Simple Expressions
Chapter 2: Arithmetic Expressions
Page 1 of 6: Simple Expressions
Welcome to the world of arithmetic expressions! Before we dive into complex algebra, we must master the language of numbers and operations. This page will teach you how to read, write, and compare the basic mathematical phrases that form the foundation of all calculations.
Imagine you're at a book fair. You buy a comic book for ₹50 and a bookmark for ₹10. To find the total cost, your brain instantly performs a calculation. The mathematical phrase you think of, 50 + 10, is a perfect example of an arithmetic expression. It’s a combination of numbers and a mathematical operation that represents a single value. In this case, its value is 60. Throughout this chapter, we'll explore how these simple phrases can describe real-world situations and how to work with them accurately.
Definitions & Key Terms
Let's define the fundamental building blocks of our topic.
| Term | Meaning | Example |
|---|
| Arithmetic Expression | A mathematical phrase that combines numbers and at least one operator (+, –, ×, ÷). | 18 + 7, 100 – 45, 9 × 8, 63 ÷ 3 |
| Value | The single number that an arithmetic expression evaluates to. | The value of the expression 9 × 8 is 72. |
| Operator | A symbol that represents a mathematical operation. | + (Addition), – (Subtraction), × (Multiplication), ÷ (Division) |
| Comparison Operators | Symbols used to compare the values of two expressions. | =, <, > (Equal to, Less than, Greater than) |
The Logic of Expressions
Understanding how expressions are formed and compared is crucial. It's a logical process that moves from simple phrases to powerful comparisons.
-
Forming an Expression: An expression starts with a real-world problem or a mathematical statement. For example, if Reena has 15 chocolates and eats 4, the number of chocolates she has left is represented by the expression:
15 – 4
-
Finding the Value: Every simple arithmetic expression has a definite value. We find this by performing the indicated operation.
15 – 4 = 11
Here, the value of the expression is 11.
-
Equivalence of Expressions: Different expressions can have the same value. This is a key concept. The number 20 can be represented in many ways:
15 + 5
25 – 5
10 × 2
40 ÷ 2
All these expressions are different, but their value is the same: 20.
-
Direct Comparison: We can compare two expressions by first finding their individual values and then using a comparison operator (<, >, =).
- To compare
13 – 2 and 4 × 3:
- Value of first expression:
13 – 2 = 11
- Value of second expression:
4 × 3 = 12
- Since 11 is less than 12, we can write:
13 – 2 < 4 × 3
-
Logical Comparison (Without Full Calculation): Sometimes, we can compare expressions just by reasoning about the numbers involved. This is a very powerful skill.
- Consider comparing
113 – 25 and 112 – 24.
- Think: Raja starts with 113 marbles (
+1 more than Joy's 112). But Raja also loses 25 marbles (+1 more loss than Joy's 24).
- The extra marble Raja started with is cancelled out by the extra marble he lost.
- Therefore, they end up with the same amount. No calculation needed!
113 – 25 = 112 – 24
{{KEY: type=concept | title=Comparison by Reasoning | text=To compare expressions like A+B and C+D without full calculation, analyze the differences between A and C, and B and D. This 'mental math' technique helps in quickly estimating which expression is larger, smaller, or if they are equal.}}
Solved Examples
Let's work through some problems to solidify our understanding.
Example 1: Writing an Expression (Easy)
Given: A bus has a seating capacity of 52 passengers. On a trip, 45 passengers boarded the bus.
To Find: An expression for the number of empty seats.
Solution:
- We start with the total number of seats available.
52
- We subtract the number of seats that are occupied by passengers.
52 – 45
- This expression represents the number of empty seats. The value is
52 – 45 = 7.
Final Answer: The expression for the number of empty seats is 52 – 45.
Example 2: Comparing Expressions by Value (Medium)
Given: Two expressions: 35 + 25 and 120 ÷ 2.
To Find: Which expression has a greater value.
Solution:
- First, calculate the value of the first expression.
35 + 25 = 60
- Next, calculate the value of the second expression.
120 ÷ 2 = 60
- Now, compare the two values.
60 = 60
- Since the values are equal, the expressions are equal.
Final Answer: The expressions are equal: 35 + 25 = 120 ÷ 2.
Example 3: Ordering Expressions (Hard)
Given: The expressions (a) 67 – 19, (b) 5 × 11, (c) 120 ÷ 3, and (d) 35 + 25.
To Find: Arrange these expressions in ascending (increasing) order of their values.
Solution:
-
Calculate the value of each expression individually.
- (a)
67 – 19 = 48
- (b)
5 × 11 = 55
- (c)
120 ÷ 3 = 40
- (d)
35 + 25 = 60
-
List the calculated values.
48, 55, 40, 60
-
Arrange these values in ascending order.
40 < 48 < 55 < 60
-
Now, replace the values with their original expressions to get the final order.
120 ÷ 3 < 67 – 19 < 5 × 11 < 35 + 25
Final Answer: The expressions in ascending order are: 120 ÷ 3, 67 – 19, 5 × 11, 35 + 25.
Example 4: Comparing by Logic (Tricky)
Given: Two expressions: 364 + 587 and 363 + 589.
To Find: Compare the expressions using > , <, or = without performing the full addition.
Solution:
- Let's analyze the first expression:
364 + 587.
- Now let's analyze the second expression:
363 + 589.
- Compare the first numbers in each expression:
363 is 1 less than 364.
- Compare the second numbers in each expression:
589 is 2 more than 587.
- Think: The second expression takes away 1 from the first number but adds 2 to the second number. The net effect is an increase of
2 – 1 = 1.
- This means the second expression's value will be 1 greater than the first expression's value.
- Therefore, the first expression is less than the second expression.
- Verification (optional):
364 + 587 = 951 and 363 + 589 = 952. Indeed, 951 < 952.
Final Answer: 364 + 587 < 363 + 589.
Tips & Tricks
Use these shortcuts to compare expressions faster and more accurately.
| Trick | Explanation | Example |
|---|
| The "Give and Take" Rule | If you decrease one number in a sum by k and increase the other by k, the value remains the same. | 100 + 50 is equal to (100 – 10) + (50 + 10), which is 90 + 60. Both equal 150. |
| Subtraction Balance | If you add (or subtract) the same number from both parts of a subtraction, the value remains unchanged. | 150 – 50 is equal to (150 + 5) – (50 + 5), which is 155 – 55. Both equal 100. |
| Focus on the Change | When comparing A+B and A+C, you only need to compare B and C. The common part A doesn't affect the comparison. | To compare 124 + 245 and 129 + 245, just notice that 124 < 129, so the first expression is smaller. |
Common Mistakes to Avoid
Many students make these simple errors. Be careful!
| ❌ Wrong Approach | ✅ Right Approach | Why it's a Mistake |
|---|
100 + 20 > 100 + 30 | 100 + 20 < 100 + 30 | Confusing the < (less than) and > (greater than) signs. 20 is less than 30. |
273 – 145 looks similar to 272 – 144, so they must be equal. | Analyze the change: (273–1) – (145–1). Since 1 is subtracted from both numbers, the difference remains the same. So 273–145 = 272–144. | In this specific case they are equal, but you must reason it out. Never assume equality based on looks. For example, 273-145 > 272-145. |
Comparing 213 – 77 and 214 – 76 by guessing. | You start with 1 more (214 vs 213) but you also subtract 1 less (76 vs 77). The first makes it bigger, the second also makes it bigger. So 214-76 is definitely greater. | Logical comparison requires careful step-by-step thinking, not just a quick glance. The final value is greater by 2. |
| Writing the answer to a word problem as just a number. | Write the expression first, then its value. The question asked for the expression. | Always read the question carefully. If it asks for an expression, provide the mathematical phrase like 5 × 25, not just the final value 125. |
Brain-Teaser Questions
Challenge yourself with these higher-order thinking problems.
