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Arithmetic Expressions

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Chapter Notes

Overview
This chapter introduces arithmetic expressions as mathematical phrases involving numbers and operations like addition, subtraction, multiplication, and division. It explains how to evaluate expressions correctly using brackets and the concept of terms, and how to compare expressions without full calculation. The big idea is that an expression has a definite value, and the order of operations (determined by brackets and terms) is essential to get that value correctly.
Key Concepts (12)
Arithmetic Expression
A mathematical phrase that involves numbers and operations such as +, –, ×, and ÷.
e.g. 13 + 2, 20 – 4, 12 × 5, and 18 ÷ 3 are all arithmetic expressions.
Value of an Expression
The number that an arithmetic expression evaluates to.
e.g. The value of 13 + 2 is 15.
Equality Sign (=)
The symbol used to denote the relationship between an arithmetic expression and its value.
e.g. 13 + 2 = 15.
Comparing Expressions
Comparing expressions based on their values using the signs '=', '<', or '>'.
e.g. 10 + 2 > 7 + 1 because 12 > 8.
Brackets (Parentheses)
Symbols used in expressions to specify the order of operations; the expression inside brackets must be evaluated first.
e.g. In 30 + (5 × 4), first find 5 × 4 = 20, then add 30 to get 50.
Terms
The parts of an expression separated by a '+' sign. Subtractions are converted to additions of the inverse to identify terms.
e.g. In 30 + 5 × 4, the terms are 30 and 5 × 4. In 83 – 14, the terms are 83 and –14.
Inverse of a Number
A number with the opposite sign; subtracting a number is the same as adding its inverse.
e.g. The inverse of 14 is –14, and the inverse of –14 is 14.
Commutative Property of Addition
Swapping the order of terms does not change the sum.
e.g. 6 – 4 = –4 + 6, both equal 2.
Associative Property of Addition
Grouping terms in different ways does not change the sum.
e.g. (–7) + 10 + (–11) can be grouped as (–7 + 10) + (–11) or –7 + (10 + (–11)), both equal –8.
Distributive Property
Multiplying a number by a sum (or difference) is the same as multiplying the number by each term inside the brackets and then adding (or subtracting) the products.
e.g. 2 × (43 + 24) = 2 × 43 + 2 × 24. Also, 14 × 10 – 6 × 10 = (14 – 6) × 10.
Removing Brackets (Negative Sign)
When removing brackets preceded by a negative sign, the signs of the terms inside the brackets change.
e.g. 200 – (40 + 3) = 200 – 40 – 3. 500 – (250 – 100) = 500 – 250 + 100.
Removing Brackets (Positive Sign)
When removing brackets not preceded by a negative sign, the signs of the terms inside the brackets do not change.
e.g. 28 + (35 – 10) = 28 + 35 – 10.
Formulas & Equations (5)
a – b = a + (–b)
Reads as: a minus b is equal to a plus the inverse of b
Variables: a and b are any numbers
Use when: When converting subtraction to addition to identify terms in an expression
⚠ Students forget to change the sign of the subtracted number when writing it as a term.
a × (b + c) = a × b + a × c
Reads as: a times the sum of b and c is equal to a times b plus a times c
Variables: a, b, c are any numbers
Use when: When expanding or simplifying expressions involving multiplication over addition
⚠ Students forget to multiply the term outside with every term inside the brackets.
a × (b – c) = a × b – a × c
Reads as: a times the difference of b and c is equal to a times b minus a times c
