Explore: Applications for Rational Expressions Solutions

Applications for Rational Expressions Solutions

Complex fractions are commonly used with ratios that: 
- find a geometric probability . 
- use formulas for efficiency ratings . 
A geometric probability is written as: $P (area of figure) = \frac{area you want}{total area}$
An efficiency ratio is comparing two options to find the best one. 
- First simplify the rational expression, then compare ratios to find the best choice. 
- A lower efficiency rating is considered best because it means there is less risk .

Example 7

Benji finished wiping the kitchen island countertop (trapezoid) and tosses the rag toward the sink (rectangle).

1. Determine the probability of the rag landing in the sink instead of the counter.
2. Determine the probability of the rag landing anywhere on the counter except in the sink.

Plan
Find the area of the rectangle

Find the area of the trapezoid

Find the area of the (trapezoid – rectangle)

Find P(rectangle)

Find P(not rectangle)

Rectangle

$\begin{array}{rcl} A & = & b h \\ A & = & (x + 6) (x + 2) \\ A & = & x^{2} + 8 x + 12 \end{array}$

Trapezoid

$\begin{array}{rcl} A & = & \frac{1}{2} h (b_{1} + b_{2}); b_{1} = 2 x + 5, b_{2} = 3 x + 8 \\ A & = & \frac{1}{2} (2 x + 4) (2 x + 5 + 3 x + 8) \\ A & = & (x + 2) (5 x + 13) \\ A & = & 5 x^{2} + 23 x + 26 \end{array}$

Trapezoid — Rectangle

$(5 x^{2} + 23 x + 26) - (x^{2} + 8 x + 12) 5 x^{2} + 23 x + 26 - x^{2} - 8 x - 12 4 x^{2} + 15 x + 14 (4 x + 7) (x + 2)$

$\begin{array}{rcl} P (rectangle) & = & \frac{rectangle area}{total area} \\ = & \frac{x^{2} + 8 x + 12}{5 x^{2} + 23 x + 26} \end{array}$

$\begin{array}{rcl} = & \frac{(x + 6) (x + 2)}{(x + 2) (5 x + 13)} \\ P (rectangle) & = & \frac{x + 6}{5 x + 13}, x \neq - \frac{13}{5}, - 2 \end{array}$

$P (not rectangle) = \frac{trapezoid - rectangle}{total area}$

$\begin{array}{rcl} = & \frac{4 x^{2} + 15 x + 14}{5 x^{2} + 23 x + 26} \\ = & \frac{(4 x + 7) (x + 2)}{(x + 2) (5 x + 13)} \\ P (not rectangle) & = & \frac{4 x + 7}{5 x + 13}, x \neq - \frac{13}{5}, - 2 \end{array}$

Example 8

Three companies, X, Y, and Z, are being compared based on efficiency ratios. The operating costs are determined using the formulas:

$C = \frac{m^{2}}{2} (3 m - 1) (4 c + 5)$

$T = \frac{5 m}{2} {(3 m - 1)}^{2} (6 c + 7)$

Determine each company’s efficiency rating by comparing operating cost to total income, $\frac{C}{T}$ .
Using the table, find the best efficiency rating for the three companies.

	Company X	Company Y	Company Z
m	7.8	9.2	4.7
c	5.2	1.8	2.3

$\begin{array}{rcl} \frac{C}{T} & = & \frac{\frac{m^{2}}{2} (3 m - 1) (4 c + 5)}{\frac{5 m}{2} (3 m - 1) (3 m - 1) (6 c + 7)} = \frac{\frac{m^{2}}{2}}{\frac{5 m}{2}} \cdot \frac{(3 m - 1) (4 c + 5)}{(3 m - 1) (3 m - 1) (6 c + 7)} \end{array}$

$\frac{m^{2}}{2} \div \frac{5 m}{2} \frac{m^{2}}{2} \cdot \frac{2}{5 m} = \frac{m}{5}$

$\frac{C}{T} = \frac{m (3 m - 1) (4 c + 5)}{5 (3 m - 1) (3 m - 1) (6 c + 7)} \frac{C}{T} = \frac{m (4 c + 5)}{5 (3 m - 1) (6 c + 7)}$

Note

Explain

Write the complex fraction using C and T
Simplify the fractional monomials
Simplify the entire expression
Use the table to find the efficiency rating
To find the answer as a percent, multiply the decimal by 100 (or move the decimal point two spaces right)

$\frac{C}{T} = \frac{m (4 c + 5)}{5 (3 m - 1) (6 c + 7)}$

Company X

$\frac{() (4 () + 5)}{5 (3 () - 1) (6 () + 7)} =$

$\frac{(7.8) (4 (5.2) + 5)}{5 (3 (7.8) - 1) (6 (5.2) + 7)} = 0.047036$

Company X rating: 4.70%

Company Y

$\frac{(9.2) (4 (1.8) + 5)}{5 (3 (9.2) - 1) (6 (1.8) + 7)} = 0.0474$

Company Y rating: 4.74%

Company Z

$\frac{(4.7) (4 (2.3) + 5)}{5 (3 (4.7) - 1) (6 (2.3) + 7)} = 0.048987$

Company Z rating: 4.90%

Explain

Company X has the lowest rating, therefore it is the most efficient.