This question aims to find the number of hours in which each factory produces **48 tables** and **24 chairs**.

Suppose there are two factories and we label these factories as **factory A** and **factory B**. Factory A produces **12 tables** and **6 chairs** in one hour while the other factory that is factory B produces **8 tables** and **4 chairs** in one hour.

Now we have to calculate the number of hours in which a factory produces a certain number of tables and chairs.

## Expert Answer

If we assume that factory A works for **x hours** and factory B works for **y hours** then according to the equation:

**Factory A = x hours **

**Factory B = y hours**

The equations are as follows:

\[ 12 x + 8 y = 48 ……. Eq1 \]

\[ 6 x + 4 y = 24 …….. Eq2 \]

By dividing equation 2 by equation 1, we get:

\[ \frac { 12 } { 6 } + \frac { 8 } { 4 } = \frac { 48 } { 24 } \]

\[ \frac { 2 } { 1 } + \frac { 2 } { 1 } = \frac { 2 } { 1 } \]

These equations are the same. It means these equations will have finite solutions. **Finite solutions** mean the type of solutions in which the elements of the solution are **finite** and **countable**.

\[ 6 x + 4 y = 2 \]

\[ 3 x + 2 y = 12 \]

\[ x \geq 0 \]

\[ y \geq 0 \]

## Numerical Solution

There are **three** types of solutions that are possible for this question. These are:

For **x-terms:**

\[ x = 0 \]

\[ x = 2 \]

\[ x = 4 \]

For **y-terms:**

\[ y = 6 \]

\[ y = 3 \]

\[ y = 0 \]

## Example

If we take the same question and take the **ratio** of the **tables** produced by **factory** **A** and the tables produced by **factory** **B,** we can find the **number** of **hours.**

If **factory A** produces **12 tables** and we want to calculate the **number** of **hours** in which **48 tables** are produced by the same **factory.** Then, we will take the **ratio** of both **tables:**

\[ \frac { 48 } { 12 } = 4 \]

\[ \frac { 24 } { 6 } = 4 \]

If **factory A** produces **8 tables** and we want to calculate the **number** of **hours** in which **48 tables** are produced by the same **factory.** Then, we will take the **ratio** of both **tables:**

\[ \frac { 48 } { 8 } = 6 \]

\[ \frac { 24 } { 4 } = 6 \]

Factory A must work for **4 hours** to produce **48 tables and 24 chairs.**

Factory B must work for **6 hours** to produce **48 tables and 24 chairs.**

*Image/Mathematical drawings are created in Geogebra.*