Variables: a, b, c are any numbers
Use when: When expanding or simplifying expressions involving multiplication over subtraction
⚠ Students incorrectly write a × (b – c) = a × b – c, forgetting to multiply a with c.
(a + b) × c = a × c + b × c
Reads as: the sum of a and b times c is equal to a times c plus b times c
Variables: a, b, c are any numbers
Use when: When expanding expressions where a sum is multiplied by a number
⚠ Students think (a + b) × c = a × c + b, forgetting to multiply b by c.
(a – b) × c = a × c – b × c
Reads as: the difference of a and b times c is equal to a times c minus b times c
Variables: a, b, c are any numbers
Use when: When expanding expressions where a difference is multiplied by a number
⚠ Students incorrectly write (a – b) × c = a × c – b, missing the multiplication of b by c.
How It Works
Evaluating an Expression Using Terms
  1. Step 1: Convert all subtractions to additions of the inverse (e.g., 83 – 14 becomes 83 + (–14)).
  2. Step 2: Identify the terms — parts separated by '+' signs.
  3. Step 3: Evaluate each term individually (e.g., 5 × 4 becomes 20).
  4. Step 4: Add all the evaluated terms together to get the final value.
💡 Multiplication and division are evaluated within a term before adding terms. For example, in 30 + 5 × 4, the term 5 × 4 is evaluated first to 20, then added to 30.
Removing Brackets Preceded by a Negative Sign
  1. Step 1: Identify the expression inside the brackets.
  2. Step 2: Remove the brackets and the negative sign before them.
  3. Step 3: Change the sign of every term that was inside the brackets (positive becomes negative, negative becomes positive).
  4. Step 4: Write the resulting expression without brackets.
💡 For example, 100 – (15 + 56) becomes 100 – 15 – 56, not 100 – 15 + 56. The sign of every term inside changes.
Removing Brackets Not Preceded by a Negative Sign
  1. Step 1: Identify the expression inside the brackets.
  2. Step 2: Remove the brackets (and the positive sign before them, if any).
  3. Step 3: Keep the signs of all terms inside the brackets unchanged.
  4. Step 4: Write the resulting expression without brackets.
💡 For example, 28 + (35 – 10) becomes 28 + 35 – 10. The signs of 35 and –10 remain the same.
Comparing Expressions Without Full Calculation
  1. Step 1: Imagine a real-life situation for the expressions (e.g., marbles, money).
  2. Step 2: Compare the starting quantities and the changes (additions or subtractions).
  3. Step 3: Reason which final quantity is greater, smaller, or equal based on the changes.
  4. Step 4: Write the appropriate comparison sign ('>', '<', or '=').
💡 For example, to compare 1023 + 125 and 1022 + 128: Raja starts with 1 more marble than Joy, but Joy gets 3 more marbles today, so Joy ends up with more. Hence, 1023 + 125 < 1022 + 128.
Exam Traps — Don't Make These Mistakes
⚠ Adding before multiplying in expressions without brackets
✗ 30 + 5 × 4 = 35 × 4 = 140
→ The student added 30 and 5 first, but multiplication must be done within the term before addition. The terms are 30 and 5 × 4, so 5 × 4 is evaluated first.
✓ 30 + 5 × 4 = 30 + 20 = 50
⚠ Incorrectly removing brackets with a negative sign
✗ 100 – (15 + 56) = 100 – 15 + 56 = 141
→ When removing brackets preceded by a negative sign, the signs of all terms inside must change. The student kept the '+' sign of 56 unchanged.
✓ 100 – (15 + 56) = 100 – 15 – 56 = 29
⚠ Forgetting to multiply every term inside brackets
✗ 2 × (43 + 24) = 2 × 43 + 24 = 86 + 24 = 110