-
Rohan scored 125 runs in his first cricket match and 78 runs in his second. Sameer scored 130 runs in his first match and 72 in his second. Write an expression for each player's total runs and determine who scored more, without calculating the totals.
💡 Answer:
Rohan's expression: 125 + 78. Sameer's expression: 130 + 72.
To compare, notice Sameer scored +5 runs in the first match (130 vs 125), but –6 runs in the second match (72 vs 78).
The net effect is +5 – 6 = –1. So, Sameer scored 1 run less than Rohan. 125 + 78 > 130 + 72.
-
Fill in the blank with a single number to make the statement true: 8 × 12 = (10 × 12) – ____.
💡 Answer:
8 × 12 means "8 groups of 12". 10 × 12 means "10 groups of 12".
To get from 10 groups of 12 down to 8 groups of 12, you need to subtract 2 groups of 12.
2 groups of 12 is 2 × 12 = 24. The missing number is 24.
-
Which is greater: 4567 – 1234 or 4566 – 1235? Explain your reasoning without finding the exact values.
💡 Answer:
Let's compare the second expression to the first. The starting number is 1 less (4566 vs 4567). The number being subtracted is 1 more (1235 vs 1234).
Both of these changes make the final result smaller. You start with less, and you take away more.
Therefore, the second expression is significantly smaller. 4567 – 1234 > 4566 – 1235.
Mini Cheatsheet
A quick summary of this page for your last-minute revision.
| Concept | Description | Example |
|---|
| Expression | Numbers and operators combined to represent a value. | 75 + 15 |
| Value | The result of an expression's calculation. | The value of 75 + 15 is 90. |
Greater Than (>) | The value on the left is larger than the value on the right. | 10 + 5 > 10 + 4 |
Less Than (<) | The value on the left is smaller than the value on the right. | 50 – 10 < 50 – 5 |
| Logical Comparison | Comparing expressions by analyzing the change in numbers, not by full calculation. | 100+5 > 99+5 because the starting number is larger. |
Reading and Evaluating Complex Expressions — Brackets
{{FORMULA: expr=Rule of Brackets | symbols=():Parentheses, {}:Braces, []:Square Brackets → Solved Innermost First}}
Reading and Evaluating Complex Expressions
Have you ever given instructions to a friend, only for them to misunderstand? In language, a misplaced comma can change the entire meaning of a sentence.
Consider this: "Let's eat grandpa!" vs. "Let's eat, grandpa!"
Punctuation saves lives! In mathematics, we have a similar tool to prevent confusion and save our calculations from disaster. This tool is the bracket. Without a clear order, a simple expression like 30 + 5 × 4 could mean two different things. Did you mean to add 30 and 5 first, or multiply 5 and 4 first? Brackets remove this doubt, acting as the punctuation of mathematics.
What are Brackets?
Brackets are symbols used in pairs to group parts of a mathematical expression. The part of the expression inside the brackets is treated as a single unit and must be solved first, before any operations outside the brackets are performed.
| Symbol | Name | Function |
|---|
( ) | Parentheses or Round Brackets | The most commonly used type of bracket for grouping. |
{ } | Braces or Curly Brackets | Used to group expressions that already contain parentheses. |
[ ] | Square Brackets | Used to group expressions that already contain braces. |
The standard order for these nested brackets is to solve from the inside out: [ { ( ) } ].
{{KEY: type=concept | title=The Golden Rule of Brackets | text=When evaluating an expression, you must ALWAYS calculate the value of the expression inside the brackets first. Think of brackets as creating a mini-problem that needs to be solved before you can tackle the main problem.}}
The Logic: Why Order Matters
Let's see why a fixed rule is necessary. We'll use the example from the NCERT textbook to understand the confusion.
The Problem: Mallesh has 30 marbles. Arun brings 5 bags, each with 4 marbles. How many in total?
The expression is 30 + 5 × 4.
-
Purna's Approach (Incorrect)
Purna reads the expression from left to right. He first adds 30 and 5.
30 + 5 = 35
Then, he multiplies the result by 4.
35 × 4 = 140
This result doesn't match the story. Purna's calculation implies that both Mallesh's 30 marbles and Arun's 5 bags were multiplied by 4.
-
Mallesh's Approach (Correct)
Mallesh understands the context. Arun's total marbles must be calculated first.
5 × 4 = 20
Then, this total is added to Mallesh's marbles.
30 + 20 = 50
This result correctly represents the situation.
-
Using Brackets for Clarity
To make sure everyone gets the right answer without needing the full story, we use brackets. The operation that needs to be done first is placed inside parentheses.
30 + (5 × 4)
-
Evaluating with Brackets
Now, the rule is clear: solve the inside of the bracket first.
30 + (20)
Then, perform the remaining operation.
30 + 20 = 50
The brackets force the correct order of operations, eliminating any ambiguity.
Solved Examples
Example 1: Simple Grouping (Easy)
Given: The expression (12 + 8) ÷ 4.
To Find: The value of the expression.
Solution:
-
The rule of brackets states we must solve the expression inside the parentheses first.
12 + 8 = 20
-
Now, we replace the bracketed expression with its calculated value and solve the rest.
20 ÷ 4 = 5
Final Answer: 5
Example 2: Real-World Calculation (Medium)
Given: Sanya buys a notebook for ₹45 and a set of pens for ₹80. She pays the shopkeeper with a ₹200 note.
To Find: An expression with brackets for the change Sanya receives, and calculate the amount.
Solution:
-
First, let's find the total amount Sanya spent. This needs to be calculated before finding the change. We can group this as (45 + 80).
45 + 80 = 125
-
The change is the amount paid minus the total cost. The full expression is 200 - (45 + 80).
-
We substitute the value calculated in step 1.
200 - 125
-
Now, we perform the final subtraction.
200 - 125 = 75
Final Answer: The expression is 200 - (45 + 80), and the change is ₹75.
Example 3: Nested Brackets (Hard)
Given: The expression 150 - [40 + {15 - (10 - 5)}].
To Find: The value of the expression.
Solution:
-
We must start with the innermost bracket, which is (10 - 5).
10 - 5 = 5
-
Substitute this value back into the expression. It now becomes 150 - [40 + {15 - 5}]. The next innermost bracket is the curly brace {15 - 5}.
15 - 5 = 10
-
Substitute this value back. The expression is now 150 - [40 + 10]. The final bracket is the square bracket [40 + 10].
40 + 10 = 50
-
Finally, perform the last operation.
150 - 50 = 100
Final Answer: 100
Example 4: Distributing a Negative Sign (Tricky)
Given: The expression 50 - (20 - 12).
To Find: The value of the expression.
Solution:
-
First, solve the expression inside the parentheses as per the rule.
20 - 12 = 8
-
Now, substitute this value back into the main expression.
50 - 8
-
Perform the final subtraction.
50 - 8 = 42
Note: A common mistake is to remove the bracket incorrectly. It is NOT 50 - 20 - 12. The subtraction applies to the result of the bracket.
Final Answer: 42
Tips & Tricks
| Technique | Description | Example |
|---|
| Inside-Out Method | For nested brackets [ { ( ) } ], always start with the innermost pair ( ) and work your way outwards. | In [10 + {5 - (2)}], solve (5-2) first. |
| Rewrite & Replace | After solving a bracket, rewrite the entire expression with the new value. This prevents confusion. | 100 - (12+8) becomes 100 - 20. |
| Brackets as a Box | Think of everything inside a bracket as being locked in a box. You must figure out what's in the box first. | In 5 × (4+3), the box contains 7. So, 5 × 7. |
Common Mistakes
| ❌ Wrong Method | ✅ Correct Method | Why it's Wrong |
|---|
100 – 15 + 56 = 85 + 56 = 141 | 100 – (15 + 56) = 100 – 71 = 29 | The total cost (15+56) must be calculated first and then subtracted from 100. |
50 – (10 + 5) = 50 – 10 + 5 = 45 | 50 – (10 + 5) = 50 – 15 = 35 | The minus sign applies to the entire sum (15), not just the first number (10). |
8 × (2 + 3) = 8 × 2 + 3 = 16 + 3 = 19 | 8 × (2 + 3) = 8 × 5 = 40 | You must resolve the addition inside the bracket before performing the multiplication. |
[20 - {10 - 2}] = [20 - 10 - 2] = 8 | [20 - {10 - 2}] = [20 - 8] = 12 | Nested brackets have their own priority. The inner {} must be solved to a single number first. |
Brain-Teaser Questions
-
Insert one pair of brackets () in the expression 8 + 6 ÷ 2 - 1 to get the smallest possible result.
