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Posted on : 03-01-2019 Posted by : Admin

In mathematics, to arrive at final answer, we need to solve various types of expressions. For solving these expressions, we need different techniques. Simplification is one of the basic techniques to solve the given complex expression into a simpler form.

Generally, an arithmetic expression involves various mathematical operations like addition, subtraction, division, multiplication arranged in various types of brackets. These expressions must be solved in a specific defined order from left to right. The mnemonic of this order is VBODMAS. It is also known as VBODMAS rule.

**V** is vinculum or bar

**B** is brackets

**O** is order of exponent roots

**D** is division

**M** is multiplication

**A** is addition

**S** is subtraction

Even the operations in various brackets must be solved in an order. The order of simplifying brackets is as follows,

- Small circular brackets, Ci ( )
- Middle curly brackets, Cu { }
- Big square brackets, Sq [ ]

So the sequence of solving the brackets is ViCiCuSq, Let us understand it better with the below example,

There are many types of simplification questions. A few of them are explained below,

These types of questions strictly follow VBODMAS rule. And if the rule is followed correctly, we can arrive at the answer quickly and easily.

**For example: **

**Question:** What will come in th eplace of question mark in the below given question?

$1246+\sqrt[4]{\left(256\right)}=\left({?}^{2}+19\frac{1}{3}\right)\times 15$

**Solution:**

$\Rightarrow 1246+\sqrt[4]{256}=\left({x}^{2}+19\frac{1}{3}\right)\times 15\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow 1246+4=\left({x}^{2}+\frac{58}{8}\right)\times 15\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{1250}{15}={x}^{2}+\frac{58}{3}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{250}{3}-\frac{58}{3}={x}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{192}{3}={x}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow x=\sqrt{64}=8$

**Addition and subtraction of decimals**

While performing addition or subtraction of the decimals, each of the given decimal numbers re written under each other such that the decimal points line in one column. The numbers on either side of the decimal point must be arranged carefully. Finally the addition and subtraction is performed conventionally.

**For example:**

**Question:** What will come in place of question mark in the below given question?

6435.9+7546.4+1203.5=?

**Solution: **The numbers on either side of the decimal point must be arranged carefully

$\begin{array}{l}\text{6435.9}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}7546.4}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\frac{\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+1203.5}}{\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}15185.8}}\end{array}$

**Multiplication of decimals**

While performing multiplication, the given decimal numbers are multiplied without considering the decimal point. Finally in the product the decimal point is marked from right hand side to as many places as the total sum of the number of decimal places in multiplier and multiplicand together.

**For example:**

**Question:** What will come in place of question mark in the below given question?

$\text{2.5}\times \text{3.25=?}$

**Solution: **The decimal point is marked from right hand side to as many places as the total sum of the number of decimal places in multiplier and multiplicand together.

$\begin{array}{l}\text{2.5}\times \text{3.25}\end{array}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=8.125$

**Division of decimals**

While performing division, the dividend and the divisor both are multiplied by a suitable multiple of 10 to convert the divisor into to a whole number and then the conventional division is carried out.

**For example:**

**Question:** What will come in place of question mark in the below given question?

$\text{(833.25}-\text{384.45)}\xf7\text{24=?}$

**Solution: **The dividend and the divisor both are multiplied by a suitable multiple of 10 to convert the divisor into to a whole number and then the conventional division is carried out.

$\frac{\text{(833.25}-384.45)}{24}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{448.8)}{24}=18.7$

Fractions are integral parts of a whole number. In a fraction the number above the line is called numerator and the number below the line is called denominator. Sometimes the fractions may also be in the form of mixed fractions which first need to be converted into a fraction and then simplified.

For simplifying the fractions, the integral part and the fraction are added separately. And VBODMAS rule is followed.

**For example:**

**Question:** What will come in place of question mark in the below given question?

$4\frac{1}{2}-2\frac{5}{6}=?-1\frac{7}{12}$

**Solution: **For simplifying the fractions, the integral part and the fraction are added separately. And VBODMAS rule is followed.