→ The student multiplied only the first term (43) by 2 and left 24 unchanged. The distributive property requires multiplying every term inside the brackets.
✓ 2 × (43 + 24) = 2 × 43 + 2 × 24 = 86 + 48 = 134
⚠ Confusing subtraction with addition of inverse when identifying terms
✗ In 83 – 14, the terms are 83 and 14.
→ The student did not convert subtraction to addition of the inverse. The correct terms are 83 and –14, because 83 – 14 = 83 + (–14).
✓ In 83 – 14, the terms are 83 and –14.
⚠ Incorrectly applying distributive property to subtraction
✗ 5 × (9 – 2) = 5 × 9 – 2 = 45 – 2 = 43
→ The student subtracted 2 directly instead of multiplying it by 5. The correct expansion is 5 × 9 – 5 × 2.
✓ 5 × (9 – 2) = 5 × 9 – 5 × 2 = 45 – 10 = 35
Compare & Contrast
Effect on signs of terms inside the brackets
Removing brackets preceded by a negative sign
Removing brackets not preceded by a negative sign
When removing brackets preceded by a negative sign, the signs of all terms inside change (e.g., 100 – (15 + 56) becomes 100 – 15 – 56). When removing brackets not preceded by a negative sign, the signs remain the same (e.g., 28 + (35 – 10) becomes 28 + 35 – 10).
What is changed in the expression
Commutative property of addition
Associative property of addition
Commutative property allows swapping the order of terms (e.g., 6 – 4 = –4 + 6). Associative property allows changing the grouping of terms (e.g., (–7 + 10) + (–11) = –7 + (10 + (–11))). Both do not change the sum.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA (3 marks) on evaluating expressions or removing brackets, 1 LA (5 marks) on writing expressions from word problems and evaluating them
Most asked: Evaluating expressions using the correct order of operations (BODMAS/Brackets and Terms) · Applying the distributive property to expand or simplify expressions · Removing brackets correctly, especially with negative signs · Comparing expressions without full calculation using reasoning
Don't skip: Word problems requiring students to write an arithmetic expression from a real-life situation and then evaluate it using terms and brackets; Questions that ask to compare two expressions using reasoning without full calculation
One-Line Revision (read aloud before the exam)
  1. An arithmetic expression is a mathematical phrase like 13 + 2 or 5 × 4.
  2. The value of an expression is the number it evaluates to.
  3. Brackets tell us which part of an expression to evaluate first.
  4. Terms are parts of an expression separated by '+' signs; subtraction is converted to addition of the inverse.
  5. When removing brackets preceded by a negative sign, change the sign of every term inside.
  6. When removing brackets not preceded by a negative sign, keep the signs of terms inside unchanged.
  7. The commutative property says swapping terms does not change the sum.
  8. The associative property says grouping terms differently does not change the sum.
  9. The distributive property: a × (b + c) = a × b + a × c.
  10. The distributive property also works for subtraction: a × (b – c) = a × b – a × c.
  11. To compare expressions without calculating, imagine a real-life situation and reason about the changes.
  12. Always evaluate multiplication and division within a term before adding terms together.
  13. An expression like 30 + 5 × 4 is evaluated as 30 + 20 = 50, not 35 × 4 = 140.
  14. The inverse of a number has the opposite sign; subtracting is the same as adding the inverse.