💡 Answer:
The expression is 8 + 6 ÷ (2 - 1).
8 + 6 ÷ 1 = 8 + 6 = 14. Let's re-evaluate.
How about (8 + 6) ÷ 2 - 1? This gives 14 ÷ 2 - 1 = 7 - 1 = 6.
How about 8 + (6 ÷ 2) - 1? This gives 8 + 3 - 1 = 10.
What if we subtract a larger number? 8 + 6 ÷ (2 - 1) isn't it. Let's try (8 + 6) ÷ (2-1) - not allowed, only one pair.
The answer is placing brackets around 8 + 6: (8 + 6) ÷ 2 - 1 = 14 ÷ 2 - 1 = 7 - 1 = 6.
Wait, let me try one more: 8 + 6 ÷ (2-1) is invalid. How about making the divisor large? Not possible.
What about making the subtracted part large? 8 + 6 ÷ 2 - 1.
Let's try 8 + (6 ÷ 2 - 1) = 8 + (3 - 1) = 8 + 2 = 10.
Let's try (8 + 6) ÷ 2 - 1 = 14 ÷ 2 - 1 = 7 - 1 = 6. This seems small.
The smallest is likely by making a negative part larger. 8 - (something large). Let's try 8 + 6 ÷ 2 - 1... this question is tricky. 8 - (6/2 + 1)? That changes operators.
Let's stick to the prompt. Insert one pair of brackets () in 8 + 6 ÷ 2 - 1.
(8+6)÷2-1 = 14÷2-1 = 7-1=6.
8+(6÷2)-1 = 8+3-1 = 10.
8+6÷(2-1) is invalid as per BODMAS which isn't taught yet. But if we follow order, (2-1)=1, so 8+6÷1 = 14.
The smallest is 6.
(8 + 6) ÷ 2 - 1 gives 14 ÷ 2 - 1 = 7 - 1 = 6. This seems correct.
-
What is the difference in value between 3 × (8 - 2) and (3 × 8) - 2?
💡 Answer:
First expression: 3 × (8 - 2) = 3 × 6 = 18.
Second expression: (3 × 8) - 2 = 24 - 2 = 22.
The difference is 22 - 18 = 4.
-
Can you remove the brackets from the expression (20 × 5) + 10 without changing its value? Explain why or why not.
💡 Answer:
Yes, you can. The expression becomes 20 × 5 + 10. In an expression with both multiplication and addition, multiplication is performed first by convention (a topic we will explore next). So, 20 × 5 would be calculated first anyway, giving 100, and then 100 + 10 = 110. The original expression (20 × 5) + 10 also evaluates to 100 + 10 = 110. The brackets here are for clarity but are not strictly necessary to get the correct answer.
Solved Numericals
This section provides extra practice with word problems that require brackets for correct evaluation.
Hero Rule: The Rule of Brackets
To evaluate an expression, always calculate the value of the part inside any brackets first, starting from the innermost pair.
Numerical 1: Calculating Average Score
GIVEN: A cricket batsman scores 45, 60, and 15 runs in three different matches. He gets out in all three innings.
FORMULA: Average Score = (Total Runs) ÷ (Number of Innings)
SUBSTITUTION:
-
First, we need to find the total runs. This must be calculated before dividing. So, we group the scores in brackets: (45 + 60 + 15).
(45 + 60 + 15) = 120
-
The number of innings is 3. The expression for the average is (45 + 60 + 15) ÷ 3.
-
Substitute the total runs into the expression.
120 ÷ 3 = 40
ANSWER with units: The batsman's average score is 40 runs.
Numerical 2: Monthly Budget Calculation
GIVEN: A family has a monthly income of ₹25,000. Their expenses are: Rent ₹8,000, Groceries ₹6,500, and Bills ₹2,500.
FORMULA: Savings = Income - (Total Expenses)
SUBSTITUTION:
- First, calculate the total expenses by summing up all the individual expenses. This group of operations must be done first.
| Expense | Amount (₹) |
|---|
| Rent | 8,000 |
| Groceries | 6,500 |
| Bills | 2,500 |
| Total | 17,000 |
The calculation is `(8000 + 6500 + 2500) = 17000`.
2. The full expression for savings is 25000 - (8000 + 6500 + 2500).
-
Substitute the total expenses into the expression.
25000 - 17000 = 8000
ANSWER with units: The family's monthly savings are ₹8,000.
Try It Yourself
- A bus starts with 45 passengers. At the first stop, 12 people get off and 7 people get on. Write a single expression using brackets to find the number of passengers now on the bus.
- Calculate the value of
[100 - {50 - (20 + 10)}].
- Reema buys 4 chocolates for ₹15 each and 2 juices for ₹25 each. She pays with a ₹200 note. How much change does she get back? (Write a full expression).
Answer Key
1. 45 - (12 - 7) = 40 passengers OR (45 - 12) + 7 = 40 passengers. The second one is more intuitive.
2. 80
3. 200 - [(4 × 15) + (2 × 25)] = ₹90
Mini Cheatsheet
| Concept | Rule / Identity | Example |
|---|
| Priority | Operations inside brackets () are always done first. | In 10 × (2+3), solve 2+3 first to get 5. |
Parentheses () | Innermost and most common type of bracket. | (5 - 2) × 4 = 3 × 4 = 12 |
Braces {} | Used to enclose parentheses. | {10 + (4-1)} = {10 + 3} = 13 |
Square Brackets [] | Used to enclose braces. | [2 × {5 + (6÷2)}] = [2 × {5+3}] = [2×8] = 16 |
| Order of Solving | Inside-Out: ( ) → { } → [ ] | Solve () first, then {}, then []. |
Reading and Evaluating Complex Expressions — Terms and Addition Properties
Reading and Evaluating Complex Expressions
Welcome back! So far, we've seen how brackets can bring clarity to mathematical expressions. But what happens when there are no brackets? If you see an expression like 30 + 5 × 4, should you add first or multiply first? It can be confusing, just like a sentence without punctuation.
Consider the sentence: "Shalini sat next to a friend with toys." Who has the toys? Shalini or her friend? The meaning is ambiguous. Adding a comma changes everything: "Shalini sat next to a friend, with toys." Now it's clear Shalini has the toys. In mathematics, we have a set of rules that act like punctuation to remove confusion and ensure everyone gets the same answer from the same expression. The first and most important rule involves understanding what a term is.
{{FORMULA: expr=a + b = b + a | symbols=a:First term, b:Second term}}
The Building Blocks of Expressions: Terms
At its heart, an expression is a story told with numbers and operations. The main characters in this story are the terms.
A term is a part of an expression that is separated by a + (plus) sign. This seems simple, but there's a crucial trick when subtraction is involved.
To identify terms correctly, we must first rewrite any subtraction as the addition of a negative number. Remember, subtracting a number is the same as adding its opposite.
For example, the expression 83 – 14 is the same as 83 + (–14).
Now, it's easy to see the terms separated by the + sign. The terms are 83 and –14. The sign is part of the term!
{{KEY: type=concept | title=The Golden Rule of Terms | text=To correctly identify terms, always convert subtractions into additions of negative numbers. For example, in the expression 83 – 14, the terms are not 83 and 14. The expression is 83 + (–14), so the terms are 83 and –14. The sign always sticks with the number that follows it!}}
Once we've identified the terms, we unlock a powerful ability: we can rearrange them and group them in any way we like, thanks to the properties of addition.
