CEUET Mathematics — Arithmetic — Multiples, Factors, PEMDAS, Fractions & DecimalsExam Answer Templates

Marks

2

Topic

Multiples and Factors

Difficulty

easy

Template Id

T1

Examiner Tip

Show alternative method (listing multiples) for bonus appreciation

Model Answer

Method 1 (Prime Factorization): 12 = 2² × 3 18 = 2 × 3² LCM = 2² × 3² = 4 × 9 = 36 Therefore, LCM of 12 and 18 = 36

Question Type

short_answer

Answer Structure

• Line 1: Write prime factorization of both numbers [1 mark]
• Line 2: Apply LCM formula and calculate [1 mark]

Scoring Breakdown

Common Mark Deductions

• Incomplete prime factorization
• Calculation errors
• Missing final statement

Key Phrases To Include

• Prime factorization
• LCM
• Therefore

Marks

3

Topic

Divisibility Rules

Difficulty

medium

Template Id

T2

Examiner Tip

Always verify your final step calculation to ensure accuracy

Model Answer

Given: Number = 847 To check: Divisibility by 7 Using divisibility rule for 7: Step 1: Double the last digit: 7 × 2 = 14 Step 2: Subtract from remaining digits: 84 - 14 = 70 Step 3: Check if 70 is divisible by 7: 70 ÷ 7 = 10 Since 70 is divisible by 7, therefore 847 is divisible by 7.

Question Type

short_answer

Answer Structure

• Line 1: State the divisibility rule [1 mark]
• Line 2-3: Apply the rule step by step [1 mark]
• Line 4: State conclusion clearly [1 mark]

Scoring Breakdown

Common Mark Deductions

• Wrong divisibility rule
• Calculation errors
• No clear conclusion

Key Phrases To Include

• Divisibility rule
• Step by step
• Therefore

Marks

3

Topic

PEMDAS

Difficulty

medium

Template Id

T3

Examiner Tip

Label each step clearly to show you understand the order of operations

Model Answer

Given: 2 + 3 × (4² - 2 × 6) ÷ 2 Using PEMDAS: Step 1: Solve brackets first 4² - 2 × 6 = 16 - 12 = 4 Step 2: Expression becomes: 2 + 3 × 4 ÷ 2 Step 3: Multiplication and Division (left to right): 3 × 4 = 12 12 ÷ 2 = 6 Step 4: Addition: 2 + 6 = 8 Therefore, the answer is 8.

Question Type

numerical

Answer Structure

• Line 1: Identify PEMDAS order [1 mark]
• Line 2-4: Solve step by step following order [1 mark]
• Line 5: State final answer [1 mark]

Scoring Breakdown

Common Mark Deductions

• Wrong order of operations
• Arithmetic errors
• Not showing working

Key Phrases To Include

• PEMDAS
• Step by step
• Therefore

Marks

2

Topic

Fractions and Decimals

Difficulty

easy

Template Id

T4

Examiner Tip

Always check if your fraction can be simplified further

Model Answer

Given: 3.75 Step 1: Write as fraction 3.75 = 375/100 Step 2: Simplify by dividing by GCF GCF of 375 and 100 = 25 375 ÷ 25 = 15 100 ÷ 25 = 4 Therefore, 3.75 = 15/4

Question Type

short_answer

Answer Structure

• Line 1: Convert decimal to fraction [1 mark]
• Line 2: Simplify to lowest terms [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not simplifying completely
• Wrong GCF calculation

Key Phrases To Include

• Simplest form
• GCF
• Therefore

Marks

5

Topic

Application of GCF

Difficulty

medium

Template Id

T5

Examiner Tip

Always verify your answer by checking if the division works out correctly

Model Answer

Given: Number of boys = 24 Number of girls = 18 To Find: Maximum number of groups with equal boys and girls in each group Solution: For equal distribution, we need to find GCF of 24 and 18. Finding GCF using prime factorization: 24 = 2³ × 3 = 8 × 3 18 = 2 × 3² = 2 × 9 GCF = 2¹ × 3¹ = 2 × 3 = 6 Therefore, maximum number of groups = 6 Verification: Boys per group = 24 ÷ 6 = 4 Girls per group = 18 ÷ 6 = 3 Total students per group = 4 + 3 = 7 Answer: Maximum 6 groups can be formed, each having 4 boys and 3 girls.

