Tom and Kathy together have 30 chocolates.Both of them lost 6 chocolates each,and the product of the number of chocolates now they have is 80.Find out how many chocolates do they have at the beginning.
t+k=30
(t-6)(k-6)=80
(t-6)((30-t)-6) = 80
(t-6)(24-t) = 80
24t-t^2-144+6t = 80
t^2 - 30t + 224 = 0
(t-14)(t-16) = 0
...
Let's solve this step-by-step.
Let's assume that Tom had x chocolates at the beginning and Kathy had y chocolates.
According to the problem, we know that the sum of their chocolates is 30. So, we can write the equation: x + y = 30.
After they both lost 6 chocolates each, Tom would have x - 6 chocolates and Kathy would have y - 6 chocolates.
We are also given that the product of the number of chocolates they have now is 80. So, we can write the equation: (x - 6)(y - 6) = 80.
Now we have a system of equations:
x + y = 30
(x - 6)(y - 6) = 80
We can solve this system of equations to find the values of x and y.
From the first equation, we can solve for y: y = 30 - x.
Substituting this value of y in the second equation, we get:
(x - 6)((30 - x) - 6) = 80
(x - 6)(24 - x) = 80
24x - x^2 - 144 + 6x = 80
x^2 - 30x + 224 = 0
Now we can solve this quadratic equation to find the values of x.
Factoring or using the quadratic formula, we get: (x - 4)(x - 28) = 0.
This gives us two possible solutions: x = 4 or x = 28.
If x = 4, then y = 30 - x = 26.
If x = 28, then y = 30 - x = 2.
So, there are two possible scenarios:
1. Tom had 4 chocolates and Kathy had 26 chocolates at the beginning.
2. Tom had 28 chocolates and Kathy had 2 chocolates at the beginning.
To solve this problem, let's assign variables to the unknown quantities in the question. Let's say Tom initially had t chocolates, and Kathy had k chocolates.
According to the given information, Tom and Kathy together have 30 chocolates: t + k = 30 --------(1)
Both Tom and Kathy lost 6 chocolates each, which means their new quantities of chocolate are (t - 6) and (k - 6), respectively.
The product of the number of chocolates they now have is 80: (t - 6)(k - 6) = 80 --------(2)
We now have a system of equations with two unknowns (t and k). We can solve this system of equations to find the values of t and k, which will give us the initial quantities of chocolates that Tom and Kathy had.
Let's start by rearranging equation (1) to express k in terms of t:
k = 30 - t
Now we substitute this value for k in equation (2):
(t - 6)(30 - t - 6) = 80
Simplifying the equation, we have:
(t - 6)(24 - t) = 80
24t - t^2 - 144 + 6t = 80
-t^2 + 30t - 144 - 80 = 0
-t^2 + 30t - 224 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, let's use the quadratic formula:
t = (-b ± √(b² - 4ac)) / (2a)
For our equation, a = -1, b = 30, and c = -224. Substituting these values into the quadratic formula, we get:
t = (-(30) ± √((30)² - 4(-1)(-224))) / (2(-1))
t = (-30 ± √(900 - 896)) / (-2)
t = (-30 ± √4) / (-2)
t = (-30 ± 2) / (-2)
Now, we have two possible values for t:
1. t = (-30 + 2) / (-2) = -28 / (-2) = 14
2. t = (-30 - 2) / (-2) = -32 / (-2) = 16
Since we are dealing with quantities of chocolates, the values must be positive. Therefore, the correct value for t is 14.
Now, we can substitute this value back into equation (1) to find k:
k = 30 - t
k = 30 - 14 = 16
So, Tom initially had 14 chocolates, and Kathy initially had 16 chocolates.