Key Takeaways for GCSE Arithmetic, Multiples, and Factors

1. Order of Operations (BODMAS)

Rule:
Follow the order: Brackets → Orders (powers/roots) → Division/Multiplication (left to right) → Addition/Subtraction (left to right).

Example:
20−12÷2×3=20−6×3=20−18=220−12÷2×3=20−6×3=20−18=2

Common Mistakes:

  • Doing addition before multiplication.
  • Ignoring left-to-right order for division/multiplication.

Tip:
Circle operations in the order they should be performed.

2. Negative Numbers

Rules:

  • Adding a negative = subtracting: −5+(−2)=−7−5+(−2)=−7
  • Subtracting a negative = adding: 1−(−4)=51−(−4)=5
  • Multiplication/Division:
    • Same signs → positive: (−5)×(−8)=40(−5)×(−8)=40
    • Different signs → negative: 24÷(−6)=−424÷(−6)=−4

Example:
−4×3=−12and(−15)÷(−3)=5−4×3=−12and(−15)÷(−3)=5

Tip:
Use a number line to visualise negative values.

3. Decimals

Addition/Subtraction:
Align decimal points and add trailing zeros.

Example: 4.53+1.60 6.13 4.53+1.60 6.13​​

Multiplication:
Convert to whole numbers first:
0.32×0.6=(32×6)÷1000=0.1920.32×0.6=(32×6)÷1000=0.192

Division:
Multiply both numbers to make the divisor whole:
0.516÷0.8=5.16÷8=0.6450.516÷0.8=5.16÷8=0.645

Common Mistakes:

  • Misplacing decimal points.

4. Multiples and Factors

Multiples: Numbers a number divides into.
Factors: Numbers that divide into a number.

Examples:

  • Multiples of 5 (23–43): 25, 30, 35, 40.
  • Factors of 18: 1, 2, 3, 6, 9, 18.

Prime Numbers: Numbers with only two factors (1 and itself).

  • Prime numbers <10: 2, 3, 5, 7.

Prime Factorisation:
Break numbers into primes.
Example:
12=22×312=22×3

Tip:
Use factor trees to systematically find primes.

5. LCM and HCF

LCM (Least Common Multiple):
Smallest number that is a multiple of all.
HCF (Highest Common Factor):
Largest number that divides into all.

Using Prime Factors:

  • LCM: Multiply the highest powers of all primes.
  • HCF: Multiply the lowest powers of common primes.

Example:

  • LCM of 4, 6, 8:
    4=22, 6=2×3, 8=234=22, 6=2×3, 8=23
    LCM = 23×3=2423×3=24.
  • HCF of 60 and 72:
    60=22×3×5, 72=23×3260=22×3×5, 72=23×32
    HCF = 22×3=1222×3=12.

Common Mistakes:

  • Missing primes in LCM.
  • Including non-common primes in HCF.

6. Word Problems

Example (LCM):
Laurence (8 mins/lap) and Naima (12 mins/lap) cycle. When do they meet?
LCM of 8 and 12 = 24 minutes.

Example (HCF):
A chef has 63 carrots and 91 parsnips. Largest equal plates?
HCF of 63 and 91 = 7.

  • Carrot plates: 63÷7=963÷7=9.
  • Parsnip plates: 91÷7=1391÷7=13.

Tip:
For LCM, think “when will they align?” For HCF, think “how to divide equally?”

7. Practice Tips

  1. Estimate first to check decimal answers.
  2. Verify prime factors by multiplying them back.
  3. Use grids for LCM/HCF questions:
    • List primes vertically.
    • Compare powers.
  4. Double-check negatives:
    • Even negatives → positive result.
    • Odd negatives → negative result.

Common Exam Phrases:

  • “Write in index form” → Use exponents (e.g., 2323).
  • “Product of prime factors” → Simplify fully.

