Solving Earning Problems: Men and Women Earnings

This problem involves calculating the individual daily earnings of women and men based on given information about their combined earnings over a certain number of days. We can set up a system of linear equations to find the daily earning rate for each person.

Let's define our variables:

• Let \(w\) be the daily earning of one woman (in Rs.).
• Let \(m\) be the daily earning of one man (in Rs.).

From the first statement, "4 women and 7 men earn a total of Rs. 11,480 in 7 days", we can find their total daily earning:

Total daily earning of 4 women and 7 men =  \(\frac{11480}{7}\) Rs.  =  1640 Rs.

This gives us our first equation:

\(4w + 7m = 1640\)   (Equation 1)

From the second statement, "10 women and 17 men earn a total of Rs. 36,360 in 9 days", we can find their total daily earning:

Total daily earning of 10 women and 17 men  =  \(\frac{36360}{9}\) Rs.  =  4040 Rs.

This gives us our second equation:

\(10w + 17m = 4040\)   (Equation 2)

Solving the System of Equations to Find Daily Earning

Now we have a system of two linear equations with two variables:

\(4w + 7m = 1640\)
\(10w + 17m = 4040\)

We can solve this system using the elimination method. Multiply Equation 1 by 10 and Equation 2 by 4 to make the coefficients of \(w\) equal:

• Multiply Equation 1 by 10: \(10 \times (4w + 7m) = 10 \times 1640\)
• This gives: \(40w + 70m = 16400\)   (Equation 3)
• Multiply Equation 2 by 4: \(4 \times (10w + 17m) = 4 \times 4040\)
• This gives: \(40w + 68m = 16160\)   (Equation 4)

Subtract Equation 4 from Equation 3:

\((40w + 70m) - (40w + 68m) = 16400 - 16160\)

\(40w - 40w + 70m - 68m = 240\)

\(2m = 240\)

Divide by 2 to find the value of \(m\):

\(m = \frac{240}{2}\)

\(m = 120\)

So, the daily earning of one man is Rs. 120.

Now substitute the value of \(m\) (120) back into Equation 1 to find the value of \(w\):

\(4w + 7m = 1640\)

\(4w + 7(120) = 1640\)

\(4w + 840 = 1640\)

Subtract 840 from both sides:

\(4w = 1640 - 840\)

\(4w = 800\)

Divide by 4 to find the value of \(w\):

\(w = \frac{800}{4}\)

\(w = 200\)

So, the daily earning of one woman is Rs. 200.

Calculating Total Earning for 11 Women and 9 Men in 13 Days

We need to find out how much 11 women and 9 men will earn together in 13 days. First, let's calculate their combined daily earning:

Combined daily earning  =  \(11w + 9m\)

Substitute the values of \(w = 200\) and \(m = 120\):

Combined daily earning  =  \(11(200) + 9(120)\)

Combined daily earning  =  \(2200 + 1080\)

Combined daily earning  =  3280 Rs. per day

Now, calculate the total earning over 13 days:

Total earning in 13 days  =  Combined daily earning \(\times\) Number of days

Total earning in 13 days  =  \(3280 \times 13\)

Let's perform the multiplication:

\(3280 \times 13 = 3280 \times (10 + 3)\)
\(             = 3280 \times 10 + 3280 \times 3\)
\(             = 32800 + 9840\)
\(             = 42640\)

So, 11 women and 9 men together will earn Rs. 42,640 in 13 days.

Revision Table: Earning Calculation Steps

Additional Information on System of Linear Equations

A system of linear equations is a set of two or more linear equations that involve the same variables. In this problem, we used two equations with variables \(w\) and \(m\). Solving such a system means finding the values of the variables that satisfy all equations simultaneously.

Common methods to solve a system of linear equations include:

• Substitution Method: Solve one equation for one variable, then substitute that expression into the other equation.
• Elimination Method: Multiply one or both equations by constants so that the coefficients of one variable are opposites. Then, add the equations to eliminate that variable.
• Graphical Method: Graph both equations. The point of intersection of the lines represents the solution. (Less practical for non-integer solutions or complex numbers).
• Matrix Method: Use matrices (like Cramer's rule or Gaussian elimination) to solve the system, particularly useful for larger systems.

In this solution, we used the elimination method, which is efficient when you can easily make the coefficients of one variable match or become opposites by multiplication.

Understanding how to set up and solve systems of equations is crucial for many types of word problems in mathematics and real-world applications involving multiple unknown quantities.