Answer key for educational exercises covering arithmetic, patterns, and geometric shapes.

It looks like you've uploaded a page from an answer key (page 97) that contains solutions to problems from Unit 1 and Unit 2 of a math or reasoning workbook. The content includes:

- Numerical calculations in Unit 1.
- Visual pattern recognition and shape-based logic puzzles in Unit 2.

Let me go through each section and explain the solutions, especially focusing on how they were derived — particularly for the visual parts, which may be trickier.

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🔢 UNIT 1: Arithmetic and Logical Problems

#### Question 1: Number Pairs
These are likely about identifying patterns:
- (a) 25, 30 → possibly multiples or sequences
- (b) 15, 21 → maybe increasing by 6?
- (c) 92, 86 → decreasing by 6?
- (d) 81, 243 → 81 × 3 = 243

✔ These are examples of number patterns or multiples.

#### Question 2: Area/Basic Operations
- (a) A = 12×2 = 24; B = 10×2 = 20 → likely area of rectangles
- (b) 5+7=12; 12×4=48 → perhaps perimeter or multiplication chain
- (c) 1+3=4; 4×5=20 → arithmetic steps

✔ This is basic arithmetic with two-step operations.

#### Question 3: Missing Numbers?
Answer: 19, 21
→ Likely sequence: 19, 20, 21? Or odd numbers?

#### Question 4: Negative Numbers
- −14 × 2 = −28
- −28 − 6 = −34
✔ Correct application of negative number rules.

#### Question 5: Simple Addition/Subtraction
- 20 + 6 = 26
- 26 − 7 = 19
✔ Straightforward.

#### Question 6: Multiplication & Addition
- (a) 24×5 = 120
- (b) 15, 18 → maybe next in sequence?
- (c) 7+2=9, 4+1=5 → simple sums
- (d) 48×5 = 240

✔ Basic operations.

#### Question 7: Multi-step Calculations
- (a) A = 12×6 = 72; B = 10−7 = 3
- (b) 8+3=11, then A = 11−4 = 7, etc.
- (c) A = 36÷3 = 12; B = 13×4 = 52
- (d) A = 8×6 = 48; B = 27÷9 = 3

✔ Shows understanding of order of operations.

#### Question 8: Algebraic Reasoning
- (a) A = 4+5+7 = 16
- (b) 8−1=7; B = 7×3 = 21

✔ Sum and product.

#### Question 9: Solving for Variables
- (a) 4×3=12, 8×3=12 → A×3=48 → A = 16
- (b) 4+10=14, 14+6=20 → 18+B=38 → B=20
- (c) 3×3=9, 6×6=36, 9×9=81 → C = 81
- (d) 100−97=3, 66−63=3 → 40−3=37 → D = 37

✔ Each part uses logical deduction based on given equations.

#### Question 10: Multiples
- 4th row: multiples of 4 → A = 4×2 = 8
- 5th row: multiples of 5 → B = 5×3 = 15
- 6th row: multiples of 6 → C = 6×3 = 18, D = 6×6 = 36

✔ Clearly identifies patterns in rows based on multiplication tables.

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🖼️ UNIT 2: Visual Pattern Recognition

This unit involves interpreting diagrams and shapes.

#### Question 1: Shapes and Patterns

(a) Two vertical columns of stars:
- Left: 4 stars
- Right: 5 stars
→ Possibly asking for count or comparison.

(b) Single star
→ Maybe a symbol or representation.

(c) Smiley face and heart
→ Could represent emotions or categories.

(d) Grid (4×2) with shaded squares:
```
□ □
□ □
■ □
□ ■
```
- Shaded cells: bottom-left and middle-right
→ Might be showing symmetry or specific positions.

(e) 2×4 grid, shaded rightmost column:
```
| | |■■|
| | |■■|
```
→ Entire right half shaded → possibly "half" or "column-wise shading"

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#### Question 2: Shape Classification and Overlapping

(a) 2×2 grid with different shapes:
- Top-left: circle
- Top-right: rectangle
- Bottom-left: triangle
- Bottom-right: heart

→ Likely asks to identify common properties or classify shapes.

(b) Square divided into 4 triangles by diagonals, with:
- One small square shaded in top-left
- One small triangle shaded at bottom-center

→ Possible question: Which region is shaded? Or what fraction?

Let’s suppose the original problem was asking:
> “Which shape is shaded?” or “What fraction of the square is shaded?”

But since it's an answer key, we can infer the correct answers are already shown.

For example:
- In (b), if the square is split into 4 equal triangles, but there’s a smaller square and triangle shaded, the shaded areas might not be equal — so the answer could involve area comparison.

However, without the actual questions, we interpret this as visual logic tasks, possibly involving:
- Counting
- Symmetry
- Fractional parts
- Shape classification

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✔ Summary of Key Concepts Covered

| Topic | Example |
|------|--------|
| Arithmetic | Multiplication, addition, subtraction |
| Negative Numbers | −14 × 2 = −28 |
| Order of Operations | Step-by-step calculations |
| Patterns | Sequences, multiples of numbers |
| Visual Logic | Shape recognition, grids, shading |
| Fractions & Areas | Shaded regions in grids |

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💡 Final Notes

Since this is an answer key, it confirms correct responses. However, to fully solve the original task, you’d need the actual questions (which aren't visible here). But based on the answers:

- Unit 1 focuses on basic arithmetic and number patterns.
- Unit 2 emphasizes visual-spatial reasoning and shape analysis.

If you can provide the original questions or clarify which specific problem you'd like explained (e.g., “How do I solve Unit 2, Question 1(e)?”), I can give a more targeted explanation.

Would you like me to reconstruct the possible original questions based on these answers?