By mastering these rules and practising mixed problems, you’ll tackle GCSE questions confidently! 🎓

50 GCSE Practice Questions

Section 1: Order of Operations (BODMAS)

  1. Work out:
    20−12÷2×3+420−12÷2×3+4
  2. Calculate:
    (5+1×3)2÷4(5+1×3)2÷4
  3. Simplify:
    2×(8+4)−7÷12×(8+4)−7÷1
  4. Evaluate:
    6+(11−8)7−5×37−56+(11−8)​×3

Section 2: Negative Numbers

  1. Solve:
    −4+3×(−2)−4+3×(−2)
  2. Work out:
    (−5)×(−8)−(−12)÷3(−5)×(−8)−(−12)÷3
  3. Simplify:
    −6−(−2)+(−5)×2−6−(−2)+(−5)×2
  4. Calculate:
    (−3)×7−21+4−21(−3)×7​+4

Section 3: Decimals

  1. Add:
    12.74+7+0.49212.74+7+0.492
  2. Subtract:
    23−18.59123−18.591
  3. Multiply:
    0.32×0.60.32×0.6
  4. Divide:
    0.516÷0.80.516÷0.8
  5. A ribbon is 2.72 m long. How many 0.08 m pieces can be cut?
  6. Petrol costs £1.35 per litre. Calculate the cost of 9.2 litres.

Section 4: Multiples and Factors

  1. List the first five multiples of 16.
  2. Write all factors of 36.
  3. Identify the common factors of 24 and 32.
  4. List the prime numbers between 20 and 50.
  5. Is 51 a prime number? Explain.

Section 5: Prime Factorisation

  1. Write 42 as a product of prime factors.
  2. Express 120 in index form.
  3. Find the prime factors of 255.
  4. What is the smallest number to multiply 75 by to make a square number?

Section 6: LCM and HCF

  1. Find the LCM of 6 and 8.
  2. Calculate the HCF of 36 and 60.
  3. Determine the LCM of 5, 7, and 10.
  4. Find the HCF of 150 and 250 using prime factors.
  5. Two buses depart every 12 and 18 minutes. When will they next coincide?

Section 7: Mixed Word Problems

  1. At midday, the temperature was 6°C. By midnight, it fell by 7°C. What was the temperature?
  2. Asha buys 2 CDs at £11.95 each and 3 at £6.59 each. She pays with £50. How much change?
  3. A school raises £412.86. Split equally among 3 charities. How much per charity?
  4. Mike visits Oscar every 4 days; Narinda every 5 days. If both visit today, when next?
  5. A chef has 63 carrots and 91 parsnips. What’s the largest equal plates he can make?

Section 8: Advanced Problems

  1. Simplify:
    −4.2–(1.5×−0.3)−4.2–(1.5×−0.3)
  2. Work out:
    8×2÷45−6+75−6+78×2÷4​
  3. Calculate:
    (3+2)×(9−4)÷5(3+2)×(9−4)÷5
  4. Solve:
    [(−24)÷8]÷3+(−5)[(−24)÷8]÷3+(−5)
  5. A block of wood is 4.2 m. If 2.75 m is cut off, what length remains?
  6. Convert 3.5 miles to km (1 mile = 1.6 km).

Section 9: Multi-Step Calculations

  1. Simplify:
    (18÷(9−12÷4))2(18÷(9−12÷4))2
  2. Calculate:
    36÷(11−2)8−8÷28−8÷236÷(11−2)​
  3. Work out:
    (0.61×0.6)+(5.2×0.09)(0.61×0.6)+(5.2×0.09)
  4. Evaluate:
    1.4×2.3−0.9461.4×2.3−0.946

Section 10: Real-World Applications

  1. A multipack of 3 S&V and 3 C&O crisps costs £3.19. Buying individually (70p and 65p), how much saved?
  2. A car travels 2.3 km to shops and 4.6 km to town. Total distance?
  3. Jay’s meal costs £66.49. With a £15.25 voucher, what’s left to pay?
  4. Pears cost £6.93 for 3.5 kg. What’s the cost per kg?
  5. A printer ink lasts 216 days; another 188 days. When will both be replaced on the same day?
  6. A wall has plates divisible by 40 and 70, between 240–300. How many plates?
  7. Jess swims every 21 days; Seamus every 35 days. When next together?

