$5$ men and $4$ women can earn ₹ $20000$ in $8$ days. $10$ men and $7$ women can earn ₹ $23,750$ in $5$ days. In how many days will $5$ men and $6$ women earn ₹ $12,000$?
This problem involves calculating the number of days required for a group of men and women to earn a certain amount, based on their combined earnings over different periods. We need to determine the individual daily earning rates of men and women first.
Calculating Men and Women Earnings Rates
Let's define the variables:
- Let $m$ be the amount (in ₹) earned by one man in one day.
- Let $w$ be the amount (in ₹) earned by one woman in one day.
From the first statement: 5 men and 4 women can earn ₹ 20000 in 8 days.
This means the total earning in 8 days is ₹ 20000. The daily earning of the group (5 men and 4 women) is ₹ $\frac{20000}{8}$.
So, the equation is:
$5m + 4w = \frac{20000}{8}$ $5m + 4w = 2500 \quad \cdots (1)$From the second statement: 10 men and 7 women can earn ₹ 23,750 in 5 days.
The daily earning of this group (10 men and 7 women) is ₹ $\frac{23750}{5}$.
So, the equation is:
$10m + 7w = \frac{23750}{5}$ $10m + 7w = 4750 \quad \cdots (2)$Solving for Individual Earnings
Now we have a system of two linear equations with two variables:
- $5m + 4w = 2500$
- $10m + 7w = 4750$
To solve this system, we can use the method of elimination or substitution. Let's use elimination. We can multiply equation (1) by 2 to make the coefficients of $m$ the same:
$2 \times (5m + 4w) = 2 \times 2500$ $10m + 8w = 5000 \quad \cdots (3)$Now, subtract equation (2) from equation (3):
$(10m + 8w) - (10m + 7w) = 5000 - 4750$ $10m + 8w - 10m - 7w = 250$ $w = 250$So, one woman earns ₹ 250 per day.
Substitute the value of $w$ back into equation (1):
$5m + 4(250) = 2500$ $5m + 1000 = 2500$ $5m = 2500 - 1000$ $5m = 1500$ $m = \frac{1500}{5}$ $m = 300$So, one man earns ₹ 300 per day.
Calculating Earnings for 5 Men and 6 Women
We need to find out how many days it will take for 5 men and 6 women to earn ₹ 12,000.
First, let's calculate the total daily earning of this new group:
Daily earning = (Number of men $\times$ Earning per man) + (Number of women $\times$ Earning per woman)
Daily earning = $5m + 6w$
Daily earning = $5(300) + 6(250)$
Daily earning = $1500 + 1500$
Daily earning = $3000$
Thus, 5 men and 6 women together earn ₹ 3000 per day.
Finding the Number of Days
The target amount to be earned is ₹ 12,000.
Let $D$ be the number of days required.
The relationship is:
$\text{Daily Earning} \times \text{Number of Days} = \text{Total Amount}$ $3000 \times D = 12000$Now, solve for $D$:
$D = \frac{12000}{3000}$ $D = 4$Therefore, 5 men and 6 women will earn ₹ 12,000 in 4 days.
Important Questions from Time & Work (Notes)
- If 6 men and 8 boys can do a piece of work in 10 days and 26 men and 48 boys can do the same work in 2 days, then the time taken by 15 men and 20 boys to do the same work will be
- A fresh water tap fills a fish tank in 40 minutes. The same tank is filled by a salt water tap in 120 minutes. If both the taps are open, how many minutes will it take to fill the tank?
Three pipes A, B and C can fill a tank in $10$, $15$ and $20$ hours respectively. Pipe A was opened at $6$ AM, pipe B at $7$ AM and pipe C at $8$ AM. At what time was the tank completely filled, if pipe C needs a break of $1$ hour after remaining open for $3$ hours?
A tank has four pipes $P_1$, $P_2$, $P_3$ and $P_4$. The tank can be filled in $15$ minutes by pipes $P_1$, $P_2$, $P_3$ together. It can be filled in $20$ minutes by pipes $P_2$, $P_3$, $P_4$ together and it can be filled by pipes $P_1$, $P_4$ together in $30$ minutes. If all the pipes are opened together, then in how much time will the tank be filled?
- $6$ men or $5$ women earn ₹ $16,800$ in $4$ days. How much will $4$ women and $6$ men earn in one day?
