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Suppose that factory a produces 12 tables.

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.

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