General terms related to spherical mirrors: A mirror with the shape of a portion cut out of a spherical surface or substance is known as a...

General Terms Related to Spherical Mirrors

April 11, 2024**Brackets** imply a sort of grouping, the operators in the sub-expression take precedence over the operators in the surrounding expression. Using brackets are commonly seen in mathematical functions. We have seen expressions like (3×2)–1. (3×2)–1. We can consider two parts in the expression: the portion within the bracket and the portion outside the bracket.

Remember, if we miscalculate it, we may get the incorrect answer. How will we go if more than one bracket is there? We use the arithmetic operations according to their precedence. Similarly, we use brackets according to their precedence. Let us discuss here different types of brackets and their uses.

In Mathematics, brackets are the symbols often used to create groups or explain the order in which operations are to be done in an expression. Some brackets have multiple particular uses in mathematics.

Generally, we use brackets for grouping in math. The different types of brackets we generally use are:

2. \(\left\{{} \right\}\) called the curly braces

3. \(\left[{} \right]\) called the square braces

We always use a pair of brackets that have an opening and a closing. Brackets are used to give transparency in the order of operations.

For instance, let us assume that you have the expression \(2 + 5 \times 7 – 2.\) We know that the order of mathematical operations is division, multiplication, addition and subtraction. So, we will be moving towards right starting from multiplication before performing additions and subtractions, and we get \(2 + 35 – 2 = 35.\)What will we do if we want to perform the addition and subtraction first and multiply the results? This can be achieved by using the brackets appropriately.

BODMAS is a short form used for Brackets, Order, Division, Multiplication, Addition and Subtraction. Sometimes people use PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition and Subtraction), similar to BODMAS.

It describes the order of mathematical operations be performed while solving a mathematical expression. According to this rule, if multiple brackets are present in the expression, start simplifying the innermost round bracket or parenthesis first, followed by the square bracket and the curly bracket and then solve according to the precedence arithmetic operations.

In other words, according to the BODMAS rule, to solve any mathematical expression, first, solve the terms written inside the brackets and then simplify exponential terms. After that, we solve division and multiplication operations, then, at last, addition and subtraction. So, the BODMAS rule evaluates mathematical expressions and deals with complex calculations much easier and correctly.

Then, perform division or multiplication (go from left to right).

Then, perform addition or subtraction (go from left to right).

According to the BODMAS rule, to solve any mathematical expression, first, solve the terms written inside the brackets, simplify exponential terms and move ahead to division and multiplication operations, then, at last, addition and subtraction.

If there is any bracket in the expression, open the bracket and add or subtract the terms.\(a + \left({b + c} \right) = a + b + c,\quad a + \left({b – c} \right) = a + b – c\)

If there is a negative sign, open the bracket, multiply the negative sign with each term inside the bracket. \(a – \left({b + c} \right) = a – b – c\) If there is any term outside the bracket, multiply that outside term with each term inside the bracket. \(a\left({b + c} \right) = ab + ac\)

Example, \(11 – \left({3 – 2} \right) = 11 – 3 + 2 = 11 – 1 = 10\)

Example, \(3\left({5 – 2} \right) = 15 – 6 = 9\)

Let us understand through an example. In yesterdays’ storm, some guavas have fallen from the tree in our garden. My mother has picked it up in a bag. I counted that there are \(40\) guavas in the bag. From there, I took \(5\) guavas, and my sister took \(6.\) After some time, my aunt found \(10\) more guavas in the garden. She divided all of them equally among \(13\) girls and boys from the neighbourhood. How many guavas did each get? Let us find out.

**Step 1**: How many guavas were there initially? \(40\) guavas.**Step 2**: How many guavas did my sister, and I take?

\(\left({5 + 6} \right)\) Let us place this in the first bracket \(\left({} \right).\)**Step 3**: After we took guavas, how many were left in the bag?

\(\left\{{40 – \left({5 + 6} \right)} \right\}\) We place in the second bracket \(\left\{{} \right\}.\)

If aunt found \(10\) more guavas, the total number of guavas would be \(\left\{{40 – \left({5 + 6} \right)} \right\} + 10\)

We have more tasks left. So, we need another bracket. We will call this bracket the square bracket.**Step 4**: Divide equally among \(13\) people. Each will get, \(\left[{\left\{{40 – \left({5 + 6} \right)} \right\} + 10} \right] \div 13\)

\(\left[{\left\{{40 – \left({5 + 6} \right)} \right\} + 10} \right] \div 13\)

\( = \left[{\left\{{40 – 11} \right\} + 10} \right] \div 13\) (Simplify inside round bracket)

\( = \left[{29 + 10} \right] \div 13\) (Simplify inside curly bracket)

\( = 39 \div 13\) (Simplify inside square bracket)

\( = 3\) (Divide)

Therefore, each will get \(3\) guavas.

This is how the brackets help us in solving an instance.

When more calculations are involved, and the order of mathematical operations should be altered, then we take the help of brackets.

**10 Maths Tricks for Fast Calculation**

**Q.1. Solve \(8 + 9 \div 9 + 5 \times 2 – 7\).Ans:** The given expression is \(8 + 9 \div 9 + 5 \times 2 – 7.\)

Since there are no brackets here, the order of the mathematical operations will be division, multiplication, addition and then subtraction.