Definitions & Properties
Let's formalize the rules that govern how we can manipulate terms. These properties only apply to addition and multiplication, but since we convert all subtractions to additions, we can use them much more widely.
| Property Name | Algebraic Form | Meaning |
|---|
| Term | a, b, -c | A single number or a product of numbers/variables in an expression, separated by + signs. The sign (+ or –) is part of the term. |
| Commutative Property of Addition | a + b = b + a | You can swap the order of terms being added without changing the final sum. Think commute like traveling back and forth. |
| Associative Property of Addition | (a + b) + c = a + (b + c) | You can change the grouping of terms being added without changing the final sum. Think associate like choosing which friends to group with first. |
The Logic: Why Can We Rearrange Terms?
Why is it safe to shuffle terms around, even when subtraction is involved? The logic lies in our "Golden Rule" of converting every subtraction into an addition. Let's see it in action.
-
Start with a basic expression involving subtraction.
Let's use 6 – 4.
-
Convert the subtraction to the addition of an inverse.
Subtracting 4 is identical to adding its inverse, which is –4.
6 – 4 = 6 + (–4)
-
Identify the terms.
Now that the expression is purely addition, we can clearly see the terms separated by the + sign.
The terms are 6 and –4.
-
Apply the Commutative Property of Addition.
This property states that a + b = b + a. We can swap the order of our terms 6 and –4.
6 + (–4) = (–4) + 6
-
Simplify the expression.
Both versions of the expression give the same result.
6 + (–4) = 2
(–4) + 6 = 2
-
Conclusion.
Because 6 – 4 can be rewritten as 6 + (–4), and because the order of addition doesn't matter, we can freely move the –4 term around. This proves that you can reorder the terms of an expression as long as you keep the sign attached to its number.
Solved Examples
Let's apply this concept of terms to solve some problems, from easy to tricky.
Example 1: Basic Reordering (Easy)
Given: The expression 15 – 9 + 4.
To Find: The value of the expression by first identifying and reordering the terms.
Solution:
-
First, rewrite the expression to show only addition.
15 – 9 + 4 = 15 + (–9) + 4
-
Identify the terms. The terms are 15, –9, and 4.
-
Apply the Commutative Property to reorder the terms. Let's move the positive terms together.
15 + (–9) + 4 = 15 + 4 + (–9)
-
Now, perform the addition from left to right.
(15 + 4) + (–9) = 19 + (–9)
-
Calculate the final sum.
19 + (–9) = 19 – 9 = 10
Final Answer:
10
Example 2: Grouping Positives and Negatives (Medium)
Given: The expression 75 – 20 + 15 – 40.
To Find: The value of the expression by grouping positive and negative terms.
Solution:
-
Rewrite the expression as a sum of its terms.
75 + (–20) + 15 + (–40)
-
The terms are 75, –20, 15, and –40.
-
Use the Commutative and Associative properties to group all positive terms together and all negative terms together.
(75 + 15) + (–20 + –40)
-
Calculate the sum of the positive group and the negative group separately.
(75 + 15) = 90
(–20 + –40) = –60
-
Add the results of the two groups.
90 + (–60) = 90 – 60 = 30
Final Answer:
30
Example 3: A Real-World Scenario (Hard)
Given: A drone pilot starts on a terrace. The drone goes 12 m up, then 7 m down, then 5 m up, and finally 8 m down.
To Find: An expression for the drone's final height relative to the terrace, and its value.
Solution:
-
Represent the movements as an arithmetic expression. "Up" is positive, and "down" is negative.
12 – 7 + 5 – 8
-
Identify the terms by converting subtractions to additions.
12 + (–7) + 5 + (–8)
The terms are 12, –7, 5, and –8.
-
Group the upward movements (positive terms) and downward movements (negative terms).
(12 + 5) + (–7 + –8)
-
Calculate the total upward movement and total downward movement.
Total upward = 12 + 5 = 17 m
Total downward = –7 + –8 = –15 m
-
Find the net displacement by adding the two results.
17 + (–15) = 17 – 15 = 2 m
Final Answer:
The drone is 2 m above the terrace.
Example 4: Finding Friendly Pairs (Tricky)
Given: The expression 148 + 75 – 48 – 25.
To Find: The value of the expression using clever grouping.
Solution:
-
Identify the terms of the expression.
148 + 75 + (–48) + (–25)
The terms are 148, 75, –48, and –25.
-
Instead of just grouping positives and negatives, look for "friendly pairs" — numbers that are easy to calculate together. Notice that 148 and –48 are a good pair, and 75 and –25 are another.
-
Reorder and group these friendly pairs using the Commutative and Associative properties.
(148 + (–48)) + (75 + (–25))
-
Simplify the expression inside each bracket.
(148 – 48) = 100
(75 – 25) = 50
-
Add the results from the simplified groups.
100 + 50 = 150
This method was much faster than calculating 148 + 75 first!
Final Answer:
150
Tips & Tricks
Mastering expressions is about working smarter, not harder. Here are some shortcuts based on the properties of addition.
| Trick | Description | Example |
|---|
| 1. Group and Conquer | Rearrange the expression to put all positive terms first and all negative terms last. Add each group separately, then combine. | For 50 - 10 + 20 - 5, calculate (50+20) - (10+5) = 70 - 15 = 55. |
| 2. Look for Friendly Pairs | Scan the expression for numbers that add or subtract to make a round number (like 10, 50, 100). | For 88 + 45 - 38, pair 88 - 38 first to get 50, then add 45 to get 95. |
| 3. The Sign Sticks! | Never forget that the sign (+ or –) to the left of a number is part of its identity. When you move the number, you must move its sign too. | In 10 - 20 + 5, the terms are 10, –20, and 5. If you move 20, you must move the – sign with it. |
Common Mistakes
It's easy to make small errors when reordering terms. Here are some common pitfalls to avoid.
| ❌ Wrong Method | ✅ Right Method | Why it's Wrong |
|---|
Reordering 25 – 10 + 5 as 25 + 10 – 5. | The terms are 25, –10, 5. Reordering gives 25 + 5 – 10. | The – sign belongs to the 10. When moving the 10, the sign must travel with it. The wrong method changed -10 into +10. |
Thinking 15 – 8 is the same as 8 – 15. | 15 – 8 = 7, but 8 – 15 = –7. Commutative property doesn't apply to subtraction directly. | To use the property, you must convert to addition first: 15 + (–8) is the same as (–8) + 15. Both equal 7. |
Calculating 100 – (20 + 30) as 100 – 20 + 30. | 100 – (20 + 30) = 100 – 50 = 50. | When removing brackets with a minus sign in front, you must change the sign of every term inside. 100 – (20+30) becomes 100 – 20 – 30. |
Brain-Teaser Questions
Ready for a challenge? Use your knowledge of terms and properties to solve these.
-
Find the missing number: 125 + ____ + 50 – 15 = 150.
💡 Answer:
Let the missing number be x. The expression is 125 + x + 50 – 15 = 150.
Grouping the knowns: (125 + 50 – 15) + x = 150 → (175 – 15) + x = 150 → 160 + x = 150.
Therefore, x = 150 – 160 = –10. The missing number is –10.
-
You know that addition is commutative (a + b = b + a). Is division also commutative? That is, is a ÷ b always equal to b ÷ a? Explain with an example.
💡 Answer:
No, division is not commutative. For example, let a = 10 and b = 2.
a ÷ b = 10 ÷ 2 = 5.
b ÷ a = 2 ÷ 10 = 0.2.
Since 5 ≠ 0.2, division is not commutative.
-
Without doing heavy calculations, find the value of: 1 – 2 + 3 – 4 + 5 – 6 + ... + 99 – 100. (Hint: Look for a pattern by grouping pairs).
