Of course! I’ll solve each of the 12 function machine problems by finding the mathematical relationship (equation) between the input and output values.
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Problem 1
Input (e): 10, 2, 7, 9, 4
Output (h): 40, 8, 28, 36, 16
Look at the pattern:
- 10 → 40 → 10 × 4 = 40
- 2 → 8 → 2 × 4 = 8
- 7 → 28 → 7 × 4 = 28
✔ All outputs are
4 times the input.
Equation: `h = 4e`
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Problem 2
Input (y): 21, 18, 19, 106, 96
Output (z): 4, 1, 2, 89, 79
Check differences:
- 21 – 4 = 17
- 18 – 1 = 17
- 19 – 2 = 17
- 106 – 89 = 17
- 96 – 79 = 17
✔ Output = Input – 17
Equation: `z = y - 17`
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Problem 3
Input (m): 20, 37, 10, 67, 92
Output (p): 14, 31, 4, 61, 86
Check differences:
- 20 – 14 = 6
- 37 – 31 = 6
- 10 – 4 = 6
- 67 – 61 = 6
- 92 – 86 = 6
✔ Output = Input – 6
Equation: `p = m - 6`
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Problem 4
Input (m): 4, 1, 10, 60, 37
Output (n): 24, 21, 30, 80, 57
Check differences:
- 24 – 4 = 20
- 21 – 1 = 20
- 30 – 10 = 20
- 80 – 60 = 20
- 57 – 37 = 20
✔ Output = Input + 20
Equation: `n = m + 20`
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Problem 5
Input (y): 18, 6, 27, 21, 15
Output (z): 6, 2, 9, 7, 5
Check division:
- 18 ÷ 3 = 6
- 6 ÷ 3 = 2
- 27 ÷ 3 = 9
- 21 ÷ 3 = 7
- 15 ÷ 3 = 5
✔ Output = Input ÷ 3
Equation: `z = y / 3` or `z = y ÷ 3`
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Problem 6
Input (e): 7, 9, 6, 2, 3
Output (f): 42, 54, 36, 12, 18
Check multiplication:
- 7 × 6 = 42
- 9 × 6 = 54
- 6 × 6 = 36
- 2 × 6 = 12
- 3 × 6 = 18
✔ Output = Input × 6
Equation: `f = 6e`
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Problem 7
Input (s): 90, 80, 100, 30
Output (t): 9, 8, 10, 3
Check division:
- 90 ÷ 10 = 9
- 80 ÷ 10 = 8
- 100 ÷ 10 = 10
- 30 ÷ 10 = 3
✔ Output = Input ÷ 10
Equation: `t = s / 10`
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Problem 8
Input (n): 63, 45, 90, 36
Output (b): 7, 5, 10, 4
Check division:
- 63 ÷ 9 = 7
- 45 ÷ 9 = 5
- 90 ÷ 9 = 10
- 36 ÷ 9 = 4
✔ Output = Input ÷ 9
Equation: `b = n / 9`
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Problem 9
Input (k): 49, 32, 58, 44
Output (l): 37, 20, 46, 32
Check difference:
- 49 – 37 = 12
- 32 – 20 = 12
- 58 – 46 = 12
- 44 – 32 = 12
✔ Output = Input – 12
Equation: `l = k - 12`
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Problem 10
Input (l): 10, 7, 8, 5
Output (m): 20, 14, 16, 10
Check multiplication:
- 10 × 2 = 20
- 7 × 2 = 14
- 8 × 2 = 16
- 5 × 2 = 10
✔ Output = Input × 2
Equation: `m = 2l`
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Problem 11
Input (e): 1, 2, 32, 10
Output (d): 16, 17, 47, 25
Check pattern:
- 1 → 16 → 1 + 15 = 16
- 2 → 17 → 2 + 15 = 17
- 32 → 47 → 32 + 15 = 47
- 10 → 25 → 10 + 15 = 25
✔ Output = Input + 15
Equation: `d = e + 15`
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Problem 12
Input (y): 15, 20, 35, 50
Output (z): 3, 4, 7, 10
Check division:
- 15 ÷ 5 = 3
- 20 ÷ 5 = 4
- 35 ÷ 5 = 7
- 50 ÷ 5 = 10
✔ Output = Input ÷ 5
Equation: `z = y / 5`
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✔ Final Answers:
1. `h = 4e`
2. `z = y - 17`
3. `p = m - 6`
4. `n = m + 20`
5. `z = y / 3`
6. `f = 6e`
7. `t = s / 10`
8. `b = n / 9`
9. `l = k - 12`
10. `m = 2l`
11. `d = e + 15`
12. `z = y / 5`
You can now fill these into your "Answers" column on the worksheet!
Let me know if you’d like a visual graph or explanation for any one of them!
Parent Tip: Review the logic above to help your child master the concept of patterns and functions worksheet.