Question Type

long_answer

Answer Structure

• Line 1: Write Given and To Find [1 mark]
• Line 2-3: Identify that GCF is needed [1 mark]
• Line 4-5: Calculate GCF correctly [2 marks]
• Line 6: Verify and state final answer [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not writing Given/To Find
• Confusing GCF with LCM
• No verification
• Wrong prime factorization

Key Phrases To Include

• Given
• To Find
• GCF
• Prime factorization
• Verification
• Therefore

Marks

2

Topic

Factors

Difficulty

easy

Template Id

T6

Examiner Tip

List factors in ascending order to avoid missing any

Model Answer

Given: Number = 36 To find factors, we find all numbers that divide 36 exactly: 36 ÷ 1 = 36 ✓ 36 ÷ 2 = 18 ✓ 36 ÷ 3 = 12 ✓ 36 ÷ 4 = 9 ✓ 36 ÷ 6 = 6 ✓ Therefore, factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36

Question Type

short_answer

Answer Structure

• Line 1: Show systematic division method [1 mark]
• Line 2: List all factors correctly [1 mark]

Scoring Breakdown

Common Mark Deductions

• Missing factors
• Including non-factors
• No systematic method

Key Phrases To Include

• Factors
• Divide exactly
• Therefore

Marks

3

Topic

Prime and Composite Numbers

Difficulty

medium

Template Id

T7

Examiner Tip

Always show the factorization when proving a number is composite

Model Answer

Given: Number = 91 To check if 91 is prime or composite: Step 1: Find √91 ≈ 9.5 So we check prime factors up to 9 Step 2: Check divisibility by prime numbers 2, 3, 5, 7: 91 ÷ 2 = 45.5 (not exact) 91 ÷ 3 = 30.33... (not exact) 91 ÷ 5 = 18.2 (not exact) 91 ÷ 7 = 13 (exact!) Step 3: Since 91 = 7 × 13, and both 7 and 13 are primes Therefore, 91 is composite because it has factors other than 1 and itself.

Question Type

short_answer

Answer Structure

• Line 1: Apply systematic method to check primality [1 mark]
• Line 2: Show division tests correctly [1 mark]
• Line 3: State conclusion with justification [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not checking systematically
• Calculation errors
• Wrong conclusion

Key Phrases To Include

• Prime
• Composite
• Factors
• Therefore

Marks

2

Topic

Fraction Operations

Difficulty

easy

Template Id

T8

Examiner Tip

Always check if the final fraction can be simplified

Model Answer

Given: 2/3 + 5/8 Step 1: Find LCM of denominators 3 and 8 LCM of 3 and 8 = 24 Step 2: Convert to equivalent fractions 2/3 = (2 × 8)/(3 × 8) = 16/24 5/8 = (5 × 3)/(8 × 3) = 15/24 Step 3: Add the fractions 16/24 + 15/24 = 31/24 Therefore, 2/3 + 5/8 = 31/24

Question Type

short_answer

Answer Structure

• Line 1: Find common denominator [1 mark]
• Line 2: Add correctly and simplify if possible [1 mark]

Scoring Breakdown

Common Mark Deductions

• Wrong LCM
• Calculation errors
• Not in simplest form

Key Phrases To Include

• LCM
• Equivalent fractions
• Therefore

Marks

5

Topic

Application of Multiples and Remainders

Difficulty

hard

Template Id

T9

Examiner Tip

Show several values to demonstrate systematic approach, then verify your answer

Model Answer

Given: Number ÷ 15 = quotient + remainder 7 Number ÷ 25 = quotient + remainder 17 To Find: Smallest positive number satisfying both conditions Solution: Let the number be N. Then: N = 15q₁ + 7 = 25q₂ + 17 From first condition: N = 15q₁ + 7 This means N can be: 7, 22, 37, 52, 67, 82, 97, 112, 127, 142, 157, 172, 187, 202, 217, 232, 247, 262, 277, 292, 307, 322, 337, 352, 367, 382, 397, 412, 427, 442, 457, 472, 487, 502, 517, 532... Checking which of these satisfies second condition: 517 ÷ 25 = 20 remainder 17 ✓ Verification: 517 ÷ 15 = 34 remainder 7 ✓ 517 ÷ 25 = 20 remainder 17 ✓ Therefore, the smallest such number is 517.

Question Type

long_answer

Answer Structure

• Line 1: Set up the problem using remainder theorem [1 mark]
• Line 2-3: Generate possible values from first condition [2 marks]
• Line 4: Test values against second condition [1 mark]
• Line 5: Verify and state final answer [1 mark]

Scoring Breakdown

Common Mark Deductions

• No systematic approach
• Calculation errors
• No verification
• Missing working

Key Phrases To Include

• Remainder theorem
• Systematic checking
• Verification
• Therefore

Marks

1

Topic

Fractions and Decimals

Difficulty

easy

Template Id

T10

Examiner Tip

Remember to simplify by dividing by GCF

Model Answer

0.375 = 375/1000 = 3/8

Question Type

very_short_answer

Answer Structure

• Line 1: Direct conversion and simplification [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not simplifying
• Wrong conversion

Key Phrases To Include

• Simplest form

Marks

2

Topic

PEMDAS

Difficulty

easy

Template Id

T11

Examiner Tip

Show each operation clearly to demonstrate understanding of order

Model Answer

Given: 3² + 2³ - 4 × 5 ÷ 2 Using PEMDAS: Step 1: Exponents first 3² = 9, 2³ = 8 Expression becomes: 9 + 8 - 4 × 5 ÷ 2 Step 2: Multiplication and Division (left to right) 4 × 5 = 20, 20 ÷ 2 = 10 Expression becomes: 9 + 8 - 10 Step 3: Addition and Subtraction (left to right) 9 + 8 = 17, 17 - 10 = 7 Therefore, the answer is 7.

