.
• Set up an equation: Use the information and ratios to establish a mathematical model or equation.
• Use techniques like cross-multiplication to solve the equation.
• Double-check your work: Always verify your solution to ensure it makes sense in the context.

Remember, practice makes solving problems easier and enhances your ability to think mathematically. Every step you take in solving a problem builds your confidence and skills. Proportions are particularly useful in real life for tasks such as map reading, cooking, and construction, where maintaining consistent ratios is essential.