Detailed Answers

  1. Answer:
    20−12÷2×3+4=20−6×3+4=20−18+4=620−12÷2×3+4=20−6×3+4=20−18+4=6​
  2. Answer:
    (5+3)2÷4=82÷4=64÷4=16(5+3)2÷4=82÷4=64÷4=16​
  3. Answer:
    2×12−7=24−7=172×12−7=24−7=17​
  4. Answer:
    6+32×3=92×3=4.5×3=13.526+3​×3=29​×3=4.5×3=13.5​
  5. Answer:
    −4+(−6)=−10−4+(−6)=−10​
  6. Answer:
    40−(−4)=40+4=4440−(−4)=40+4=44​
  7. Answer:
    −6+2−10=−14−6+2−10=−14​
  8. Answer:
    −21−21+4=1+4=5−21−21​+4=1+4=5​
  9. Answer:
    12.74+7.00+0.492=20.23212.74+7.00+0.492=20.232​
  10. Answer:
    23.000−18.591=4.40923.000−18.591=4.409​
  11. Answer:
    0.32×0.6=0.1920.32×0.6=0.192​
  12. Answer:
    0.516÷0.8=5.16÷8=0.6450.516÷0.8=5.16÷8=0.645​
  13. Answer:
    2.72÷0.08=342.72÷0.08=34​
  14. Answer:
    1.35×9.2=£12.421.35×9.2=£12.42​
  15. Answer:
    16,32,48,64,8016,32,48,64,80
  16. Answer:
    1,2,3,4,6,9,12,18,361,2,3,4,6,9,12,18,36
  17. Answer:
    Common factors of 24 and 32: 1,2,4,81,2,4,8
  18. Answer:
    Primes between 20–50: 23,29,31,37,41,43,4723,29,31,37,41,43,47
  19. Answer:
    No. 51=3×1751=3×17 (not prime).
  20. Answer:
    42=2×3×742=2×3×7
  21. Answer:
    120=23×3×5120=23×3×5
  22. Answer:
    255=3×5×17255=3×5×17
  23. Answer:
    Multiply by 3: 75=3×5275=3×52 → 32×52=22532×52=225.
  24. Answer:
    LCM of 6 and 8 = 2424​
  25. Answer:
    HCF of 36 and 60 = 1212​
  26. Answer:
    LCM of 5, 7, 10 = 7070​
  27. Answer:
    150=2×3×52150=2×3×52; 250=2×53250=2×53. HCF = 2×52=502×52=50​
  28. Answer:
    LCM of 12 and 18 = 3636​ minutes.
  29. Answer:
    6−7=−1∘C6−7=−1∘C​
  30. Answer:
    Total: 2×11.95+3×6.59=23.90+19.77=£43.672×11.95+3×6.59=23.90+19.77=£43.67. Change: 50−43.67=£6.3350−43.67=£6.33​.
  31. Answer:
    412.86÷3=£137.62412.86÷3=£137.62​.
  32. Answer:
    LCM of 4 and 5 = 2020​ days.
  33. Answer:
    HCF of 63 and 91 = 7. Carrot plates: 9; Parsnip plates: 13.
  34. Answer:
    −4.2−(−0.45)=−4.2+0.45=−3.75−4.2−(−0.45)=−4.2+0.45=−3.75​
  35. Answer:
    46=2364​=32​​
  36. Answer:
    5×5÷5=55×5÷5=5​
  37. Answer:
    [−3]÷3+(−5)=−1−5=−6[−3]÷3+(−5)=−1−5=−6​
  38. Answer:
    4.2−2.75=1.45 m4.2−2.75=1.45 m​
  39. Answer:
    3.5×1.6=5.6 km3.5×1.6=5.6 km​
  40. Answer:
    (18÷(9−3))2=(18÷6)2=32=9(18÷(9−3))2=(18÷6)2=32=9​
  41. Answer:
    36÷98−4=44=18−436÷9​=44​=1​
  42. Answer:
    0.366+0.468=0.8340.366+0.468=0.834​
  43. Answer:
    3.22−0.946=2.2743.22−0.946=2.274​
  44. Answer:
    Individual cost: 6×70p+6×65p=£8.106×70p+6×65p=£8.10. Saving: 8.10−3.19=£4.918.10−3.19=£4.91​.
  45. Answer:
    2.3+4.6=6.9 km2.3+4.6=6.9 km​
  46. Answer:
    66.49−15.25=£51.2466.49−15.25=£51.24​
  47. Answer:
    6.93÷3.5=£1.98 per kg6.93÷3.5=£1.98 per kg​
  48. Answer:
    LCM of 216 and 188 = Prime factors:
    216=23×33216=23×33; 188=22×47188=22×47. LCM = 23×33×47=10,15223×33×47=10,152​ days.
  49. Answer:
    LCM of 40 and 70 = 280. Between 240–300: 280280​ plates.
  50. Answer:
    LCM of 21 and 35 = 105 days.