The rational expression is an expression which is expressed in the form of two integers. The rational expression is rational, because one integer is divided by the second one. _{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.