Two Individuals with Equal and Unequal Cash

The problem we're considering involves two individuals, John and Tony, with an interesting scenario involving exchanging cash. Let's go through the problem step-by-step to reach a conclusion.

John and Tony both have some cash in their pockets, but no coins. If John gives Tony 98 units of cash, both will have the same amount. If Tony gives John 98 units of cash, John will have nine times as much cash as Tony. How much cash did each start with?

Algebraic Solutions

We can set up algebraic equations to solve this problem. Let's define the amount of cash John has as J and the amount of cash Tony has as T.

The first condition can be written as:

John gives Tony 98 units:

J - 98 T 98

Rearranging the equation, we get:

J - T 196 ---- equation (A)

The second condition can be written as:

Tony gives John 98 units:

9(T - 98) J 98

Rearranging the equation, we get:

9T - 98 J 98

Substituting the value of J from equation (A) into this equation:

9T - 98 (196 T) 98

Simplifying further:

9T - 98 196 T 98

9T - 98 294 T

8T 294 98

8T 392

T 49

Verification and Solution

Let's check the value of T (Tony's initial amount) and then find J (John's initial amount).

Substituting T 49 into equation (A):

J - 49 196

J 196 49

J 245

Clearly, there was an arithmetic mistake in the initial solution provided. Let's correct it and find the accurate values for John and Tony using the correct steps.

Correct Approach

Let's use the corrected algebraic approach with proper substitution and simplification steps.

The two equations from the problem are:

J - 98 T 98 (John gives 98 to Tony) 9(T - 98) J 98 (Tony gives 98 to John)

From the first equation, we have:

J - T 196 ---- (A)

From the second equation, we have:

9T - 882 J 98

9T - J 980 ---- (B)

Now, we have two linear equations:

J - T 196

9T - J 980

Adding these two equations:

J - T 9T - J 196 980

8T 1176

T 147

Substituting T 147 into equation (A):

J - 147 196

J 196 147

J 343

Verification

Let's verify the solution by substituting back into the original statements:

If John gives Tony 98 units: 343 - 98 245 147 98 245 If Tony gives John 98 units: 9(147 - 98) 9(49) 441 343 98 441

The solution is consistent with the given conditions, confirming that Tony started with 147 units of cash and John started with 343 units of cash.

Conclusion

This problem demonstrates the use of algebraic equations to solve practical cash distribution problems. By setting up and solving the equations, we can find the initial amounts of cash held by John and Tony, illustrating the power of algebra in solving real-world scenarios.

The keywords for this problem are: cash distribution, algebraic equations, cash transactions.