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Chapter 4: Problem 281

Road grade. A local road has a grade of \(6 \%\). The grade of a road is its slope expressed as a percent. Find the slope of the road as a fraction and then simplify. What rise and run would reflect this slope or grade?

Short Answer

Expert verified
The slope as a fraction is \(\frac{3}{50}\), with a rise of 3 units and a run of 50 units.

Step by step solution

01

Understand the Meaning of Road Grade

The grade of a road, given as a percentage, represents the steepness of the road. The percentage means that for every 100 units of horizontal distance (run), the vertical distance (rise) changes by that percentage. In this case, a 6% grade means that for every 100 units of horizontal distance, the road rises 6 units.
02

Convert Percent to Fraction

Convert the grade percentage into a fraction. A 6% grade can be expressed as the fraction \(\frac{6}{100}\).
03

Simplify the Fraction

Simplify the fraction \(\frac{6}{100}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 6 and 100 is 2. Therefore: \(\frac{6 \div 2}{100 \div 2} = \frac{3}{50}\)
04

Interpret the Simplified Fraction

The simplified fraction \(\frac{3}{50}\) represents the slope of the road. This means that for every 50 units of horizontal distance (run), the road rises by 3 units (rise).
05

State the Rise and Run Values

Given the slope \(\frac{3}{50}\), the corresponding rise is 3 units and the run is 50 units. These values reflect the same 6% grade described in the problem.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Identification
In order to calculate the grade or slope of a road, it鈥檚 essential to understand what 'slope' actually means. The slope is a measure of how steep a surface is. It is calculated as the ratio of the vertical change to the horizontal change.

Imagine you are walking up a hill. The slope tells you how much you go up for every distance you go forward.

Here鈥檚 a key to identify the slope:
  • **Vertical Change (Rise)**: The vertical distance you have gone up or down.
  • **Horizontal Change (Run)**: The horizontal distance you鈥檝e walked forward.
So, in our road grade example, a 6% grade means for every 100 units you move horizontally, you rise 6 units vertically. Understanding this concept makes calculating and interpreting slopes easier.
Fraction Simplification
Simplifying fractions is a core math skill and crucial when dealing with slopes and grades. It involves reducing fractions to their simplest form.

Here's how to do it:
  • **Identify the Greatest Common Divisor (GCD)**: Find the largest number that divides both the numerator and the denominator.
  • **Divide Both Terms by the GCD**: This reduces the fraction while maintaining its value.
Let鈥檚 take our initial fraction from the problem: \(\frac{6}{100}\). Both 6 and 100 are divisible by 2, their GCD.
\(\frac{6 \mathop{div} 2}{100 \mathop{div} 2} = \frac{3}{50}\)
The simplified fraction is \(\frac{3}{50}\), meaning for every 50 units of run, the road rises 3 units, retaining the slope's integrity.
Grade Percentage Conversion
Converting a grade percentage into a fraction is a straightforward but critical step. The grade percentage indicates how steep the slope is in terms of percent change over a standard distance.

Here鈥檚 how to perform this conversion:
  • **Take the Percentage Value**: The percentage directly equivalent to the rise over 100 run units.
  • **Convert to a Fraction**: Simply place the percentage as the numerator and 100 as the denominator.
For instance, with a 6% grade: \(\frac{6}{100}\)
This indicates that for every 100 units of horizontal movement (run), the vertical movement (rise) is 6 units.

This fractional conversion is the first step towards further simplifying and understanding the slope in 'fraction' terms, aiding calculations in various real-world applications like road construction and even roofing.

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Most popular questions from this chapter

Find the equation of each line. Write the equation in slope-intercept form. Perpendicular to the line \(y-1=0\), point (-2,6)

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