💡 Answer:
Group the terms in pairs: (1 – 2) + (3 – 4) + (5 – 6) + ... + (99 – 100).
Each pair evaluates to –1.
How many pairs are there? From 1 to 100, there are 100 numbers, which make 100 ÷ 2 = 50 pairs.
So the sum is 50 × (–1) = –50.
Mini Cheatsheet
Here's a quick summary of the key ideas from this page. Screenshot this for your last-minute revision!
| Concept | Key Idea | Example |
|---|
| Identifying Terms | Convert all subtractions to additions of negative numbers. The terms are separated by +. | 5 – 3 + 8 becomes 5 + (–3) + 8. Terms are 5, –3, 8. |
| The Sign Sticks | When reordering, the sign (+ or –) to the left of a number must move with it. | 10 – 20 can be reordered to –20 + 10. |
| Commutative Property | You can swap the order of terms being added. a + b = b + a. | –7 + 12 is the same as 12 + (–7). |
| Associative Property | You can change the grouping of terms being added. (a+b)+c = a+(b+c). | (5+10)+2 is the same as 5+(10+2). |
| Smart Grouping | Group positive numbers, negative numbers, or "friendly pairs" to simplify calculations. | In 23+50–13, calculate (23–13)+50 first. |
Reading and Evaluating Complex Expressions — Evaluating Terms with Multiplication/Division
Reading and Evaluating Complex Expressions
Welcome back! In our last session, we saw how ambiguity can sneak into mathematical expressions, just like it can in sentences. The expression 30 + 5 × 4 gave two different answers, 140 and 50, depending on what you did first.
We learned that brackets ( ) are a powerful tool to remove this confusion. By writing 30 + (5 × 4), we make it crystal clear that the multiplication must happen first. But what if there are no brackets? Is the expression meaningless? Not at all! Mathematics has a built-in set of rules to ensure everyone gets the same answer.
{{KEY: type=concept | title=The Golden Rule of Expressions | text=To evaluate an expression, we first find the value of each term individually. Only after all terms have been simplified do we perform the final additions.}}
What Are Terms, Again?
Let's quickly refresh our memory on the most important idea for this topic: Terms.
As we saw, terms are the parts of an expression that are separated by addition (+) signs. To identify terms correctly, we must first convert all subtractions into additions of their negative counterparts.
For example, in 23 – 2 × 4 + 16, we first rewrite it as 23 + (–2 × 4) + 16.
Now, it's easy to see the terms:
- The first term is
23.
- The second term is
–2 × 4.
- The third term is
16.
Notice that –2 × 4 is a single term because the multiplication "glues" the numbers together. The + signs are the only thing that separates terms.
The Order of Operations: Evaluating Terms
The rule for evaluating expressions without brackets flows directly from this idea of terms. The process is a two-step dance:
-
Step 1: Simplify Each Term
Look at each term you have identified. If a term involves multiplication or division, solve that part first. A term must be simplified down to a single number.
- In our example,
23 + (–2 × 4) + 16, we first simplify the second term: –2 × 4 = –8.
- Our expression now becomes
23 + (–8) + 16.
-
Step 2: Add the Simplified Terms
Once every term is a single number, you can simply add them all up.
- Continuing our example:
23 + (–8) + 16.
23 – 8 = 15.15 + 16 = 31.
This simple two-step process—evaluate terms, then add them—is the fundamental rule that ensures consistency in mathematics. It's why multiplication and division are always performed before addition and subtraction.
Solved Examples
Let's walk through some examples together, from easy to tricky, to master this concept.
Example 1: Basic Multiplication and Addition
Given: The expression 15 + 7 × 3.
To Find: The value of the expression.
Solution:
-
First, we identify the terms. The expression has a + sign separating 15 and 7 × 3.
-
Next, we evaluate each term. Term 1 is already a single number. We need to evaluate Term 2.
7 × 3 = 21
-
Now, we replace the term 7 × 3 with its value and add the terms together.
15 + 21 = 36
Final Answer: 36
Example 2: Subtraction and Multiplication
Given: The expression 40 – 5 × 6.
To Find: The value of the expression.
Solution:
-
To identify the terms correctly, we first convert the subtraction to addition. 40 – 5 × 6 is the same as 40 + (–5 × 6).
- Term 1:
40
- Term 2:
–5 × 6
-
We evaluate the second term, which involves multiplication.
–5 × 6 = –30
-
Now, we substitute this value back into the expression and perform the addition.
40 + (–30) = 40 – 30 = 10
Final Answer: 10
Example 3: Multiple Operations (The Full Gauntlet)
Given: The expression 18 + 24 ÷ 3 – 2 × 5.
To Find: The value of the expression.
Solution:
-
First, let's rewrite the expression to clearly see the terms. The subtraction – 2 × 5 becomes + (–2 × 5).
So, 18 + 24 ÷ 3 + (–2 × 5).
- Term 1:
18
- Term 2:
24 ÷ 3
- Term 3:
–2 × 5
-
Now, we evaluate each term that has an operation within it.
24 ÷ 3 = 8
–2 × 5 = –10
-
Finally, we substitute the simplified values back and add all the terms together.
18 + 8 + (–10)
-
Adding from left to right:
18 + 8 = 26
26 – 10 = 16
Final Answer: 16
Example 4: Brackets and Terms Combined
Given: The expression 70 – (10 + 4 × 5).
To Find: The value of the expression.
Solution:
-
Whenever we see brackets, our first priority is to solve the expression inside them. Let's focus on (10 + 4 × 5).
-
Inside the bracket, we follow our rules. We identify the terms: 10 and 4 × 5.
-
We evaluate the term with multiplication first.
4 × 5 = 20
-
Now we solve the expression inside the bracket.
10 + 20 = 30
-
The original expression now becomes much simpler. We replace the entire bracket with its value, 30.
70 – 30
-
Perform the final subtraction.
70 – 30 = 40
Final Answer: 40
Tips & Tricks
| Tactic | Description | Example |
|---|
| 1. Underline the Terms | Before you start calculating, take a pencil and lightly draw a line under each complete term. This visually separates the calculation steps. | In 10 + 5 × 2 – 8 ÷ 4, you would underline 10, 5 × 2, and 8 ÷ 4. |
| 2. The "Doer" Pass | Do a first "pass" over the expression and only solve the multiplications and divisions. Write the simplified expression on a new line. | 7 + 8 × 2 – 6 becomes 7 + 16 – 6 after the first pass. |
| 3. The "Adder" Pass | Do a second "pass" on the new expression, performing all the additions and subtractions from left to right. | 7 + 16 – 6 becomes 23 – 6, which equals 17. |
Common Mistakes
| ❌ Wrong Method | ✅ Correct Method | Why it's Wrong |
|---|
5 + 3 × 10<br>= 8 × 10<br>= 80 | 5 + 3 × 10<br>= 5 + 30<br>= 35 | The mistake is adding before multiplying. 3 × 10 is a single term and must be evaluated first. |
24 – 8 + 2<br>= 24 – 10<br>= 14 | 24 – 8 + 2<br>= 16 + 2<br>= 18 | When only addition and subtraction remain, you must work from left to right. You can't add 8 + 2 first. |
36 ÷ 6 × 3<br>= 36 ÷ 18<br>= 2 | 36 ÷ 6 × 3<br>= 6 × 3<br>= 18 | For division and multiplication (which are part of the same term-evaluation step), always work from left to right. |
50 – (10 + 5)<br>= 50 – 10 + 5<br>= 40 + 5 = 45 | 50 – (10 + 5)<br>= 50 – 15<br>= 35 | When removing a bracket with a minus sign in front, you must subtract the entire value of the bracket, not just the first number. |
Brain-Teaser Questions
Test your understanding with these slightly more challenging problems.
-
Insert one pair of brackets ( ) to make the following equation true: 8 + 4 × 5 – 2 = 58.
