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. _{1}(x) = x^{2} + 3x + 2 and f_{2} (x) = x + 2.In these kind of problems, we have to solve for x. |

### 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:**

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:**

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:**

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.

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 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 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.