How can I model two changing quantities that always add to the same total?

Name the fixed total T, and call the two quantities x and y. Then write:

x + y = T

This equation says exactly what stays unchanged: the sum of x and y is always T. Rearrange by subtracting x from both sides:

y = T − x

So, if the total is known, every chosen value of x determines the corresponding value of y. OpenStax illustrates the general method of naming quantities and writing their fixed-sum relationship as an equation; its example uses two numbers, so applying that form to your situation depends on defining what your quantities mean. [S1]

Relationship diagram:

Fixed total T
   /       \
  x   +    y
            |
       y = T − x

The branches represent the two parts making up the total, not a scale drawing. No length, area, or other geometric measurement is implied.

Suppose, as an illustrative assumption, a class has 30 activity tokens to divide between two groups. Let x be the number given to Group A and y the number given to Group B. The fixed-sum model is:

x + y = 30

y = 30 − x

If Group A gets 12 tokens, then y = 30 − 12 = 18, so Group B gets 18. If x increases by 1 while the total remains 30, y must decrease by 1 to compensate. The equation describes that relationship; it does not say which allocation is fair or appropriate.

The assumed context also sets practical limits: if tokens are whole items, x and y should be whole numbers from 0 through 30. Those limits come from this example’s assumptions, not from the equation alone. For a context where parts can be fractional, different allowed values may make sense.

First check your definitions: x and y should identify the two quantities clearly, and they should use compatible units. Then check the total and the rearrangement. For example, substituting the results back into x + y = T should reproduce T. Finally, compare the values with the limits of your real situation. If a group cannot receive more than 20 tokens, for instance, a result of 23 may satisfy the algebra but violate that assumed limit.

A fixed-sum equation models the stated relationship; it does not establish that the total really stays fixed, explain why it stays fixed, or supply any missing measurements. Those must come from the situation you are modeling.

Use this template:

Fixed total: T = [total and units]
First quantity: x = [what it represents and units]
Second quantity: y = [what it represents and units]
Relationship: x + y = T
Solve for the second quantity: y = T − x
Allowed values: [limits from the situation]
Check: [chosen x] + [calculated y] = [total]

For the token example, the entries are T = 30 tokens, x = tokens for Group A, y = tokens for Group B, and y = 30 − x. Keep the total and both quantities in matching units, and state any limits separately. The equation alone does not determine a particular allocation until you provide a value for one quantity.