🔢 Arithmetic — Problem Types

Master the common arithmetic problem patterns and their systematic solution approaches.

Basic Calculation Problems

Single Operation Problems

Recognition: One arithmetic operation required Template:

1. Identify the operation
2. Set up the calculation
3. Perform the operation
4. Check your answer

Example: What is $47 + 23$?

1. Operation: Addition
2. Setup: $47 + 23$
3. Calculate: $47 + 23 = 70$
4. Check: $70 - 23 = 47$ ✓

Common variations:

• Addition: $a + b = ?$
• Subtraction: $a - b = ?$
• Multiplication: $a \times b = ?$
• Division: $a \div b = ?$

Multiple Operation Problems

Recognition: Order of operations required Template:

1. Identify all operations
2. Apply PEMDAS/BODMAS
3. Work step by step
4. Check each step

Example: What is $2 + 3 \times 4^2 - (6 + 2) \div 2$?

1. Operations: Addition, multiplication, exponentiation, subtraction, division
2. PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction
3. Steps:
   • Parentheses: $6 + 2 = 8$
   • Exponents: $4^2 = 16$
   • Multiplication: $3 \times 16 = 48$
   • Division: $8 \div 2 = 4$
   • Addition: $2 + 48 = 50$
   • Subtraction: $50 - 4 = 46$
4. Answer: $46$

Number Property Problems

Even/Odd Problems

Recognition: Questions about even/odd properties Template:

1. Identify the numbers involved
2. Apply even/odd properties
3. Determine the result
4. Check your reasoning

Example: Is $47 + 23$ even or odd?

1. Numbers: 47 (odd), 23 (odd)
2. Property: Odd + Odd = Even
3. Result: Even
4. Check: $47 + 23 = 70$ (ends in 0, so even) ✓

Common variations:

• Sum of even/odd numbers
• Product of even/odd numbers
• Powers of even/odd numbers
• Combinations of operations

Prime Problems

Recognition: Questions about prime numbers Template:

1. Identify the number
2. Check if it’s prime
3. Apply prime properties
4. Solve the problem

Example: Is 47 prime?

1. Number: 47
2. Check: Test divisibility by primes up to $\sqrt{47} \approx 6.86$
   • 2: 47 is odd, so not divisible by 2
   • 3: $4 + 7 = 11$, not divisible by 3
   • 5: Doesn’t end in 0 or 5
3. Result: 47 is prime
4. Check: 47 has exactly two factors (1 and 47) ✓

Common variations:

• Prime identification
• Prime factorization
• GCD/LCM problems
• Prime counting problems

Divisibility Problems

Recognition: Questions about divisibility Template:

1. Identify the number and divisor
2. Apply divisibility rules
3. Determine divisibility
4. Check your answer

Example: Is 1,234 divisible by 3?

1. Number: 1,234, Divisor: 3
2. Rule: Sum of digits must be divisible by 3
3. Calculate: $1 + 2 + 3 + 4 = 10$
4. Check: 10 is not divisible by 3, so 1,234 is not divisible by 3

Common variations:

• Single divisibility checks
• Multiple divisibility checks
• Finding divisors
• Divisibility in word problems

Estimation Problems

Rounding Problems

Recognition: Questions about rounding Template:

1. Identify the number and place to round
2. Look at the digit to the right
3. Apply rounding rules
4. Check your answer

Example: Round 3,247 to the nearest hundred

1. Number: 3,247, Place: Hundreds
2. Digit to right: 4 (in tens place)
3. Rule: 4 < 5, so round down
4. Answer: 3,200

Common variations:

• Round to nearest ten, hundred, thousand
• Round decimals to specified places
• Round to significant figures
• Round in word problems

Approximation Problems

Recognition: Questions about approximate values Template:

1. Identify what needs to be approximated
2. Choose appropriate estimation method
3. Perform the estimation
4. Check reasonableness

Example: Estimate $347 + 256$

1. Operation: Addition
2. Method: Front-end estimation
3. Calculate: $300 + 200 = 500$
4. Check: Actual answer is 603, so estimate is reasonable

Common variations:

• Front-end estimation
• Compatible numbers
• Benchmark estimation
• Estimation in word problems

Word Problems

Basic Word Problems

Recognition: Real-world situations with arithmetic Template:

1. Read the problem carefully
2. Identify what’s given and what’s asked
3. Translate words to arithmetic
4. Solve and check

Example: A store has 47 apples and buys 23 more. How many apples does it have now?

1. Given: 47 apples, buys 23 more
2. Asked: Total number of apples
3. Translation: $47 + 23 = ?$
4. Solve: $47 + 23 = 70$ apples

Common variations:

• Addition word problems
• Subtraction word problems
• Multiplication word problems
• Division word problems

Multi-Step Word Problems

Recognition: Multiple operations required Template:

1. Read the problem carefully
2. Break into steps
3. Solve each step
4. Combine results

Example: A store has 47 apples. It sells 23 apples and then buys 15 more. How many apples does it have now?

1. Step 1: Start with 47 apples
2. Step 2: Sell 23: $47 - 23 = 24$ apples
3. Step 3: Buy 15 more: $24 + 15 = 39$ apples
4. Answer: 39 apples

Common variations:

• Sequential operations
• Conditional operations
• Rate problems
• Mixture problems

Pattern Recognition Problems

Number Pattern Problems

Recognition: Sequences or patterns in numbers Template:

1. Identify the pattern
2. Find the rule
3. Apply the rule
4. Check your answer

Example: What is the next number in the sequence: 2, 4, 6, 8, ?

1. Pattern: Even numbers increasing by 2
2. Rule: Add 2 to each term
3. Apply: $8 + 2 = 10$
4. Answer: 10

Common variations:

• Arithmetic sequences
• Geometric sequences
• Fibonacci-like sequences
• Custom patterns

Operation Pattern Problems

Recognition: Patterns in operations Template:

1. Identify the operation pattern
2. Find the rule
3. Apply the rule
4. Check your answer

Example: What is $2 \times 3 \times 4 \times 5$?

1. Pattern: Multiplication of consecutive integers
2. Rule: Multiply all numbers
3. Apply: $2 \times 3 = 6$, $6 \times 4 = 24$, $24 \times 5 = 120$
4. Answer: 120

Common variations:

• Factorial problems
• Power problems
• Sum problems
• Product problems

Common Mistakes and Fixes

Careless Errors

Mistake: Arithmetic mistakes like $2 + 3 = 6$ Fix: Slow down and double-check each step

Order of Operations Errors

Mistake: $2 + 3 \times 4 = 20$ (should be 14) Fix: Use PEMDAS/BODMAS systematically

Sign Errors

Mistake: Forgetting negative signs Fix: Pay attention to signs and double-check

Copying Errors

Mistake: Writing 47 instead of 74 Fix: Show work clearly and check each step

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