MATHEMATICS - Free Printable

Problem Overview:

The task involves solving mathematical expressions using the BODMAS rule (Brackets, Orders, Division/Multiplication, Addition, Subtraction). The goal is to evaluate each expression step-by-step without a calculator.

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Step-by-Step Solutions:

#### Part 1: Solving the Sequences

1. 100 − (20 × 3)
- Step 1: Solve inside the brackets: \( 20 \times 3 = 60 \)
- Step 2: Subtract: \( 100 - 60 = 40 \)
- Answer: \( \boxed{40} \)

2. (35 − 15) + (27 − 7)
- Step 1: Solve inside the first bracket: \( 35 - 15 = 20 \)
- Step 2: Solve inside the second bracket: \( 27 - 7 = 20 \)
- Step 3: Add: \( 20 + 20 = 40 \)
- Answer: \( \boxed{40} \)

3. 15 + (6 × 6)
- Step 1: Solve inside the brackets: \( 6 \times 6 = 36 \)
- Step 2: Add: \( 15 + 36 = 51 \)
- Answer: \( \boxed{51} \)

4. (4 + 5) × (3 + 6)
- Step 1: Solve inside the first bracket: \( 4 + 5 = 9 \)
- Step 2: Solve inside the second bracket: \( 3 + 6 = 9 \)
- Step 3: Multiply: \( 9 \times 9 = 81 \)
- Answer: \( \boxed{81} \)

5. (5 + 5) × (5 − 2)
- Step 1: Solve inside the first bracket: \( 5 + 5 = 10 \)
- Step 2: Solve inside the second bracket: \( 5 - 2 = 3 \)
- Step 3: Multiply: \( 10 \times 3 = 30 \)
- Answer: \( \boxed{30} \)

6. 50 − (6 × 6)
- Step 1: Solve inside the brackets: \( 6 \times 6 = 36 \)
- Step 2: Subtract: \( 50 - 36 = 14 \)
- Answer: \( \boxed{14} \)

7. (4 + 8) × (3 − 2)
- Step 1: Solve inside the first bracket: \( 4 + 8 = 12 \)
- Step 2: Solve inside the second bracket: \( 3 - 2 = 1 \)
- Step 3: Multiply: \( 12 \times 1 = 12 \)
- Answer: \( \boxed{12} \)

8. (9 − 3) + (6 × 6)
- Step 1: Solve inside the first bracket: \( 9 - 3 = 6 \)
- Step 2: Solve inside the second bracket: \( 6 \times 6 = 36 \)
- Step 3: Add: \( 6 + 36 = 42 \)
- Answer: \( \boxed{42} \)

9. (5 × 7) − (2 × 5)
- Step 1: Solve inside the first bracket: \( 5 \times 7 = 35 \)
- Step 2: Solve inside the second bracket: \( 2 \times 5 = 10 \)
- Step 3: Subtract: \( 35 - 10 = 25 \)
- Answer: \( \boxed{25} \)

10. 56 − (4 × 7)
- Step 1: Solve inside the brackets: \( 4 \times 7 = 28 \)
- Step 2: Subtract: \( 56 - 28 = 28 \)
- Answer: \( \boxed{28} \)

11. 78 − (10 × 7)
- Step 1: Solve inside the brackets: \( 10 \times 7 = 70 \)
- Step 2: Subtract: \( 78 - 70 = 8 \)
- Answer: \( \boxed{8} \)

12. (7 × 7) + (4 × 8)
- Step 1: Solve inside the first bracket: \( 7 \times 7 = 49 \)
- Step 2: Solve inside the second bracket: \( 4 \times 8 = 32 \)
- Step 3: Add: \( 49 + 32 = 81 \)
- Answer: \( \boxed{81} \)

13. (45 − 23) + (5 × 8)
- Step 1: Solve inside the first bracket: \( 45 - 23 = 22 \)
- Step 2: Solve inside the second bracket: \( 5 \times 8 = 40 \)
- Step 3: Add: \( 22 + 40 = 62 \)
- Answer: \( \boxed{62} \)

14. 38 − (5 × 7)
- Step 1: Solve inside the brackets: \( 5 \times 7 = 35 \)
- Step 2: Subtract: \( 38 - 35 = 3 \)
- Answer: \( \boxed{3} \)

15. (100 − 45) + (7 × 7)
- Step 1: Solve inside the first bracket: \( 100 - 45 = 55 \)
- Step 2: Solve inside the second bracket: \( 7 \times 7 = 49 \)
- Step 3: Add: \( 55 + 49 = 104 \)
- Answer: \( \boxed{104} \)

