Rational Equations Practice

An equation which has rational expression is known as rational equation. A fraction is also termed as a rational equation. By certain steps we can solve the rational equations. We can practice many problems through online. We are going to see some problems to practice the rational equations.

Example: `1/x` + `5/x` = `7/9` is a rational expression called rational equation.

Explanation to Rational Equations Practice

The following steps are helpful for solving the Rational Expressions.

Step 1:

If possible factor the denominator of every rational expression term in rational equation.

Step 2:

Be sure that the rational expressions in rational equation has same denominator. If not, make it by taking LCM for all rational expression.

Step 3:

Now cancelling all the common terms and simplifying for final answer.

Example Problems to Rational Equations Practice:

Example: 1

Solve: `1/6` = `x/8`

Solution:

Given,

`1/6` = `x/8`

Step 1:

1 × 8 = 6 x

Step 2:

6 x = 8

x = `8/6`

x = `4/3`

Answer: x = `4/3`

Example: 2

Solve:  `x/8` + `5/8` = `7/8`

Solution:

Given: `x/8` + `5/8` = `7/8`

Step 1:

The fractions `x/8` , `5/8` and `7/8` having a common factor 8 in their denominator.

Step 2:

`x/8` + `5/8` = `7/8`

`(x + 5)/8` = `7/8`

Step 3:

Cancel a common factor 8 on both sides of above rational equation.

x + 5 = 7

x + 5 - 5 = 7 - 5

x = 2

Answer: x = 2


Example: 3

Solve: `6/x` + `8/(x + 3)` = `(10)/(x^2 + 3x)`

Solution:

Given,on: `6/x` + `8/(x + 3)` = `(10)/(x^2 + 3x)`

Step 1:

Factor the last rational expression `10/(x^2 + 3x)` = `10/(x(x +3))`

Step 2:

LCM of the all the terms in given rational equation is x(x + 3)

`6/x` x `(x + 3)/(x + 3)` = `(6(x + 3)) /(x(x + 3))`

`8/(x+ 3) ` x `x/x` = `(8x)/(x(x+3))`

`(10)/(x^2 +3x)` x `1/1` = `(10)/(x(x + 3))`

Step 3:

Now the rational equation is,

`"(6(x + 3))/(x(x+3))` + `(8x)/(x(x+3))` = `(10)/(x(x+3))`

`(6x + 18 + 8x)/(x(x+3))` = `(10)/(x(x+3))`

Step 4:

By cancelling the common x(x + 3) we get,

6x + 18 + 8x = 10

14x + 18 = 10

14x = 10 - 18

14x = -8

x = -`8/14`

x = `-4/7`

Answer: x = `-4/7`.

Practice Problem to Rational Equations Practice:

Problem: 1

Solve: `6/x` = `5/6`

Answer: x = `36/5`

Problem: 2

Solve the rational equation, `(x -9)/5` = `21/9`

Answer: x = `62/3`

Decimals least to Greatest

We have learnt the fractions in the lower class mathematics. Now we shall study about special fractions whose denominators are 10, 100, 1000 etc. These fractions are called decimal fractions. Decimals least to greatest is also called as decimals in ascending order (Lower value to higher value)

Let us these fractions in a new way. That is

1/10 is understand writing one-tenth and is correspond to in decimal fraction as 0.1

1/100 is understand writing as one-hundredth and is correspond to in decimal fraction as 0.01

1/1000 is understand writing as one-thousandth and is correspond to in decimal fraction as 0.001

Steps for Calculating Decimals least to Greatest:

Step 1: Look the whole number first, and then look at the tenth, then the hundredth place and so on.

Step 2: Find out which number is smallest

Step 3: Arrange the decimals from least to greatest

For example, arrange the following numbers from least to greatest

0.56, 0.102, 0. 272, 0.5

Ans:  0.102, 0.272, 0.5, 0.56

Examples on Decimals least to Greastest:

Let us see some examples of decimal least to greatest:

Ex 1:  Find order of the decimals from least to greatest

5.6, 2.5, 1.2, 6.5, 0.6 and 0.2

Sol :   Step 1: First look at the tenth, and then the hundredth place so on. Next find the compare the two decimal numbers.

6.5 > 5.6

5.6 > 2.5

2.5 > 1.2

1.2 > 0.6

0.6 > 0.2

Step 2:  Arrange the decimals from least to greatest.

0.2, 0.6, 1.2, 2.5, 5.6 and 6.5

The above values are the arranged values from least to greatest.

Ex  2:   Find order of the decimals from least to greatest

5.7, 57, 0.71, 0.26

Sol :  Step 1: First look at the tenth, and then the hundredth place so on. Next find the compare the two decimal numbers.

56 > 5.7

5.7 > 0.71

0.71 > 0.26

Step 2:  Arrange the decimals from least to greatest.

0.26, 0.71, 5.7, 57

The above values are the arranged values from least to greatest.

Value Weighted Average

Arithmetic average is an average in which is defined as the summation of all the given elements divided by the total number of elements. In this all the values get same weighted.

Ex: Find the arithmetic average of 7, 16, 20, 57 and 60.


Arithmetic average = `("sum of all the elements")/("total number of elements")`

sum of all the given elements = 7 + 16 + 20 + 57+ 60 = 160

total number of elements = 5

Arithmetic average = `(160)/(5)`  = 32

so  arithmetic average = 32

In the above example we weighted to all the values.

Weighted average is same as that of arithmetic average but the only difference is that each of the element have assigned different weighted in the data given. The above example changes like that

Ex: Find the weighted  average of 7, 16, 20, 57 and 60, whose occurrences (weighted) values are 4, 3, 6, 2 and 1.

Solution is given after definition and formula

Definition of Weighted Average Value

Weighted average is defined as the summation of the element values multiplied with the occurrences (allocated weighted) which is divided by the summation of the occurrences values. This is called Value Weighted Average.

Formula for the weighted average is given by,

where

`barx_(w) ->`weighted average value

`w_(i) ->`allocated weighted value(occurrences) for the given element.

`x_(i) ->`element value

Steps to calculate weighted average value:

Step 1: Calculate the summation of the element values multiplied with the occurrences (allocated weighted).

Step 2: Calculate the summation of the occurrences values.

Step 3: Now division the value obtain by step 1 with step2 and get weighted average value.

Examples on Weighted Average Value

Ex 1: Find the weighted average of 7, 16, 20, 57 and 60 whose occurrences (weighted) values are 4, 3, 6, 2 and 1.

Sol :    Formula for the weighted average is

Step 1:     `sum_(n=1)^5(w_(i)x_(i))` =  (4) (7) + (3) (16) + (6) (20) + (2) (57) + (1) (60)

= 370

Step 2:         `sum_(n=1)^5 (x_(i))`   =  4 + 3 + 6 + 2 + 1

= 16

Step 3:     weighted average value = `barx_(w)` =   `(370)/(16)`

= 23.125                    Ans

Ex 2: Find the weighted average of 7, 2, 8 and 10, whose occurrences (weighted) values are 3, 2, 1 and 1.

Sol : Formula for the weighted average is

Step 1:   `sum_(n=1)^4(w_(i)x_(i))`  (3) (7) + (2) (2) + (1) (8) + (1) (10)

= 43

Step 2:      `sum_(n=1)^4(w_(i))`  3 + 2 + 1 + 1

= 7

Step 3:       weighted average value = `barx_(w)`   =  `(43)/(7)`

Ans =   6.142