Question Type

short_answer

Answer Structure

• Line 1: Apply PEMDAS systematically [1 mark]
• Line 2: Calculate correctly and state answer [1 mark]

Scoring Breakdown

Common Mark Deductions

• Wrong order
• Arithmetic errors

Key Phrases To Include

• PEMDAS
• Step by step
• Therefore

Marks

5

Topic

Application of Ratios and LCM

Difficulty

medium

Template Id

T12

Examiner Tip

Always verify your answer by checking both the ratio and LCM conditions

Model Answer

Given: Two numbers are in ratio 3:4 LCM of the numbers = 60 To Find: The two numbers Solution: Let the numbers be 3x and 4x where x is a positive integer. For LCM calculation: 3x = 3 × x 4x = 4 × x = 2² × x LCM of 3x and 4x = LCM of (3 × x) and (2² × x) = 3 × 2² × x = 12x Given that LCM = 60: 12x = 60 x = 60 ÷ 12 = 5 Therefore: First number = 3x = 3 × 5 = 15 Second number = 4x = 4 × 5 = 20 Verification: Ratio check: 15:20 = 3:4 ✓ LCM of 15 and 20 = 60 ✓ Answer: The two numbers are 15 and 20.

Question Type

long_answer

Answer Structure

• Line 1: Set up variables based on ratio [1 mark]
• Line 2-3: Express LCM in terms of variable [2 marks]
• Line 4: Solve for the variable [1 mark]
• Line 5: Find numbers and verify [1 mark]

Scoring Breakdown

Common Mark Deductions

• Wrong variable setup
• LCM calculation errors
• No verification

Key Phrases To Include

• Let
• Ratio
• LCM
• Verification
• Therefore

Marks

2

Topic

Mixed Number Operations

Difficulty

medium

Template Id

T13

Examiner Tip

Always convert mixed numbers to improper fractions before multiplying

Model Answer

Given: 2¼ × 1⅗ Step 1: Convert mixed numbers to improper fractions 2¼ = (2×4+1)/4 = 9/4 1⅗ = (1×5+3)/5 = 8/5 Step 2: Multiply the fractions 9/4 × 8/5 = (9×8)/(4×5) = 72/20 Step 3: Simplify 72/20 = 18/5 = 3⅗ Therefore, 2¼ × 1⅗ = 3⅗

Question Type

short_answer

Answer Structure

• Line 1: Convert to improper fractions [1 mark]
• Line 2: Multiply and simplify [1 mark]

Scoring Breakdown

Common Mark Deductions

• Wrong conversion
• Not simplifying
• Arithmetic errors

Key Phrases To Include

• Improper fractions
• Simplify
• Therefore

Marks

2

Topic

Properties of Real Numbers

Difficulty

easy

Template Id

T14

Examiner Tip

Always mention that the number must be non-zero for multiplicative inverse to exist

Model Answer

Multiplicative Inverse Property: The product of any non-zero number and its reciprocal equals 1. Mathematically: a × (1/a) = 1, where a ≠ 0 Example: 5 × (1/5) = 1 Or: ¾ × 4/3 = 1 Therefore, every non-zero number has a multiplicative inverse which is its reciprocal.

Question Type

short_answer

Answer Structure

• Line 1: State the property clearly [1 mark]
• Line 2: Give a correct example [1 mark]

Scoring Breakdown

Common Mark Deductions

• Incomplete definition
• Wrong example
• Not mentioning non-zero condition

Key Phrases To Include

• Multiplicative inverse
• Reciprocal
• Non-zero
• Example

Marks

3

Topic

Application of Arithmetic in Algebra

Difficulty

medium

Template Id

T15

Examiner Tip

Always define your variable clearly and verify your final answer

Model Answer

Given: Three consecutive even numbers with sum = 78 To Find: The three numbers Solution: Let the three consecutive even numbers be x, x+2, and x+4 According to given condition: x + (x+2) + (x+4) = 78 3x + 6 = 78 3x = 78 - 6 = 72 x = 72 ÷ 3 = 24 Therefore: First number = x = 24 Second number = x+2 = 26 Third number = x+4 = 28 Verification: 24 + 26 + 28 = 78 ✓ Answer: The three consecutive even numbers are 24, 26, and 28.

Question Type

short_answer

Answer Structure

• Line 1: Set up algebraic equation [1 mark]
• Line 2: Solve the equation correctly [1 mark]
• Line 3: Find all three numbers and verify [1 mark]

Scoring Breakdown

Common Mark Deductions

• Wrong algebraic setup
• Calculation errors
• No verification

Key Phrases To Include

• Let
• Consecutive even
• Verification
• Therefore