MCQs — 10 Questions with Answers

Q1. What is the value of the arithmetic expression 13 + 2?

  • A. 15 ✓
  • B. 11
  • C. 26
  • D. 6

Answer: A — 13 plus 2 equals 15, which is the value of this arithmetic expression.

Q2. Which of these expressions has the same value as 12?

  • A. 10 + 2 ✓
  • B. 15 − 2
  • C. 20 − 5
  • D. 6 + 7

Answer: A — 10 + 2 = 12; the other options equal 13, 15, and 13 respectively.

Q3. What does the term 'arithmetic expression' mean?

  • A. A mathematical phrase using numbers and operations that has a value ✓
  • B. A division problem only
  • C. A sentence written in words
  • D. Any number on a number line

Answer: A — An arithmetic expression is any mathematical phrase made of numbers and operations (+, −, ×, ÷) that evaluates to a single value.

Q4. In the expression 30 + 5 × 4, which operation must be done first?

  • A. 5 × 4 ✓
  • B. 30 + 5
  • C. 4 × 30
  • D. Both at the same time

Answer: A — Multiplication (5 × 4 = 20) must be done first, then addition (30 + 20 = 50).

Q5. Mallika spends ₹25 per day for lunch, Monday to Friday. Which expression shows her weekly lunch cost?

  • A. 5 × 25 ✓
  • B. 5 + 25
  • C. 25 ÷ 5
  • D. 25 − 5

Answer: A — 5 days × ₹25 per day = 5 × 25, which equals ₹125.

Q6. Irfan paid ₹100 for items costing ₹15 and ₹56. Which expression correctly shows his change?

  • A. 100 − 15 + 56
  • B. 100 − (15 + 56) ✓
  • C. 100 + 15 − 56
  • D. (100 − 15) + 56

Answer: B — The total cost is 15 + 56 = 71, so change is 100 − 71 = 29; brackets ensure we add the costs first.

Q7. What are the terms in the expression 12 + 7 − 3?

  • A. 12, 7, and 3
  • B. 12, 7, and −3 ✓
  • C. 12 and 4
  • D. 19 and 3

Answer: B — After converting subtraction to addition of the inverse, the terms are 12, 7, and −3.

Q8. Without calculating, compare 364 + 587 and 363 + 589. Which is true?

  • A. 364 + 587 > 363 + 589
  • B. 364 + 587 < 363 + 589
  • C. 364 + 587 = 363 + 589 ✓
  • D. Cannot be compared without calculating

Answer: C — First expression decreased by 1 and second increased by 2, making them equal overall.

Q9. In 5 × (3 + 2) + 7 × 8 + 3, what should be evaluated first?

  • A. 5 × 7
  • B. 3 + 2 ✓
  • C. 8 + 3
  • D. 7 × 8

Answer: B — Brackets have highest priority, so (3 + 2) = 5 must be calculated first, then 5 × 5 = 25.

Q10. Which comparison is correct based on terms and their grouping?

  • A. (−7) + 10 + (−11) equals −8 whether we add in any order ✓
  • B. Changing the order of adding terms always changes the final sum
  • C. Multiplication terms must always be added before positive terms
  • D. Negative terms cannot be grouped together in an expression

Answer: A — The commutative and associative properties mean adding terms in any order gives the same sum of −8.

Flashcards

What is an arithmetic expression?

A mathematical phrase using numbers and operations (+, −, ×, ÷) that can be evaluated to give a value.

Why do we use brackets in expressions like 100 – (15 + 56)?

Brackets tell us the exact order of operations—calculate inside brackets first, then do the rest.

What are terms in an expression?

Terms are the parts of an expression separated by a '+' sign after converting all subtractions to additions with negative numbers.

In the expression 30 + 5 × 4, which operation do we do first?

Multiplication first: 5 × 4 = 20, then add 30 to get 50.

Does the order of adding terms change the final sum?

No, adding terms in any order always gives the same sum (commutative and associative property).

What is the inverse of a number and why do we use it?

The inverse of a number has the opposite sign; subtracting a number means adding its inverse, which helps identify terms.

How do we identify all terms in the expression 6 × 5 + 3?

The terms are (6 × 5) and 3, because multiplication is completed before we look at addition.

Compare without calculating: 245 + 289 and 246 + 285. Which is greater?

They are equal because the first increased by 1 and the second decreased by 4, which balance out to the same total increase of +0.

In the expression 23 – 2 × 4 + 16, how many terms are there?

Three terms: 23, (−2 × 4) which equals −8, and 16.

Why is 113 − 25 = 112 − 24 true without calculating?

Raja had 1 more marble and lost 1 more marble, so both end with the same amount.

Important Board Questions

What is the value of the expression 10 + 2? [1 mark]

Simply add the two numbers together: 10 + 2 = ?

Mallesh brought 30 marbles and Arun brought 5 bags with 4 marbles each. Write an expression for the total number of marbles. What is the total? [2 marks]

First, identify what Arun brought: 5 × 4. Then add Mallesh's 30 marbles. Use brackets to show order: 30 + (5 × 4).

