A number of men subscribed a certain amount to make up a deficit of $100 but 5 men failed to pay and thus increased the share of the others by $1 each. Find the amount that each man paid.
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Let's assume the number of men is x and the amount each man originally subscribed is y.
According to the information given, the total deficit is $100. Therefore, the total amount subscribed by all men is:
xy = 100 (Equation 1)
After 5 men failed to pay, the remaining men had to increase their share by $1 each. The total amount subscribed by the remaining men is:
(x-5)(y+1) = 100 (Equation 2)
We can solve this system of equations to find the values of x and y.
Expanding Equation 2:
xy + x - 5y - 5 = 100
Substituting Equation 1 into Equation 2:
100 + x - 5y - 5 = 100
Simplifying:
x - 5y - 5 = 0
Rearranging the equation:
x = 5y + 5 (Equation 3)
Substituting Equation 3 into Equation 1:
(5y + 5)y = 100
5y^2 + 5y - 100 = 0
Now we can solve this quadratic equation for y using the quadratic formula:
y = (-b ± sqrt(b^2 - 4ac)) / (2a)
where a = 5, b = 5, and c = -100.
y = (-5 ± sqrt(5^2 - 4(5)(-100))) / (2(5))
y = (-5 ± sqrt(25 + 2000)) / 10
y = (-5 ± sqrt(2025)) / 10
y = (-5 ± 45) / 10
Taking both solutions:
y1 = (-5 + 45) / 10 = 40/10 = 4
y2 = (-5 - 45) / 10 = -50/10 = -5
Since the amount each man paid cannot be negative, we discard y2 = -5.
Therefore, each man originally paid $4.
Let's break down the problem step by step to find the solution:
Step 1: Let's assume the total number of men who subscribed is 'x'.
Step 2: We know that the total deficit is $100, so each man should have paid an equal amount.
Step 3: Initially, the amount each man should have paid is $100/x.
Step 4: However, since 5 men failed to pay, the remaining men had to cover their share, resulting in an increase of $1 per man.
Step 5: So, the new amount each man paid is $100/x + $1.
Step 6: Now, we have two equations based on the given information:
Equation 1: x * ($100/x) = $100 (the total deficit should be $100)
Equation 2: (x - 5) * [($100/x) + $1] = $100 (each man's new payment should cover the deficit)
Step 7: Simplifying Equation 1, we get:
x * ($100/x) = $100
$100 = $100 (simplify)
Step 8: Simplifying Equation 2, we get:
(x - 5) * [($100/x) + $1] = $100
(x - 5) * ($100/x) + (x - 5) * $1 = $100
$100 - 5 * ($100/x) + x - 5 = $100 (simplify)
Step 9: Now, cancel out the $100 terms:
5 * ($100/x) - x + 5 = 0
Step 10: Multiply through by 'x' to get rid of the fraction:
5 * $100 - x^2 + 5x = 0
500 - x^2 + 5x = 0
Step 11: Rearrange the equation:
x^2 - 5x + 500 = 0
Step 12: By factoring or using the quadratic formula, we find that the solutions to this equation are x = 20 and x = 25.
Step 13: The correct solution is x = 20 because if all 20 men paid, there would be no deficit.
Step 14: Substituting x = 20 back into Equation 1, we find:
$100/x = $100/20 = $5
Hence, each man initially paid $5 to make up the deficit.
let the number of men be x
original share of debt = 100/x
men actually paying back = x-5
new share = 100/(x-5)
100/(x-5) - 100/x = 1
100x - 100x + 500 = x(x-5)
x^2 - 5x - 500 = 0
(x - 25)(x+20) = 0
x = 25 or x = a negative, which is not admissible
So there were 25 men, of which 20 paid 100/20 or $5.00 each