💡 Answer:
The brackets should be placed as (8 + 4) × 5 – 2. This becomes 12 × 5 – 2 = 60 – 2 = 58.
-
Rohan buys 4 notebooks for ₹25 each and 3 pens for ₹10 each. He pays the shopkeeper with a ₹200 note. Write a single mathematical expression to find the change he gets back and solve it.
💡 Answer:
The expression is 200 – (4 × 25 + 3 × 10).
Solving: 200 – (100 + 30) = 200 – 130 = ₹70.
-
Find the missing number (?) in the equation: 15 + 4 × ? – 10 = 17.
💡 Answer:
The missing number is 3.
Let's work backwards. 15 + (something) – 10 = 17.
This means 5 + (something) = 17, so the something must be 12.
If 4 × ? = 12, then the missing number is 12 ÷ 4 = 3.
Mini Cheatsheet
Here's a quick summary of everything on this page. Screenshot this for your last-minute revisions!
| Concept | Rule | Example |
|---|
| 1. Identify Terms | Convert all subtractions to adding a negative. Terms are parts separated by +. | 50 – 2 × 9 becomes 50 + (–2 × 9). The terms are 50 and –2 × 9. |
| 2. Evaluate Terms First | Solve all multiplications and divisions within each term to simplify them to single numbers. | In 50 + (–2 × 9), evaluate –2 × 9 to get –18. |
| 3. Add Terms Last | Once all terms are single numbers, perform all additions (and subtractions) from left to right. | The expression becomes 50 + (–18) or 50 – 18, which equals 32. |
| 4. Brackets First | If an expression has brackets, solve everything inside the brackets first, following rules 1-3. | In 100 / (10 + 10), solve 10 + 10 = 20 first. Then 100 / 20 = 5. |
| 5. Left-to-Right Rule | For operations of the same priority (like × and ÷, or + and –), always work from left to right. | 40 ÷ 10 × 2 is 4 × 2 = 8, not 40 ÷ 20 = 2. |
Reading and Evaluating Complex Expressions — Removing Brackets and Distributive Property
Reading and Evaluating Complex Expressions
In our last session, we saw how brackets ( ) are like punctuation marks in mathematics. They bring clarity and tell us which part of an expression to solve first, resolving any confusion. Just like a comma can change the meaning of a sentence, brackets define the order of operations.
But what happens when we need to simplify an expression that already contains brackets? We can't always leave them in. We need a set of reliable rules to remove the brackets without changing the expression's value. This is especially important when expressions become more complex, involving multiple operations or unknown values (which you will see in algebra). Learning to correctly open brackets is a foundational skill, much like learning to merge onto a highway—you need to know the rules to do it safely and reach your destination correctly.
{{FORMULA: expr=a × (b + c) = (a × b) + (a × c) | symbols=a:multiplier, b:first term in bracket, c:second term in bracket}}
Rules for Removing Brackets
Removing brackets systematically allows us to simplify complex expressions into a solvable form. The rule you apply depends entirely on what is directly in front of the bracket: a plus sign, a minus sign, or a number (implying multiplication).
| Rule / Property | Description | General Form |
|---|
| Positive Sign Prefix | If a bracket is preceded by a + sign, you can remove the bracket without changing the signs of the terms inside. | x + (y + z) = x + y + z <br> x + (y - z) = x + y - z |
| Negative Sign Prefix | If a bracket is preceded by a - sign, you must change the sign of every term inside the bracket when you remove it. | x - (y + z) = x - y - z <br> x - (y - z) = x - y + z |
| Distributive Property | If a bracket is preceded by a number (or a variable), you must multiply that number by every term inside the bracket. | a × (b + c) = a × b + a × c <br> a × (b - c) = a × b - a × c |
The Logic Behind the Rules
Let's understand why these rules work. They aren't magic; they are based on the fundamental properties of numbers.
-
The Positive Sign Rule: A + sign in front of a bracket means "add the value of the expression inside". Adding (5 + 2) to 10 is simply 10 + 7. This is the same as adding 5 and then adding 2 to 10. The result is the same.
10 + (5 + 2) is the same as 10 + 5 + 2
-
The Negative Sign Rule: This is the most crucial rule. A - sign means "subtract the entire value of the expression inside". Think of - as (-1) ×.
When you see 10 - (5 + 2), it means you need to subtract the total of 7 from 10. Subtracting 7 is the same as subtracting 5 and then subtracting 2.
10 - (5 + 2) = 10 - 7 = 3
10 - 5 - 2 = 5 - 2 = 3
Similarly, for 10 - (5 - 2), you are subtracting 3 from 10. This is the same as subtracting 5 but then adding back the 2 that was taken away from 5.
10 - (5 - 2) = 10 - 3 = 7
10 - 5 + 2 = 5 + 2 = 7
So, the minus sign effectively "distributes" itself, flipping the sign of each term inside.
-
The Distributive Property: This property connects multiplication and addition. Imagine you are buying 4 combo meals, each containing a burger (₹80) and fries (₹30).
You could find the cost of one combo meal first: ₹80 + ₹30 = ₹110. Then multiply by 4: 4 × ₹110 = ₹440.
This is 4 × (80 + 30).
Alternatively, you could calculate the cost of all 4 burgers and all 4 fries separately and then add them.
Cost of 4 burgers: 4 × ₹80 = ₹320.
Cost of 4 fries: 4 × ₹30 = ₹120.
Total cost: ₹320 + ₹120 = ₹440.
This is (4 × 80) + (4 × 30).
Both methods give the same result, proving that 4 × (80 + 30) = (4 × 80) + (4 × 30).
{{VISUAL: diagram: An area model showing a rectangle of dimensions 'a' by '(b+c)'. The rectangle is split into two smaller rectangles with dimensions 'a' by 'b' and 'a' by 'c', visually demonstrating that a × (b+c) = a × b + a × c.}}
{{KEY: type=concept | title=The Golden Rule of the Minus Sign | text=Think of a minus sign outside a bracket as a switch that flips the sign of every single term inside. A positive term becomes negative, and a negative term becomes positive.}}
Solved Examples
Let's apply these rules to solve some problems, starting from simple cases and moving to more complex ones.
Example 1: Simple Bracket Removal (Easy)
Given: The expression 50 + (15 - 5)
To Find: The simplified value of the expression.
Solution:
- Identify the sign before the bracket. It is a
+ sign.
- According to the rule for a positive prefix, we can remove the brackets without changing the signs of the terms inside.
50 + 15 - 5
- Now, perform the operations from left to right.
65 - 5
- Calculate the final result.
60
Final Answer: 60
Example 2: Negative Sign Prefix (Medium)
Given: The expression 100 - (60 + 25)
To Find: The simplified value of the expression.
Solution:
- Identify the sign before the bracket. It is a
- sign.
- According to the rule for a negative prefix, we remove the brackets and flip the sign of every term inside.
+60 becomes -60, and +25 becomes -25.
100 - 60 - 25
- Now, solve the expression from left to right.
40 - 25
- Calculate the final result.
15
Final Answer: 15
Example 3: Applying the Distributive Property (Hard)
Given: The expression 7 × (20 - 3)
To Find: The value using the distributive property.
Solution:
- Identify the number multiplying the bracket. It is
7.
- The distributive property states
a × (b - c) = (a × b) - (a × c).
- Apply this by multiplying
7 with each term inside the bracket.
(7 × 20) - (7 × 3)
- Calculate the value of each multiplication.
140 - 21
- Perform the final subtraction.
119
Self-Check: The expression inside the bracket is 20 - 3 = 17. The original problem is 7 × 17, which is 119. Our answer is correct.
Final Answer: 119
Example 4: Nested Brackets (Tricky)
Given: The expression 80 - [40 - {10 - (8 - 2)}]
To Find: The simplified value.
Solution:
- When dealing with nested brackets, always solve the innermost bracket first. In this case, it is
(8 - 2).