$=4+\frac{1}{2}-\left(2+\frac{5}{6}\right)+1+\frac{7}{12}=?\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=(4-2+1)+\left(\frac{-5}{6}+\frac{7}{12}+\frac{1}{2}\right)=?\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=?=3+\left(\frac{-10+7+6}{12}\right)=3+\frac{1}{4}=\frac{13}{4}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=3\frac{1}{4}$

**Square:** Is the number is multiplies by itself, then the product of this multiplication is called square of that number

**For example:**

**Question:** Find the value of 18^{2}?

**Solution:**

18 × 18 = 324

**Square root:** The square root of a number is that number, whose square is equal to the given number. It is represented by √ placed on the given number.

**For example:**

**Question: **What will come in place of question mark in the below given question?

**$\sqrt{\text{360}-225\times 2+379}=?$**

**Solution: **

$\begin{array}{l}\sqrt{\text{360}-\text{450+379}}\\ \\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=}\sqrt{\text{360+379}-\text{450}}\\ \\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=}\sqrt{\text{739}-\text{450}}\\ \\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=}\sqrt{\text{289}}=17\end{array}$

**Cube:** If the number is multiplied two times with itself, then the result of this multiplication is called the cube of that number.

**For example:**

**Question:** Find the value of 8^{3}?

**Solution:**

8 × 8 × 8 = 512

**Cube root:** The cube root of a number is that number, whose cube is equal to the given number. It is represented by 3√ placed on the given number.

**For example:**

**Question:** What will come in place of question mark in the below given question?

**$\sqrt[3]{804357}=?$**

**Solution:**

**$\sqrt[3]{804357}={93}^{3\mathrm{x}\frac{1}{3}}=93$**

These types of questions strictly follow VBODMAS rule. And if the rule is followed correctly, we can arrive at the answer quickly and easily.

For example:

**Question:** What will come in place of question mark in the below given question?

$\text{[(5}\sqrt{\text{7}}\text{+}\sqrt{\text{7}}\text{)}\times \text{(4}\sqrt{\text{7}}\text{+8}\sqrt{\text{7}}\text{)]}-{\text{(19)}}^{2}=?$

**Solution:**

$\begin{array}{l}\text{[(5}\sqrt{\text{7}}\text{+}\sqrt{\text{7}}\text{)}\times \text{(4}\sqrt{\text{7}}\text{+8}\sqrt{\text{7}}\text{)}-{\text{(19)}}^{\text{2}}\text{=?}\\ \\ \Rightarrow \text{[20}\times \text{7+280+28+56]}-\text{361=?}\\ \\ \Rightarrow \text{[420+84]}-\text{361=?}\\ \\ \Rightarrow \text{504}-\text{361=?}\\ \\ \text{?=143}\end{array}$

While performing addition or subtraction of the decimals, each of the given decimal numbers re written under each other such that the decimal points line in one column. The numbers on either side of the decimal point must be arranged carefully. Finally the addition and subtraction is performed conventionally.

For example:

**Question:** What will come in place of question mark?

?% of 555 + 28% of 444 = 202.02

**Solution:**

**$\begin{array}{l}\Rightarrow \frac{\mathrm{x}}{100}\times 555+\frac{28}{100}\times 444=202.02\\ \\ \Rightarrow 5.55\mathrm{x}+28\times 4.44=202.02\\ \\ \Rightarrow \text{5.55}\mathrm{x}\text{=202.02-124.32}\\ \\ \Rightarrow \text{5.55}\mathrm{x}\text{=77.7}\\ \\ \Rightarrow \text{\hspace{0.17em}}\mathrm{x}=\frac{\text{77.7}}{5.55}\text{=14}\end{array}$**

While performing multiplication, the given decimal numbers are multiplied without considering the decimal point. Finally in the product the decimal point is marked from right hand side to as many places as the total sum of the number of decimal places in multiplier and multiplicand together.

