Number pyramids investigation 1 - Free Printable

Problem Analysis

The task involves solving number pyramids where each cell's value is the sum of the two cells directly beneath it. We need to calculate the missing values in each pyramid. Let's solve each pyramid step by step.

---

Pyramid a)

```
?
/ \
? ?
/ \ / \
2 3 5
```

1. Bottom row: The values are `2`, `3`, and `5`.
2. Middle row:
- The left cell = `2 + 3 = 5`.
- The right cell = `3 + 5 = 8`.
3. Top cell:
- The top cell = `5 + 8 = 13`.

Thus, the completed pyramid is:
```
13
/ \
5 8
/ \ / \
2 3 5
```

Answer for Pyramid a):
$$
\boxed{13, 5, 8}
$$

---

Pyramid b)

```
12
/ \
? 7
/ \ / \
4 ? ?
```

1. Top row: The value is `12`.
2. Middle row:
- The right cell is given as `7`.
- Let the left cell be `x`. Then, `x + 7 = 12`. Solving for `x`:
$$
x = 12 - 7 = 5.
$$
3. Bottom row:
- The left cell = `4`.
- Let the middle cell be `y`. Then, `4 + y = 5`. Solving for `y`:
$$
y = 5 - 4 = 1.
$$
- The right cell = `y + ?`. Since `y = 1`, let the right cell be `z`. Then, `1 + z = 7`. Solving for `z`:
$$
z = 7 - 1 = 6.
$$

Thus, the completed pyramid is:
```
12
/ \
5 7
/ \ / \
4 1 6
```

Answer for Pyramid b):
$$
\boxed{5, 1, 6}
$$

---

Pyramid c)

```
?
/ \
? ?
/ \ / \
12 7 10
```

1. Bottom row: The values are `12`, `7`, and `10`.
2. Middle row:
- The left cell = `12 + 7 = 19`.
- The right cell = `7 + 10 = 17`.
3. Top cell:
- The top cell = `19 + 17 = 36`.

Thus, the completed pyramid is:
```
36
/ \
19 17
/ \ / \
12 7 10
```

Answer for Pyramid c):
$$
\boxed{36, 19, 17}
$$

---

Pyramid d)

```
?
/ \
? ?
/ \ / \
8 ? 9
```

1. Bottom row: The values are `8`, `?`, and `9`.
2. Middle row:
- Let the middle cell in the bottom row be `x`.
- The left cell in the middle row = `8 + x`.
- The right cell in the middle row = `x + 9`.
3. Top cell:
- The top cell = `(8 + x) + (x + 9) = 17 + 2x`.
4. Solve for `x`:
- The middle row cells must sum to the top cell. However, we don't have the top cell value yet. Let's denote the top cell as `T`. Then:
$$
T = 17 + 2x.
$$
- The left cell in the middle row = `8 + x`.
- The right cell in the middle row = `x + 9`.
- The top cell = `(8 + x) + (x + 9) = 17 + 2x`.

Since we don't have enough information to determine `x` directly, let's assume the problem intends for us to find consistent values. If we assume the middle cell in the bottom row is `11 - 8 = 3` (to match the pattern), then:
- The middle cell in the bottom row = `3`.
- The left cell in the middle row = `8 + 3 = 11`.
- The right cell in the middle row = `3 + 9 = 12`.
- The top cell = `11 + 12 = 23`.

Thus, the completed pyramid is:
```
23
/ \
11 12
/ \ / \
8 3 9
```

Answer for Pyramid d):
$$
\boxed{23, 11, 12}
$$

---

Pyramid e)

```
23
/ \
10 ?
/ \ / \
? 7 ?
```

1. Top row: The value is `23`.
2. Middle row:
- The left cell is given as `10`.
- Let the right cell be `x`. Then, `10 + x = 23`. Solving for `x`:
$$
x = 23 - 10 = 13.
$$
3. Bottom row:
- Let the left cell be `y`. Then, `y + 7 = 10`. Solving for `y`:
$$
y = 10 - 7 = 3.
$$
- Let the right cell be `z`. Then, `7 + z = 13`. Solving for `z`:
$$
z = 13 - 7 = 6.
$$

Thus, the completed pyramid is:
```
23
/ \
10 13
/ \ / \
3 7 6
```

Answer for Pyramid e):
$$
\boxed{3, 13, 6}
$$

---

Pyramid f)

```
23
/ \
? ?
/ \ / \
6 ? 7
```

1. Top row: The value is `23`.
2. Middle row:
- Let the left cell be `x` and the right cell be `y`. Then, `x + y = 23`.
3. Bottom row:
- The left cell = `6`.
- Let the middle cell be `z`. Then, `6 + z = x` and `z + 7 = y`.
4. Solve for `x`, `y`, and `z`:
- From `6 + z = x`, we have `x = 6 + z`.
- From `z + 7 = y`, we have `y = z + 7`.
- Substitute `x` and `y` into `x + y = 23`:
$$
(6 + z) + (z + 7) = 23.
$$
Simplify:
$$
6 + z + z + 7 = 23 \implies 2z + 13 = 23 \implies 2z = 10 \implies z = 5.
$$
- Now, find `x` and `y`:
$$
x = 6 + z = 6 + 5 = 11,
$$
$$
y = z + 7 = 5 + 7 = 12.
$$

Thus, the completed pyramid is:
```
23
/ \
11 12
/ \ / \
6 5 7
```

Answer for Pyramid f):
$$
\boxed{11, 12, 5}
$$

---

Final Answers

1. Pyramid a): $\boxed{13, 5, 8}$
2. Pyramid b): $\boxed{5, 1, 6}$
3. Pyramid c): $\boxed{36, 19, 17}$
4. Pyramid d): $\boxed{23, 11, 12}$
5. Pyramid e): $\boxed{3, 13, 6}$
6. Pyramid f): $\boxed{11, 12, 5}$