First, perform division operation, i.e., \(9 \div 9 = 1\)

Thus the expression becomes, \(8 + 1 + 5 \times 2 – 7\)

Then we will do the multiplication, i.e., \(5 \times 2 = 10\)

Now the expression becomes, \(8 + 1 + 10 – 7\)

Then perform addition i.e., \(8 + 1 + 10 = 19\)

At last, we will do the subtraction.

Now, \(19 – 7 = 12\)

Hence, the required answer is \(12.\)

**Q.2. Simplify \(\left\{ {25 – 3\left( {6 + 1} \right)} \right\} \div 4 + 9\). ****Ans:** The given expression is \(\left\{{25 – 3\left({6 + 1} \right)} \right\} \div 4 + 9.\)

We will start solving inside the round bracket or parenthesis, i.e., \(\left({6 + 1} \right) = 7\)

Next, multiply \(3\left( 7 \right)\) or \(3 \times 7 = 21\)

Now the expression becomes, \(\left\{{25 – 21} \right\} \div 4 + 9\)

Operate the curly braces, i.e., \(\left\{{25 – 21} \right\} = 4\)

So, the expression becomes \(4 \div 4 +9\)

Hence, \(4 \div 4 = 1\)

Lastly, \(1 + 9 = 10\)

Hence, the required answer is \(10\) after simplifying the expression.

**Q.3. Solve \(\left( {1/4 + 1/8} \right)\) of 32. Ans: **Here, we need to solve the expression \(\left({\frac{1}{4} + \frac{1}{8}} \right)\) of \(32.\)

First, we need to operate the expression inside the bracket, i.e., \(\left({\frac{1}{4} + \frac{1}{8}} \right) = \frac{{2 + 1}}{8} = \frac{3}{8}\)

Now the expression becomes, \(\frac{3}{8}\) of \(32\)

‘Of’ means multiply. So, \(\frac{3}{8} \times 32 = 12\)

Hence, the required answer is \(12.\)

**Q.4. Simplify \(150 \div 15\left\{ {\left( {12 – 6} \right) – \left( {14 – 12} \right)} \right\}\).Ans:**

First, we have to operate and simplify the terms inside \(\left({} \right)\) followed by \(\left\{{} \right\}.\)

Now, \(150 \div 15\left\{{\left({12 – 6} \right) – \left({14 – 12} \right)} \right\}\)

\( = 150 \div 15\left\{{6 – 2} \right\}\) (Solved inside the round bracket)

\( = 150 \div 15\left\{ 4 \right\}\) (Solved inside the curly bracket)

\( = 10\left\{ 4 \right\}\) (Divided \(150\) by \(15\),i.e.,\(12\))

\( = 10 \times 4\) (if there is no operator before bracket, consider as multiplication operator is there) \( = 40\) (Multiply \(10\) and \(4\))

Hence, the required answer is \(40.\)

**Q.5. Solve \(16\left[ {8 – \left\{ {5 – 2\left( {2 – 1 + 1} \right)} \right\}} \right]\) using BODMAS rule.Ans:**

First, solve the parenthesis.

Now, \(16\left[{8 – \left\{{5 – 2\left({1 + 1} \right)} \right\}} \right]\)

\( = 16\left[{8 – \left\{{5 – 2 \times 2} \right\}} \right]\)(solved inside the curved bracket)

\( = 16\left[{8 – \left\{{5 – 4} \right\}} \right]\) (multiplied inside the curly bracket)

\( = 16\left[{8 – 1} \right]\)(Solved inside the curly bracket)

\( = 16 \times 7\) (solved inside the square bracket)

\( = 112\) (multiplied)

Hence, the required answer is \(112.\)

**Q.1. When to use brackets?****Ans: **In Mathematics, brackets are the symbols that are often used to create groups or explain the order that operations are to be done in an expression. Some brackets have multiple particular uses in mathematics.

**Q.2. How to use the bracket formula in mathematics?Ans:** Generally, we use brackets for grouping in math. These brackets symbols are,

1. \(\left({} \right)\)

2. \(\left\{{} \right\}\)

3. \(\left[{} \right]\)

We always use a pair of brackets that have an opening and a closing. Brackets are used to give transparency in the order of operations. The order of brackets are \(\left({} \right),\left\{{} \right\}\) and \(\left[{} \right].\)

**Q.3. What do you mean by BODMAS?****Ans:** BODMAS is a short form used for Brackets, Order, Division, Multiplication, Addition and Subtraction. It describes the mathematical operations to be performed while solving a mathematical expression. According to this rule, if multiple brackets are present in the expression, start simplifying inside the round bracket, then followed by curly bracket, then square bracket and then solve concerning the precedence of the arithmetic operations.

**Q.4. Do you multiply first if there are no brackets?****Ans:** According to the BODMAS rule, the bracket should be solved first. If there is no bracket, then the next precedence will be division or multiplication (as both division and multiplication have the same order of preference) and if multiplication comes first in the mathematical expression from left to right. So, we do multiply first if no brackets are there, as multiplication comes first in the expression from left to right.

**Q.5. Do you use BODMAS when there are no brackets?****Ans:** Yes, we use the BODMAS rule to get the correct answer even if there are no brackets. If there are no brackets, start solving from ‘order’ or ‘of’ followed by division or multiplication (whatever comes first from left to right), then by addition or subtraction (whatever comes first).