Compare these two expressions without fully calculating. Explain your thinking: 273 − 145 ___ 272 − 144. Which symbol (>, <, or =) goes in the blank? [3 marks]

The first expression: a number (273) minus some amount (145). The second: that number minus 1 (272) minus an amount that is 1 less (144). If you decrease both the starting number and what you subtract by the same amount, the result stays the same.

Solve the expression 5 × (3 + 2) + 7 × 8 + 3 step by step. Identify the terms, evaluate each term, and then find the final value. [5 marks]

Step 1: Evaluate inside brackets (3 + 2) = 5. Step 2: Identify terms: [5 × 5], [7 × 8], and [3]. Step 3: Calculate each term: 25, 56, and 3. Step 4: Add the terms: 25 + 56 + 3 = 84.

Assertion-Reason Questions for Arithmetic Expressions

Q1. Assertion (A):

In the expression 83 – 14, the terms are 83 and –14.

Reason (R):

Subtraction in an expression is always converted to addition of the inverse before identifying terms, because subtracting a number is the same as adding its negative.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. Terms in expressions are separated by '+' signs, and to properly identify terms when subtraction is present, we must rewrite subtraction as addition of the inverse. This is a direct explanation of why the terms of 83 – 14 are 83 and –14.

Q2. Assertion (A):

The expression 30 + 5 × 4 equals 50.

Reason (R):

Multiplication and addition have the same priority in the order of operations.

Answer: C — A is true but R is false — The assertion is true (30 + 5 × 4 = 30 + 20 = 50), but the reason is false. Multiplication actually has higher priority than addition in the order of operations, which is why we calculate 5 × 4 first, not 30 + 5. The reason does not correctly explain why the assertion is true.

Q3. Assertion (A):

Different arithmetic expressions can have the same value.

Reason (R):

Because all mathematical expressions must have the same answer regardless of how they are written.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — The assertion is true: 10 + 2, 15 – 3, 3 × 4, and 24 ÷ 2 all equal 12. However, the reason is misleading. The reason states 'all expressions must have the same answer', which is not true. Some expressions have different values. Different expressions simply CAN have the same value, but not all do. Both parts are true, but the reason doesn't correctly explain the assertion.

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Case-Based Questions (CBQ) for Arithmetic Expressions

Case Study:

A school canteen sells lunch boxes for ₹45 each and bottles of juice for ₹15 each. Raj buys 2 lunch boxes and 3 bottles of juice. His friend Priya buys 3 lunch boxes and 2 bottles of juice. Both students had ₹200 with them. The canteen manager wants to create expressions to calculate the change each student receives.

Q1. What is the expression for the total cost of Raj's purchase?

  • A. 2 × 45 + 3 × 15 ✓
  • B. (2 + 3) × (45 + 15)
  • C. 2 + 45 × 3 × 15
  • D. 200 – 2 – 3 – 45 – 15

Answer: A

Q2. What is Raj's change from ₹200?

  • A. 200 – (2 × 45 + 3 × 15) = ₹65 ✓
  • B. 200 – 2 × 45 + 3 × 15 = ₹155
  • C. (200 – 2) × 45 – 3 × 15 = ₹8,955
  • D. 200 – 90 – 45 = ₹65

Answer: A

Q3. What is the total cost of Priya's purchase?

  • A. 3 + 45 × 2 × 15 = ₹1,353
  • B. 3 × 45 + 2 × 15 = ₹165 ✓
  • C. (3 × 45) + (2 × 15) = ₹165
  • D. 3 + 45 + 2 + 15 = ₹65

Answer: B

Q4. Who gets more change, and by how much?

  • A. Both get equal change
  • B. Priya gets ₹10 more than Raj
  • C. Raj gets ₹10 more than Priya ✓
  • D. Priya gets ₹20 more than Raj

Answer: C

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