Question

The numbers are the sum of each row and column.

(e.g., flower + tree + tree + flower = 28.) Find the value of each symbol:

Flower:
Tree:
Sunflower:
Leaf:
?
COLUMN:
Flower + tree + tree + flower = 28
Sunflower + tree + sunflower + tree = 30
Leaf + flower + leaf + leaf = 18
Leaf + tree + leaf + leaf = 20.

ROW:
Flower + sunflower + leaf + leaf = ?
tree + flower + tree + tree = 30
Leaf + leaf + sunflower + tree = 23
Leaf + leaf + tree + flower = 22

Answer 1

Let's denote:
- Flower = F
- Tree = T
- Sunflower = S
- Leaf = L

We have the following equations based on the column and row sums provided:

**From the Columns:**
1. \( 2F + 2T = 28 \) (From the first column)
2. \( 2S + 2T = 30 \) (From the second column)
3. \( 3L + F = 18 \) (From the third column)
4. \( 3L + T = 20 \) (From the fourth column)

**From the Rows:**
5. \( F + S + 2L = ? \) (From the first row)
6. \( T + 2F + 2T = 30 \) (From the second row)
7. \( 2L + S + T = 23 \) (From the third row)
8. \( 2L + F + T = 22 \) (From the fourth row)

**Now let's solve these equations:**

From the first column equation:
\[ 2F + 2T = 28 \implies F + T = 14 \quad \text{(dividing by 2)} \quad \text{(Equation 1)} \]

From the second column equation:
\[ 2S + 2T = 30 \implies S + T = 15 \quad \text{(dividing by 2)} \quad \text{(Equation 2)} \]

From the third column equation:
\[ 3L + F = 18 \quad \text{(Equation 3)} \]

From the fourth column equation:
\[ 3L + T = 20 \quad \text{(Equation 4)} \]

**Now let's express F and T from Equations 1 and 2:**
From Equation 1:
\[ F = 14 - T \quad \text{(Substituting in Equation 3)} \]

Substituting \( F \) in Equation 3:
\[ 3L + (14 - T) = 18 \]
\[ 3L - T = 4 \quad \text{(Equation 5)} \]

Now substituting \( T \) in Equation 4 as well:
From Equation 2:
\[ T = 15 - S \quad \text{(Substituting in Equation 4)} \]

Substituting \( T \) in Equation 4:
\[ 3L + (15 - S) = 20 \]
\[ 3L - S = 5 \quad \text{(Equation 6)} \]

Now we have:
Equation 5:
\[ 3L - T = 4 \]
Equation 6:
\[ 3L - S = 5 \]

**Now let's subtract Equation 5 from Equation 6:**
\[ (3L - S) - (3L - T) = 5 - 4 \]
\[ T - S = 1 \]
\[ T = S + 1 \quad \text{(Equation 7)} \]

**Now substitute Equation 7 back into Equation 2:**
\[ S + (S + 1) = 15 \]
\[ 2S + 1 = 15 \]
\[ 2S = 14 \]
\[ S = 7 \]

Using \( S \) in Equation 7:
\[ T = 7 + 1 = 8 \]

Using \( T \) in Equation 1:
\[ F + 8 = 14 \]
\[ F = 6 \]

Now we can find \( L \) using either Equation 3 or 4. Using Equation 3:
\[ 3L + 6 = 18 \]
\[ 3L = 12 \]
\[ L = 4 \]

Now, we have:
- Flower (F) = 6
- Tree (T) = 8
- Sunflower (S) = 7
- Leaf (L) = 4

**Finally, find the value of the first row:**
\[ F + S + 2L = 6 + 7 + 2(4) = 6 + 7 + 8 = 21 \]

So the answers are:
- Flower: \( 6 \)
- Tree: \( 8 \)
- Sunflower: \( 7 \)
- Leaf: \( 4 \)
- Row sum: \( 21 \)