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Rational Expressions Word Problems

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  A rational expression is an expression which is expressed in the form of two polynomials. It is nothing more than a fraction in which the numerator or denominator or both (numerator and denominator) are polynomials. In other words we can say that, an expression which is in the ratio of two polynomials.
Rational expression word problems is all about solving the given rational expressions.

We can solve the rational expressions in the following ways:

Direct problems:

$\frac{5}{4}$ + $\frac{3}{7}$ $\frac{7}{3}$ - $\frac{2}{6}$

$\frac{4}{5}$ $\times$ $\frac{6}{4}$ $\frac{7}{2}$ $\div$ $\frac{6}{5}$.

We solve these problems directly by performing the operations.

Equations which are of the form of a function:

Example:
f
1(x) = x2 + 3x + 2 and f2 (x) = x + 2.
In these kind of problems, we have to solve for x.

Rational Expressions Applications Word Problems

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Given below are some of the word problems on rational expressions applications.

Solved Examples

Question 1: Mary and Ruby are assigned a work. Mary takes 6 hours to work alone and Ruby takes 8 hours to do the same work alone. How long will it take for them to complete the job, if they work together?
Solution:
Let t be the time taken to complete the job together.

Rate of work of Mary = $\frac{1}{6}$ per hour

Rate of work of Ruby = $\frac{1}{8}$ per hour.

If they work together, time taken by them to complete the job is $\frac{1}{t}$ = $\frac{1}{6}$ + $\frac{1}{8}$
$\frac{1}{t}$
 = $\frac{14}{48}$

$\therefore$ t = $\frac{24}{7}$ hrs.

Thus, they both will complete the job in $\frac{24}{7}$ hours.

Question 2: Jinn and jack starts cleaning the house. Jinn cleans one third of the house within 1 hr. Jack takes same time to clean remaining part of the house. How much part of the house jack has cleaned?
Solution:
Jinn cleans $\frac{1}{3}$ part of the house. Whereas, Jack cleans remaining part of the house.

The part of the house cleaned by jack is: 1 - $\frac{1}{3}$ = $\frac{2}{3}$

$\therefore$ Jack completes cleaning $\frac{2}{3}$ rd part of the house.

Question 3: Mary got a $\frac{1}{4}$ th part of  water melon. She also wants to share it to 3 of her friends equally. How much part of the water melon will each get including Mary?
Solution:
Let the whole water melon be denoted by 1. The amount of water melon present with mary is $\frac{1}{4}$. Let it be distributed to four of them which also includes mary.

The amount of water melon each will be getting is:

$\frac{1}{4}$ $\div$ 4.

$\frac{1}{16}$ .

$\therefore$ Each will be getting $\frac{1}{16}$ th part of the water melon.

Solving Word Problems with Rational Expressions

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Rational numbers are terminating or recurring decimal numbers written in the form of fraction $\frac{a}{b}$ in which 'a' and 'b' are integers and the denominator 'b' is not equal to zero. Rational numbers can form expressions, when we join rational numbers with various mathematical operators namely addition, subtraction, multiplication and division. These expressions are called rational expression.

Examples of rational expressions are: $\frac{3}{5}$, $\frac{3x}{5}$, $\frac{x + 2}{x + 3}$ etc.

For word problems with rational expressions, first we have to convert the word problem into a rational expression and then, we can easily solve the rational expression.

Some of the solved examples of word problems with rational expressions are given below:

Solved Examples

Question 1: Anni and Ashley were dusting the garage. In an hour, anni dusted $\frac{3}{5}$ of the garage and ashley dusted the remaining. How much portion of the garage did ashley dust in an hour?
Solution:
Portion Anni dusted in an hour = $\frac{3}{5}$

Portion Ashley dusted in an hour = 1 - $\frac{3}{5}$ = $\frac{5 - 3}{5}$ = $\frac{2}{5}$

Question 2: The distance from Daniel's house to church is 3 km. Daniel walks at the speed of $\frac{\text{1 km}}{\text{20 min}}$ from his house to church. The church service is for an hour and while returning home from church, he walks at the speed of $\frac{\text{1km}}{\text{25 min}}$. How much time does it take for Daniel to reach his house, if he leaves the house at 8 AM for the service?
Solution:
Time taken from house to church = $\frac{20}{1}$ x 3 = 60 minutes

Time taken in the service = 1 hour = 60 minutes

Time taken from church to house = $\frac{25}{1}$ x 3 = 75 minutes

Total time taken = 60 + 60 + 75 = 195 minutes = 3 hr 15 mins

It takes 3$\frac{1}{4}$ hr for Daniel to reach house once he leaves for church. So, Daniel reaches the house at 11:15 AM.