Floor Question System of Equations

We are tasked with finding all possible values of $b$ that satisfy the following system of equations involving the floor functions:

 

$$\lfloor a \rfloor \cdot b \cdot c = 3 \tag{1}$$

$$a \cdot \lfloor b \rfloor \cdot c = 4 \tag{2}$$

$$a \cdot b \cdot \lfloor c \rfloor = 5 \tag{3}$$

 

where $a$, $b$, and $c$ are positive real numbers. We will solve this step by step by exploring possible values for $a$, $b$, and $c$.

 

### Solution By Steps

 

#### Step 1: Analyze the first equation

From Equation (1):

 

$$\lfloor a \rfloor \cdot b \cdot c = 3$$

 

The floor function $\lfloor a \rfloor$ represents the greatest integer less than or equal to $a$. Since $a$ is positive, the possible values of $\lfloor a \rfloor$ are integers.

 

We first assume $\lfloor a \rfloor = 1$, which is the smallest possible value because $a > 0$. Substituting this into Equation (1):

 

$$1 \cdot b \cdot c = 3$$

 

Thus, we have:

 

$$b \cdot c = 3 \tag{4}$$

 

#### Step 2: Analyze the second equation

Now, consider Equation (2):

 

$$a \cdot \lfloor b \rfloor \cdot c = 4$$

 

We already know $b \cdot c = 3$ from Equation (4). To satisfy this equation, we explore possible values of $\lfloor b \rfloor$.

 

Let’s assume $\lfloor b \rfloor = 1$ first. Substituting into Equation (2):

 

$$a \cdot 1 \cdot c = 4$$

 

So,

 

$$a \cdot c = 4 \tag{5}$$

 

#### Step 3: Analyze the third equation

Now consider Equation (3):

 

$$a \cdot b \cdot \lfloor c \rfloor = 5$$

 

We already have $b \cdot c = 3$ and $a \cdot c = 4$. To satisfy this equation, we explore possible values of $\lfloor c \rfloor$.

 

Let’s assume $\lfloor c \rfloor = 1$. Substituting into Equation (3):

$$a \cdot b \cdot 1 = 5$$

 

So:

 

$$a \cdot b = 5 \tag{6}$$

 

#### Step 4: Solve the system of equations

Now, we have three equations:

 

1. $b \cdot c = 3$

2. $a \cdot c = 4$

3. $a \cdot b = 5$

 

We can solve this system step by step. First, solve for $c$ from Equation (5):

 

$$c = \frac{4}{a}$$

 

Substitute this into Equation (4):

 

$$b \cdot \frac{4}{a} = 3$$

 

Simplifying:

 

$$b = \frac{3a}{4}$$

 

Now substitute this into Equation (6):

 

$$a \cdot \frac{3a}{4} = 5$$

 

Simplifying:

 

$$\frac{3a^2}{4} = 5$$

 

Multiply both sides by 4:

 

$$3a^2 = 20$$

 

Solve for $a^2$:

 

$$a^2 = \frac{20}{3}$$

 

So:

 

$$a = \sqrt{\frac{20}{3}} = \frac{2\sqrt{15}}{3}$$

 

#### Step 5: Find the value of $b$

Now that we have $a = \frac{2\sqrt{15}}{3}$, substitute this back into the expression for $b$:

 

$$b = \frac{3a}{4} = \frac{3 \times \frac{2\sqrt{15}}{3}}{4} = \frac{2\sqrt{15}}{4} = \frac{\sqrt{15}}{2}$$

 

Thus, the value of $b$ is:

$$b = \frac{\sqrt{15}}{2}$$