8 - 2 = 6
- Substitute this value back into the expression.
80 - [40 - {10 - 6}]
- Now, solve the next innermost bracket, which is
{10 - 6}.
10 - 6 = 4
- Substitute this value back.
80 - [40 - 4]
- Solve the final bracket
[40 - 4].
40 - 4 = 36
- Perform the last operation.
80 - 36
- Calculate the final result.
44
Final Answer: 44
Tips & Tricks
Use these shortcuts to solve problems faster and more accurately.
| Trick | Description | Example |
|---|
| Mental Math Multiplication | Use the distributive property to break down hard multiplications. | To calculate 18 × 102, think of it as 18 × (100 + 2). This becomes (18 × 100) + (18 × 2) = 1800 + 36 = 1836. |
| Sign Flip Chain | When you have a negative sign outside a bracket containing another negative sign, they cancel out. | 15 - (10 - (8 - 5)) → 15 - (10 - 3) → 15 - 7. Here, -(-3) did not happen. But in 15 - (10 - (-3)), it becomes 15 - (10+3). |
| Combine Like Terms First | Before applying a rule, simplify inside the bracket if possible. | In 5 × (2 + 8 + 3 - 1), it's easier to first calculate (12) inside, making it 5 × 12 = 60, than distributing 5 to all four terms. |
Common Mistakes
Be careful! These small errors are very common but easy to avoid once you know them.
| ❌ Wrong Method | ✅ Right Method | Why it's Wrong |
|---|
50 - (20 - 10) = 50 - 20 - 10 = 20 | 50 - (20 - 10) = 50 - 20 + 10 = 40 | The - sign outside the bracket must flip the sign of every term inside. -10 becomes +10. |
6 × (5 + 4) = (6 × 5) + 4 = 30 + 4 = 34 | 6 × (5 + 4) = (6 × 5) + (6 × 4) = 30 + 24 = 54 | The multiplier outside (6) must be distributed to all the terms inside the bracket, not just the first one. |
100 - [50 + {20 - 10}] = 100 - 50 + 10 = 60 | 100 - [50 + {10}] = 100 - [60] = 40 | You must solve brackets from the inside out. The + sign for the {} bracket is inside the [] bracket, which has a - sign outside. |
- (a - b + c) = -a - b - c | - (a - b + c) = -a + b - c | Again, the - outside flips every sign: +a→-a, -b→+b, +c→-c. The sign of b was missed. |
Brain-Teaser Questions
-
Simplify the expression: 12 × 105 using the distributive property.
💡 Answer:
Write it as 12 × (100 + 5).
(12 × 100) + (12 × 5) = 1200 + 60 = 1260.
-
A shopkeeper gives a discount of ₹10 on a combo of a pen (₹25) and a notebook (₹45). If you buy 5 such combos, write a single expression for the total amount you pay and solve it.
💡 Answer:
The cost of one combo after discount is (25 + 45) - 10.
The cost for 5 combos is 5 × ((25 + 45) - 10).
5 × (70 - 10) = 5 × 60 = 300.
So, you pay ₹300.
-
Find the value of: 200 - 2 × [40 + {3 × (10 - 5)}]
💡 Answer:
Innermost bracket: (10 - 5) = 5.
Expression becomes: 200 - 2 × [40 + {3 × 5}]
Next bracket: {3 × 5} = 15.
Expression becomes: 200 - 2 × [40 + 15]
Final bracket: [40 + 15] = 55.
Expression becomes: 200 - 2 × 55
Multiplication before subtraction: 2 × 55 = 110.
Final calculation: 200 - 110 = 90.
Mini Cheatsheet
| Concept | Rule / Formula | Example |
|---|
| Positive Prefix | x + (y - z) = x + y - z | 10 + (8 - 3) = 10 + 8 - 3 = 15 |
| Negative Prefix | x - (y - z) = x - y + z | 20 - (10 - 4) = 20 - 10 + 4 = 14 |
| Distributive (Addition) | a × (b + c) = (a × b) + (a × c) | 5 × (10 + 2) = 50 + 10 = 60 |
| Distributive (Subtraction) | a × (b - c) = (a × b) - (a × c) | 9 × (10 - 1) = 90 - 9 = 81 |
| Nested Brackets Rule | Solve from the innermost bracket outwards. | [ { ( ) } ] → Solve ( ) first. |
Summary & Quick Revision
Chapter 2: Arithmetic Expressions - Summary & Quick Revision
{{FORMULA: expr=Brackets → Terms (×, ÷) → Addition/Subtraction (+, -) | symbols=(): Parentheses, {}: Braces, []: Square Brackets are solved first (inside-out), Terms: Parts of an expression separated by + or - signs}}
Concept Introduction
Imagine you're managing your monthly pocket money. You receive ₹500 at the start of the month. You buy three comic books for ₹60 each and spend ₹40 on a snack with a friend. How much money is left? To calculate this, you instinctively create a mental expression: 500 - (3 × 60) - 40.
You know you must first calculate the total cost of the comics (3 × 60 = 180) before subtracting it from your initial amount. You don't do 500 - 3 first! This natural understanding of "what to do first" is the core of arithmetic expressions. Mathematics uses a clear set of rules, primarily involving brackets and terms, to ensure everyone gets the same answer from the same expression. This removes confusion and makes math a universal language for problem-solving.
Definitions & Key Concepts
Here are the fundamental building blocks for understanding and evaluating arithmetic expressions.
| Term | Meaning | Example |
|---|
| Arithmetic Expression | A combination of numbers, operators (+, –, ×, ÷), and sometimes brackets that represents a value. | 30 + 5 × 4 is an expression. |
| Terms | The parts of an expression that are separated by addition (+) signs. Subtraction is treated as adding a negative number. | In 23 – 2 × 4 + 16, the terms are 23, –2 × 4, and 16. |
| Brackets | Symbols like ( ), { }, and [ ] used to group parts of an expression. Operations inside brackets have the highest priority. | In 100 – (15 + 56), the part (15 + 56) must be solved first. |
| Evaluation | The process of finding the single numerical value of an expression by performing the operations in the correct order. | Evaluating 30 + (5 × 4) gives the value 50. |
{{KEY: type=concept | title=The "Terms First" Rule | text=Before adding or subtracting across an expression, simplify each term completely. A term is a part of the expression separated by a '+' or '-' sign. This means all multiplications and divisions within a term are done before the final additions and subtractions.}}
The Logic of Evaluating Expressions
Why do we multiply before we add? The logic comes from how we group things in the real world. The rules of evaluation ensure that mathematical expressions correctly model these real-world scenarios. Here’s the step-by-step logic.
-
Identify the Goal: The main goal is to remove ambiguity. The expression 30 + 5 × 4 could be 35 × 4 = 140 or 30 + 20 = 50. We need one consistent rule.
-
Introduce Brackets for Priority: The most powerful tool is the bracket. Whatever is inside a bracket must be solved first, period. This allows us to override the default rules. If we write (30 + 5) × 4, it means "add first," yielding 140. If we write 30 + (5 × 4), it means "multiply first," yielding 50.
-
Define Terms: When brackets are not present, we need a default rule. This is where terms come in. An expression is broken into terms separated by + signs. Remember, subtraction is just adding a negative number (a - b is the same as a + (-b)).
- Expression:
23 – 2 × 4 + 16
- As a sum of terms:
23 + (–2 × 4) + 16
- The terms are:
23, –2 × 4, and 16.
{{VISUAL: diagram: A visual breakdown of the expression 23 – 2 × 4 + 16, with boxes drawn around the three terms '23', '–2 × 4', and '16', which are then shown connected by plus signs.}}
-
The "Simplify Terms First" Rule: The fundamental rule is to evaluate each term into a single number before you perform the additions that connect them.
- In
23 + (–2 × 4) + 16, we first simplify the term (–2 × 4) to –8.
