a number of teachers were to receive 6 pens each ,but needed 4 more pens for them to be equally divided.if each teacher received 9 pens, 25 more pens would be necessary.how many pens are there
The number of pens is (6T - 4), where T is the number of teachers. But the number of pens is also (9T - 25), so (6T - 4) = (9T - 25). So 3T = 21, which means T = 7, i.e. there are 7 teachers. That means there are 38 pens. Check it: If 7 teachers received 6 pens each then you'd need 42 pens, which means you're four short: correct. Also if each of 7 teachers were to receive 9 pens, you'd need 63 pens - but you've only got 38, which means you need another 25: again, correct. So I reckon there are indeed 38 pens.
To find the number of pens, we can set up a system of equations based on the given information.
Let's assume there are 'x' teachers and 'y' pens.
According to the first condition:
The number of pens each teacher receives = 6 pens
Total pens required initially = x * 6
But for the pens to be equally divided among the teachers, we need "+4" more pens. So, the total pens required initially becomes:
Total pens required initially = x * 6 + 4
According to the second condition:
The number of pens each teacher receives = 9 pens
Total pens required with each teacher receiving 9 pens = x * 9
But for the pens to be equally divided among the teachers, we need "+25" more pens. So, the total pens required with each teacher receiving 9 pens becomes:
Total pens required with each teacher receiving 9 pens = x * 9 + 25
We can now set up the following equation using the information from the two conditions:
x * 6 + 4 = x * 9 + 25
To solve the equation and find the number of pens (y), we proceed as follows:
Step 1: Expand the equation:
6x + 4 = 9x + 25
Step 2: Rearrange the equation:
6x - 9x = 25 - 4
Step 3: Simplify:
-3x = 21
Step 4: Divide by -3:
x = -7
Since we cannot have a negative number of teachers, this solution is not valid.
Therefore, there is no valid solution to this problem based on the given information.