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?
Solving Work and Wages Ratio Problems
This problem involves calculating the share of an individual's earnings based on their contribution to a total task, considering both the number of days worked and their daily wage rate, which is given as a ratio.
Understanding the Given Information
- Total cost of the work: Rs. 3400
- Number of days X worked: 5 days
- Number of days Y worked: 7 days
- Number of days Z worked: 10 days
- Ratio of their daily wages (X : Y : Z): 4 : 5 : 3
Approach to Calculate Individual Earning
The total amount received by each person is the product of their daily wage and the number of days they worked. Since the daily wages are in a ratio, we can represent them using a common multiplier.
Let the daily wages of X, Y, and Z be $4k$, $5k$, and $3k$ respectively, where $k$ is a constant.
Step 1: Calculate the total earning units for each person
- X's total earning units = (Daily wage of X) $\times$ (Number of days X worked)
- X's total earning units = $4k \times 5 = 20k$
- Y's total earning units = (Daily wage of Y) $\times$ (Number of days Y worked)
- Y's total earning units = $5k \times 7 = 35k$
- Z's total earning units = (Daily wage of Z) $\times$ (Number of days Z worked)
- Z's total earning units = $3k \times 10 = 30k$
Step 2: Calculate the total earning units for the complete work
The total earning units for the work is the sum of the individual earning units:
Total earning units = X's units + Y's units + Z's units
Total earning units = $20k + 35k + 30k = 85k$
Step 3: Relate total earning units to the total cost
The total earning units ($85k$) represent the total cost of the work (Rs. 3400).
So, $85k = 3400$
Step 4: Find the value of the constant $k$
To find the value of $k$, divide the total cost by the total earning units:
$k = \frac{3400}{85}$
Let's simplify the fraction:
$k = \frac{3400 \div 5}{85 \div 5} = \frac{680}{17}$
$k = 40$
Step 5: Calculate the amount received by X
X's total earning is $20k$. Substitute the value of $k$ we found:
Amount received by X = $20 \times 40$
Amount received by X = $800$
Therefore, X will receive Rs. 800.
Summary Table of Earnings
| Person | Days Worked | Daily Wage Ratio | Daily Wage (with $k$) | Total Earning Units | Amount Received (with $k=40$) |
|---|---|---|---|---|---|
| X | 5 | 4 | $4k$ | $5 \times 4k = 20k$ | $20 \times 40 = 800$ |
| Y | 7 | 5 | $5k$ | $7 \times 5k = 35k$ | $35 \times 40 = 1400$ |
| Z | 10 | 3 | $3k$ | $10 \times 3k = 30k$ | $30 \times 40 = 1200$ |
Let's check the total amount: $800 + 1400 + 1200 = 3400$. This matches the given total cost of the work.
Revision Table: Key Concepts in Work and Wages
| Concept | Explanation | How it Applies Here |
|---|---|---|
| Daily Wage | Amount earned per day of work. | Given in ratio 4:5:3 for X, Y, Z. |
| Total Earning | Daily wage $\times$ Number of days worked. | Calculated as $20k, 35k, 30k$ for X, Y, Z respectively. |
| Ratio | A comparison of two or more quantities. | Used to express the relationship between daily wages (4:5:3). |
| Total Cost of Work | Sum of earnings of all individuals who contributed. | Given as Rs. 3400, equal to $85k$. |
Additional Information on Ratio and Proportion
Ratio and proportion are fundamental concepts used in various problems, including work and wages. A ratio expresses how much of one quantity there is compared to another. For instance, a ratio of daily wages 4:5:3 means that if X earns 4 units of money per day, Y earns 5 units, and Z earns 3 units for the same amount of time. A proportion is an equation stating that two ratios are equal. In this problem, we used the ratio of daily wages and the number of days worked to determine the ratio of total earnings, which was proportional to the total cost of the work.
Understanding ratios helps in distributing quantities proportionally. In this case, the total amount (Rs. 3400) is distributed among X, Y, and Z in proportion to their total earning units (20:35:30, which simplifies to 4:7:6 based on total work done adjusted for different daily rates, or directly calculated via $k$).
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?
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?
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?