-
Final Calculation: Once all terms are simplified to single numbers, the expression becomes a simple chain of additions and subtractions, which are performed from left to right.
- The expression becomes
23 + (–8) + 16, which is 23 – 8 + 16.
23 – 8 = 15.15 + 16 = 31.
This step-by-step process, focusing on Brackets first, then simplifying Terms, then Adding/Subtracting, is the foundation for evaluating any arithmetic expression correctly.
Solved Examples
Here are some worked-out problems to solidify your understanding, ranging from easy to tricky.
Example 1: Basic Operations
Given: The expression 40 + 20 ÷ 5 - 3 × 6.
To Find: The value of the expression.
Solution:
-
Identify the terms in the expression. We can rewrite it as a sum of terms: 40 + (20 ÷ 5) + (–3 × 6).
The terms are 40, 20 ÷ 5, and –3 × 6.
-
Simplify each term individually.
- First term is
40.
- Second term:
20 ÷ 5 = 4
–3 × 6 = –18
-
Combine the simplified terms.
The expression is now 40 + 4 - 18.
-
Perform the addition and subtraction from left to right.
40 + 4 = 44
44 - 18 = 26
Final Answer: 26
Example 2: Expression with Brackets
Given: The expression 120 – (45 + 5 × 3).
To Find: The value of the expression.
Solution:
-
The primary rule is to solve the expression inside the brackets first: (45 + 5 × 3).
-
Inside the bracket, we identify the terms: 45 and 5 × 3. We simplify the term with multiplication first.
5 × 3 = 15
- Now, we solve the simplified expression inside the bracket.
45 + 15 = 60
- Substitute this value back into the original expression.
120 – 60
- Perform the final subtraction.
120 – 60 = 60
Final Answer: 60
Example 3: Word Problem Application
Given: Priya buys 4 notebooks for ₹25 each and 2 bags of pens, with each bag costing ₹50. She pays the shopkeeper with a ₹500 note.
To Find: The amount of change Priya should receive.
Solution:
-
First, let's create a mathematical expression for the total cost.
- Cost of notebooks =
4 × 25
- Cost of pens =
2 × 50
- Total cost =
(4 × 25) + (2 × 50)
-
Now, let's create the expression for the change received.
- Change =
Total Money Given - Total Cost
- Expression:
500 - ((4 × 25) + (2 × 50))
-
Solve the expression inside the main bracket. We start with the multiplications.
4 × 25 = 100
2 × 50 = 100
- Now, complete the operation inside the bracket.
100 + 100 = 200
- Substitute this total cost back into the change expression.
500 - 200
- Perform the final subtraction.
500 - 200 = 300
Final Answer: Priya will receive ₹300 in change.
Example 4: Nested Brackets
Given: The expression 90 – [50 – {25 – (10 – 4)}].
To Find: The value of this complex expression.
Solution:
- The rule for nested brackets is to solve the innermost bracket first. In this case, it is
(10 – 4).
10 – 4 = 6
-
Substitute this value back. The expression becomes 90 – [50 – {25 – 6}].
-
Now, solve the next innermost bracket, which is the curly brace {25 – 6}.
25 – 6 = 19
-
Substitute this value back. The expression becomes 90 – [50 – 19].
-
Solve the final bracket, the square bracket [50 – 19].
50 – 19 = 31
- Perform the last operation.
90 – 31 = 59
Final Answer: 59
Tips & Tricks
Use these shortcuts and methods to solve expressions faster and with fewer errors.
| Tip / Trick | Description | Example |
|---|
| 1. Underline the Terms | Before you start, lightly underline each term of the expression. This visually separates the calculation blocks and prevents you from adding or subtracting prematurely. | In 15 + 2 × 8 - 10 ÷ 2, you would underline 15, 2 × 8, and 10 ÷ 2. |
| 2. Convert Subtraction | Rewrite the expression so that every subtraction is an addition of a negative number. This makes identifying terms foolproof. | 30 - 10 + 5 becomes 30 + (-10) + 5. The terms are clearly 30, -10, and 5. |
| 3. Handle Bracket Signs | When a minus sign is outside a bracket, it reverses the sign of every term inside when you open the bracket. | 100 - (20 - 5) is the same as 100 - 20 + 5. Note how -5 became +5. |
Common Mistakes to Avoid
Many students make the same small errors. Here’s how to spot and fix them.
| ❌ Wrong Method | ✅ Correct Method | Why it's a Mistake |
|---|
Mistake 1: Ignoring Order of Operations <br> 10 + 5 × 4 <br> 15 × 4 = 60 | Correct Calculation <br> 10 + 5 × 4 <br> 10 + 20 = 30 | Addition was performed before multiplication. The term 5 × 4 must be simplified first. |
Mistake 2: Incorrect Sign Change <br> 50 - (15 + 10) <br> 50 - 15 + 10 = 45 | Correct Calculation <br> 50 - (15 + 10) <br> 50 - 25 = 25 | The minus sign applies to the entire result of the bracket, not just the first number. Solve the bracket first. |
Mistake 3: Left-to-Right Confusion <br> 24 ÷ 4 × 3 <br> 24 ÷ 12 = 2 | Correct Calculation <br> 24 ÷ 4 × 3 <br> 6 × 3 = 18 | When only multiplication and division are present, work from left to right. Don't prioritize multiplication over division. |
Mistake 4: Double Negative Error <br> 12 - (-8) <br> 12 - 8 = 4 | Correct Calculation <br> 12 - (-8) <br> 12 + 8 = 20 | Subtracting a negative number is the same as adding its positive counterpart. |
Brain-Teaser Questions
-
Using the numbers 2, 4, 6, and 8 exactly once, along with the operators +, –, ×, and ÷, what is the largest possible integer you can make? You can use brackets.
💡 Answer:
The key is to make the largest number the base of a multiplication.
8 × (6 + 4 - 2) = 8 × 8 = 64
Another high value is (8+2) × 6 - 4 = 10 × 6 - 4 = 56.
(8-2) × 6 + 4 = 6 × 6 + 4 = 40.
The largest is 64.
-
Place one set of brackets ( ) in the following expression to make the equation true:
100 - 20 × 4 - 2 = 60
💡 Answer:
We need to manipulate the subtraction. If we group the end part:
100 - 20 × (4 - 2)
100 - 20 × 2
100 - 40 = 60.
The brackets go around (4 - 2).
-
A shopkeeper is calculating his profit. He writes the expression: 50 × 120 - 40 × 100. This means he sold 50 items at ₹120 each and his cost for 40 of those items was ₹100 each. He has 10 items left. A new accountant writes the expression (50-40) × (120-100). Does this new expression represent the same profit? Why or why not?
💡 Answer:
No, they are not the same.
- Shopkeeper's profit:
(50 × 120) - (40 × 100) = 6000 - 4000 = ₹2000. This correctly calculates total revenue minus total cost.
- Accountant's calculation:
(50-40) × (120-100) = 10 × 20 = ₹200. This calculates the profit per item (120-100) and multiplies it by the number of unsold items (50-40), which is logically incorrect. It doesn't represent the profit.
Mini Cheatsheet
| Concept | Rule / Identity | Example |
|---|
| Order of Priority | 1st: Brackets ( ) { } [ ] <br> 2nd: Terms (×, ÷) <br> 3rd: Addition & Subtraction (+, -) | In 5 + (6-2)×3, solve (6-2) first. |
| Identifying Terms | Terms are parts of an expression separated by + or – signs. Simplify each term before combining. | 50 + 8×2 - 9 has three terms: 50, 8×2, and -9. |
| Nested Brackets | Always solve the innermost bracket first and work your way outwards. | In [10 + {5-1}], solve {5-1} before [ ]. |
| Opening Brackets | A - sign outside a bracket changes the sign of every term inside it. | a - (b - c) = a - b + c |
| Integer Sign Rules | a - (-b) = a + b <br> -(a + b) = -a - b | 20 - (-5) = 20 + 5 = 25 |