**For example:**

**Question:** What will come in place of question mark?

$14\%\mathrm{of}250\times ?\%\mathrm{of}150=840$

**Solution:**

$\begin{array}{l}\Rightarrow \frac{14}{100}\times 250\times x\%\text{\hspace{0.17em}}of\text{\hspace{0.17em}}150=840\\ \\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}[takingxattheplaceof?]}\\ \\ \Rightarrow \frac{14}{100}\times 250\times \frac{x}{100}\times 150=840\\ \\ \Rightarrow \frac{3500}{100}\times \frac{x}{100}\times 150=840\\ \\ \Rightarrow x=\frac{840\times 100}{150\times 35}=\frac{24\times 100}{100}=16\end{array}$

While performing division, the dividend and the divisor both are multiplied by a suitable multiple of 10 to convert the divisor into to a whole number and then the conventional division is carried out.

**For example:**

**Question:** What will come in place of question mark?

$64.5\mathrm{\%}\text{\hspace{0.17em}}\mathrm{of}\text{\hspace{0.17em}}800+36.4\mathrm{\%}\mathrm{of}\text{\hspace{0.17em}}1500={(?)}^{2}+38$

**Solution:**

$\begin{array}{l}\Rightarrow \frac{64.5}{100}\times 800+\frac{36.4}{100}\times 1500={(?)}^{2}+38\\ \\ \Rightarrow \text{64.5}\times \text{8+36.4}\times \text{15=}{(?)}^{\text{2}}+38\\ \\ \Rightarrow {\text{516+546=(?)}}^{\text{2}}+38\\ \\ \Rightarrow {\text{?}}^{\text{2}}\text{=516+546}-\text{38=1062}-38\\ \\ \Rightarrow \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}?=\sqrt{1024}=32\end{array}$

In each of these questions, there is a related algebraic expression. First we need to find the related algebraic expression and then simply the question based on the expression. Below table lists out the important algebraic expressions,

- (a+b)
^{2}=a^{2}+b^{2}+2ab - (a-b)
^{2}=a^{2}+b^{2}-2ab - (a+b)(a-b)=a
^{2}-b^{2} - (a+b)
^{2}+(a-b)^{2}=2(a^{2}+b^{2}) - (a+b)
^{2}-(a-b)^{2}=4ab - (a+b)
^{3}= a^{3}+3a^{2}b+3ab^{2}+b^{3} - (a-b)
^{3}= a^{3}-3a^{2}b+3ab^{2}-b^{3} - a
^{3}+b^{3}=(a+b)( a^{2}+b^{2}-ab) - a
^{3}-b^{3}=(a-b)( a^{2}+b^{2}+ab)

**For example:**

**Question:** What will come in place of question mark?

66^{2}-34^{2}=?

Solution:

This is similar to,

$\begin{array}{l}[{\text{a}}^{\text{2}}-{\text{b}}^{\text{2}}\text{=(a+b)(a}-\text{b)}]\\ \\ \text{So,}\\ {\text{=66}}^{\text{2}}-{\text{34}}^{\text{2}}\end{array}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{=(66+34)\xd7(66}-\text{34)}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{=100}\times \text{32=3200}$

The questions of type are given in statement form. In this type of questions first we need to form an expression from the given statements and try to simplify the expression.

**For example:**

**Question:** On the annual day, sweets were to be distributed equally amongst 600 children of the school. But on that particular day, 120 children remained absent. Thus, each child got 2 extra sweets. How many sweets was each child originally supposed to get?

**Solution**

$\begin{array}{l}\text{Numberofextrasweets}\\ \\ \text{=2}\times \text{480=960}\\ \\ \text{Thesesweetsweretobedistributed}\\ \text{among120children.}\\ \\ \therefore \text{Numberofsweetstobegiventoeach}\\ \text{childonoriginally=}\frac{\text{960}}{120}\text{=8}\end{array}\phantom{\rule{0ex}{0ex}}\mathrm{Hence},\mathrm{option}\mathrm{A}\mathrm{is}\mathrm{correct}.$

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