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:
1,000
Calculating Daily Wage Based on Work Efficiency and Earnings
This problem involves calculating the daily wage of individuals based on their combined earnings over a period and their relative efficiencies. When individuals work together and earn a wage, the total wage is typically distributed among them in proportion to their efficiency or the amount of work done by each.
Understanding the Relationship Between Efficiency and Wage
Efficiency in work directly relates to how much work a person can complete in a given amount of time. If one person is more efficient than another, they can do more work or the same amount of work in less time. When two people work together for a shared income, their share of the income is proportional to their individual efficiency, assuming they work for the same duration.
In this question, A's efficiency is 5 times that of B's. This means that for the same amount of time, A does 5 times the work B does, or A is 5 times more productive than B. Consequently, A's daily wage will be 5 times B's daily wage.
The ratio of their efficiencies is A : B = 5 : 1.
Therefore, the ratio of their daily wages will also be A : B = 5 : 1.
Calculating Total Daily Wage
A and B worked together for 15 days and earned a total of Rs. 18,000. To find their combined daily wage, we divide the total earning by the number of days worked.
Total Earning = Rs. 18,000
Number of Days = 15 days
Total Daily Wage = \( \frac{\text{Total Earning}}{\text{Number of Days}} \)
Total Daily Wage = \( \frac{18000}{15} \)
Let's perform the calculation:
- \( 18000 \div 15 \)
- \( (15000 + 3000) \div 15 \)
- \( (15000 \div 15) + (3000 \div 15) \)
- \( 1000 + 200 \)
- \( 1200 \)
So, the combined daily wage of A and B is Rs. 1,200.
Distributing Daily Wage Based on Efficiency Ratio
The total daily wage of Rs. 1,200 is shared between A and B in the ratio of their daily wages, which is 5 : 1. This means that for every Rs. (5+1)=6 of the total daily wage, A gets Rs. 5 and B gets Rs. 1.
The total number of ratio parts is \( 5 + 1 = 6 \).
A's share of the daily wage is \( \frac{\text{A's ratio part}}{\text{Total ratio parts}} \times \text{Total Daily Wage} \).
A's daily wage = \( \frac{5}{6} \times 1200 \)
B's share of the daily wage is \( \frac{\text{B's ratio part}}{\text{Total ratio parts}} \times \text{Total Daily Wage} \).
B's daily wage = \( \frac{1}{6} \times 1200 \)
Calculating A's Daily Wage
Using the formula derived above for A's daily wage:
A's daily wage = \( \frac{5}{6} \times 1200 \)
A's daily wage = \( 5 \times \frac{1200}{6} \)
A's daily wage = \( 5 \times 200 \)
A's daily wage = \( 1000 \)
Thus, the daily wage of A is Rs. 1,000.
Let's quickly verify B's daily wage:
B's daily wage = \( \frac{1}{6} \times 1200 \)
B's daily wage = \( 200 \)
A's daily wage (1000) is indeed 5 times B's daily wage (200), and their sum (1000 + 200 = 1200) equals the total daily wage calculated earlier.
Summary of Steps
- Determine the ratio of daily wages based on the efficiency ratio.
- Calculate the total combined daily wage by dividing total earnings by the number of days.
- Distribute the total daily wage according to the wage ratio to find individual daily wages.
| Item | Value |
|---|---|
| Total Earnings | Rs. 18,000 |
| Number of Days | 15 days |
| Total Daily Wage | \( \frac{18000}{15} = 1200 \) Rs. |
| Efficiency Ratio (A:B) | 5:1 |
| Wage Ratio (A:B) | 5:1 |
| A's Daily Wage | \( \frac{5}{6} \times 1200 = 1000 \) Rs. |
| B's Daily Wage | \( \frac{1}{6} \times 1200 = 200 \) Rs. |
Based on the calculations, the daily wage of A is Rs. 1,000.
Revision Table: Key Concepts in Work and Wages
| Concept | Relationship with Work/Wage | Notes |
|---|---|---|
| Efficiency | Higher efficiency means more work done per unit time. | Often expressed as a ratio. |
| Work Done | Directly proportional to efficiency and time. | If efficiency doubles, work doubles (for same time). |
| Wage Distribution | Wages are distributed in proportion to work done or efficiency (if time is same). | Ratio of wages equals ratio of efficiency/work done. |
| Total Earnings | Sum of individual earnings over the period. | Can be used to calculate total daily wage. |
Additional Information: Ratio and Proportion in Wage Problems
Ratio and proportion are fundamental tools for solving problems involving the distribution of wages based on efficiency or work done. When a total amount needs to be divided into parts based on a given ratio, the total amount is divided by the sum of the ratio parts. Each person's share is then found by multiplying the value of one ratio part by their respective ratio number.
For example, if an amount is to be divided between X and Y in the ratio \( a:b \), the total ratio parts are \( a+b \). The value of one ratio part is \( \frac{\text{Total Amount}}{a+b} \). X's share is \( a \times \frac{\text{Total Amount}}{a+b} \) and Y's share is \( b \times \frac{\text{Total Amount}}{a+b} \).
This principle was applied here: the total daily wage (Rs. 1,200) was divided in the ratio 5:1. The total ratio parts were \( 5+1=6 \). The value of one ratio part was \( \frac{1200}{6} = 200 \). A's share (daily wage) was \( 5 \times 200 = 1000 \), and B's share was \( 1 \times 200 = 200 \).
Important Questions from Work and Wages
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?
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?
