5 women and 9 girls earn a total of ₹18,720 in 9 days, while 9 women and 16 girls earn a total of ₹ 52,080 in 14 days. How much will 12 women and 7 girls together earn (in ₹) in 13 days?
42120
Understanding the Problem: Calculating Earnings
This question asks us to determine the total earnings of a specific group of women and girls over a certain number of days, given information about the earnings of two other groups over different periods. To solve this, we first need to find the individual daily earning rate of a woman and a girl.
Step-by-Step Solution to Finding Daily Earnings
Let's denote the daily earning of one woman as \(w\) and the daily earning of one girl as \(g\). We are given two scenarios:
- 5 women and 9 girls earn a total of ₹18,720 in 9 days.
- 9 women and 16 girls earn a total of ₹52,080 in 14 days.
From these scenarios, we can set up equations based on the total daily earning multiplied by the number of days.
Setting up Equations for Daily Earning
In the first scenario, the total daily earning of 5 women and 9 girls is \(5w + 9g\). Over 9 days, their total earning is ₹18,720. So, we have:
\((5w + 9g) \times 9 = 18720\)
Dividing both sides by 9 to find the total daily earning:
\(5w + 9g = \frac{18720}{9}\)
\(5w + 9g = 2080\) (Equation 1)
In the second scenario, the total daily earning of 9 women and 16 girls is \(9w + 16g\). Over 14 days, their total earning is ₹52,080. So, we have:
\((9w + 16g) \times 14 = 52080\)
Dividing both sides by 14 to find the total daily earning:
\(9w + 16g = \frac{52080}{14}\)
\(9w + 16g = 3720\) (Equation 2)
Solving the System of Linear Equations
We now have a system of two linear equations with two variables:
- Equation 1: \(5w + 9g = 2080\)
- Equation 2: \(9w + 16g = 3720\)
We can solve this system using the elimination method. Let's multiply Equation 1 by 9 and Equation 2 by 5 to make the coefficients of \(w\) equal:
Multiply Equation 1 by 9:
\(9 \times (5w + 9g) = 9 \times 2080\)
\(45w + 81g = 18720\) (Equation 3)
Multiply Equation 2 by 5:
\(5 \times (9w + 16g) = 5 \times 3720\)
\(45w + 80g = 18600\) (Equation 4)
Now, subtract Equation 4 from Equation 3:
\((45w + 81g) - (45w + 80g) = 18720 - 18600\)
\(45w - 45w + 81g - 80g = 120\)
\(g = 120\)
So, the daily earning of one girl is ₹120.
Now substitute the value of \(g\) into Equation 1 to find the value of \(w\):
\(5w + 9(120) = 2080\)
\(5w + 1080 = 2080\)
\(5w = 2080 - 1080\)
\(5w = 1000\)
\(w = \frac{1000}{5}\)
\(w = 200\)
So, the daily earning of one woman is ₹200.
Calculating Earnings for 12 Women and 7 Girls in 13 Days
We need to find the total earning of 12 women and 7 girls in 13 days. First, calculate their total daily earning:
Total daily earning = \(12w + 7g\)
Substitute the values \(w = 200\) and \(g = 120\):
Total daily earning = \(12(200) + 7(120)\)
Total daily earning = \(2400 + 840\)
Total daily earning = ₹3240
Now, calculate the total earning over 13 days:
Total earning in 13 days = (Total daily earning) \(\times\) (Number of days)
Total earning in 13 days = \(3240 \times 13\)
\[ 3240 \times 13 = 3240 \times (10 + 3) = 32400 + 3240 \times 3 = 32400 + 9720 = 42120 \]
The total earning of 12 women and 7 girls in 13 days is ₹42,120.
| Person Type | Daily Earning (₹) |
|---|---|
| Woman | 200 |
| Girl | 120 |
Final Answer
12 women and 7 girls together will earn ₹42,120 in 13 days.
Revision Table: Earning Calculation Summary
| Item | Value/Formula |
|---|---|
| Woman's Daily Earning (\(w\)) | ₹200 |
| Girl's Daily Earning (\(g\)) | ₹120 |
| Total Daily Earning (12 Women, 7 Girls) | \(12w + 7g = 12(200) + 7(120) = 2400 + 840 = 3240\) |
| Total Earning (12 Women, 7 Girls in 13 days) | \(3240 \times 13 = 42120\) |
Additional Information: Work and Wages Problems
Problems involving work and wages often require you to determine the individual rates at which workers complete a task or earn money. Key concepts include:
- Individual Rate: The amount of work done or money earned by one person in one unit of time (usually a day).
- Combined Rate: The sum of individual rates when multiple people work together.
- Total Work/Earning: The combined rate multiplied by the time duration.
If different types of workers have different rates, setting up a system of linear equations, as we did here, is a common method to find the individual rates. Once the individual rates are known, you can calculate the combined rate for any group and then find the total work done or money earned for any given time period.
This type of problem is an application of simultaneous linear equations in a real-world context.
Important Questions from Work and Wages
A and B worked together and received a total of Rs. 18,000 for 15 days. A's efficiency in the work was 5 times that of B's. The daily wage of A (in Rs.) was:
Samir and Puneet can complete the same work in 10 days and 15 days respectively. The work was assigned for Rs.4500. After working together for 3 days Samir and Puneet involved Ashok. The work was completed in total 5 days. What amount (in Rs.) was paid to Ashok?
X, Y and Z completed a work costing Rs. 3,400, X worked for 5 days, Y for 7 days and Z for 10 days. If their daily wages are in the ratio of 4 ∶ 5 ∶ 3, how much amount will be received by X?
A,B and C can do a work in 10 days, 15 days, and 20 days, respectively. They finished that work together and got Rs. 2,600 as wages. Find C's wage.
4 women and 7 men earn a total of Rs. 11,480 in 7 days, while 10 women and 17 men earn a total of Rs. 36,360 in 9 days. How much will 11 women and 9 men together earn (in Rs.) in 13 days?
