suppose that a, b, c ,and d are positive integers whose sum is 100 and a-c=5.What is the greatest possible value of ab-cd answers

To find the greatest possible value of ab-cd, we need to optimize the values of a, b, c, and d based on the given conditions.

Given:
1. a, b, c, and d are positive integers.
2. The sum of a, b, c, and d is 100.
3. a - c = 5.

Let's solve this step by step:

Step 1: Expressing the value of a in terms of c using the given condition "a - c = 5".
Rearranging the equation, we get:
a = c + 5

Step 2: Rewrite the sum of a, b, c, and d in terms of c:
Since a = c + 5, the sum becomes:
(c + 5) + b + c + d = 100
Simplifying, we get:
2c + b + d = 95

Step 3: Optimizing the value of b and d:
To maximize the value of ab - cd, we need to maximize the products ab and cd. This can be achieved by maximizing the values of b and d.
Since b and d are positive integers and their sum is 95 - 2c, we should assign the maximum value to one of them while keeping the other one as small as possible.

Considering b, we can assign the largest possible value to it, which is 95 - 2c. In this case, we would assign the smallest possible value to d, which is 1.

So, we have:
b = 95 - 2c
d = 1

Step 4: Calculating the value of a and b:
Substituting the values of b and d obtained in Step 3, we can find the values of a and b:
b = 95 - 2c
d = 1
a = c + 5

Step 5: Calculating the value of ab - cd:
Substitute the values of a, b, c, and d in the expression ab - cd:
ab - cd = (c + 5) * (95 - 2c) - c * 1
= 95c + 475 - 2c^2 - c
= -2c^2 + 94c + 475

Step 6: Find the vertex of the quadratic equation -2c^2 + 94c + 475:
Since the coefficient of c^2 is negative, the graph of the quadratic equation will be a downward-opening parabola.
To find the vertex, we can use the formula: c = -b/2a, where a = -2, and b = 94.
Therefore, c = -94 / (-2 * 2) = 23.5

Step 7: Rounding the value of c:
Since c should be a positive integer, we can round the value of c to the nearest whole number.
The two nearest whole numbers to 23.5 are 23 and 24.

Step 8: Calculating the values of a and b with the rounded value of c:
Substituting the rounded value of c into the equations found in Step 4:
For c = 23:
b = 95 - 2 * 23 = 49
a = 23 + 5 = 28

For c = 24:
b = 95 - 2 * 24 = 47
a = 24 + 5 = 29

Step 9: Calculating the value of ab - cd for both cases:
For c = 23:
ab - cd = (28 * 49) - (23 * 1) = 1375

For c = 24:
ab - cd = (29 * 47) - (24 * 1) = 1333

Step 10: Finding the greatest possible value:
The greater value between 1375 and 1333 is 1375.

Therefore, the greatest possible value of ab - cd, given the conditions, is 1375.