16. 45 − (9 × 4)
- Step 1: Solve inside the brackets: \( 9 \times 4 = 36 \)
- Step 2: Subtract: \( 45 - 36 = 9 \)
- Answer: \( \boxed{9} \)

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#### Part 2: Finding Different Sums and Answers with Brackets

1. 4 + 4 × 5 − 3
- Without brackets (standard BODMAS):
- Step 1: Multiply: \( 4 \times 5 = 20 \)
- Step 2: Add: \( 4 + 20 = 24 \)
- Step 3: Subtract: \( 24 - 3 = 21 \)
- Answer: \( \boxed{21} \)
- With different bracket placements:
- Case 1: \( (4 + 4) \times (5 - 3) \)
- Step 1: Solve inside the first bracket: \( 4 + 4 = 8 \)
- Step 2: Solve inside the second bracket: \( 5 - 3 = 2 \)
- Step 3: Multiply: \( 8 \times 2 = 16 \)
- Answer: \( \boxed{16} \)
- Case 2: \( (4 + 4 \times 5) - 3 \)
- Step 1: Solve inside the brackets (multiplication first): \( 4 \times 5 = 20 \)
- Step 2: Add: \( 4 + 20 = 24 \)
- Step 3: Subtract: \( 24 - 3 = 21 \)
- Answer: \( \boxed{21} \)
- Case 3: \( 4 + (4 \times 5) - 3 \)
- Step 1: Solve inside the brackets: \( 4 \times 5 = 20 \)
- Step 2: Add: \( 4 + 20 = 24 \)
- Step 3: Subtract: \( 24 - 3 = 21 \)
- Answer: \( \boxed{21} \)

2. 8 + 5 × 1 + 3 − 6
- Without brackets (standard BODMAS):
- Step 1: Multiply: \( 5 \times 1 = 5 \)
- Step 2: Add: \( 8 + 5 = 13 \)
- Step 3: Add: \( 13 + 3 = 16 \)
- Step 4: Subtract: \( 16 - 6 = 10 \)
- Answer: \( \boxed{10} \)
- With different bracket placements:
- Case 1: \( (8 + 5) \times (1 + 3) - 6 \)
- Step 1: Solve inside the first bracket: \( 8 + 5 = 13 \)
- Step 2: Solve inside the second bracket: \( 1 + 3 = 4 \)
- Step 3: Multiply: \( 13 \times 4 = 52 \)
- Step 4: Subtract: \( 52 - 6 = 46 \)
- Answer: \( \boxed{46} \)
- Case 2: \( 8 + (5 \times 1) + (3 - 6) \)
- Step 1: Solve inside the first bracket: \( 5 \times 1 = 5 \)
- Step 2: Solve inside the second bracket: \( 3 - 6 = -3 \)
- Step 3: Add: \( 8 + 5 = 13 \)
- Step 4: Add: \( 13 + (-3) = 10 \)
- Answer: \( \boxed{10} \)
- Case 3: \( (8 + 5 \times 1 + 3) - 6 \)
- Step 1: Solve inside the brackets (multiplication first): \( 5 \times 1 = 5 \)
- Step 2: Add: \( 8 + 5 = 13 \)
- Step 3: Add: \( 13 + 3 = 16 \)
- Step 4: Subtract: \( 16 - 6 = 10 \)
- Answer: \( \boxed{10} \)

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Final Answers:

1. \( \boxed{40} \)
2. \( \boxed{40} \)
3. \( \boxed{51} \)
4. \( \boxed{81} \)
5. \( \boxed{30} \)
6. \( \boxed{14} \)
7. \( \boxed{12} \)
8. \( \boxed{42} \)
9. \( \boxed{25} \)
10. \( \boxed{28} \)
11. \( \boxed{8} \)
12. \( \boxed{81} \)
13. \( \boxed{62} \)
14. \( \boxed{3} \)
15. \( \boxed{104} \)
16. \( \boxed{9} \)

For the trickier problems:
1. \( 4 + 4 \times 5 - 3 \):
- Without brackets: \( \boxed{21} \)
- With brackets: \( \boxed{16}, \boxed{21} \)
2. \( 8 + 5 \times 1 + 3 - 6 \):
- Without brackets: \( \boxed{10} \)
- With brackets: \( \boxed{46}, \boxed{10} \)

Final Boxed Answer for Trickier Problems:
- For \( 4 + 4 \times 5 - 3 \): \( \boxed{21, 16} \)
- For \( 8 + 5 \times 1 + 3 - 6 \): \( \boxed{10, 46} \)