The board of directors for a philanthropic organization is allocating funds for the three fund-raising events they plan to hold next year: Events X, Y, and Z. The organizers of the events have requested that the following amounts be allocated to the events: Event X: $12,000 Event Y: $5,000 Event Z: $3,000 The board has a total budget of $15,000 for the three events, and they will distribute all of the $15,000 among the 3 events. Each event will receive at most the requested amount and at least half the requested amount. If the board allocates $10,000 to Event X, select for Event Y and for Event Z amounts the board could allocate to Event Y and Event Z, respectively, so that the amounts would jointly adhere to the constraints described above. Make only two selections, one in each column.

Solution: It has already been mentioned in the question stem that the board has a total budget of $15,000 for the three events, and they will distribute all of the $15,000 among the 3 events. Each event will receive at most and at least half the requested amount. The board allocates $10000 to Event X As we know Event X: $12,000 Event Y: $5,000 Event Z: $3,000 If we subtract $10000 from $15000 we will left with $5000. This amount is to be distributed between Event Y and Z. So, the only feasible option we are left with is $2000 for Event Z and $3000 for Event Y.

1. We're asked to find exactly how much of the budget was distributed among Events Y and Z . 2. Let a and b be how much is spent on Events Y and Z, respectively. The restrictions tell us that 2500 \leq a \leq 5000 and 1500 \leq b \leq 3000 . 3. It is also known that the total budget was 15000, so, 15000 = 10000 + a + b \rightarrow b = 5000 - a . Our 2nd inequality will turn into 1500 \leq 5000 - a \leq 3000 \rightarrow -3500 \leq -a \leq -2000 \rightarrow 2000 \leq a \leq 3500 . 4. Then, we know that 2500 \leq a \leq 3500 . The only answer that's in this range is a = 3000, which means b = 2000. 5. Our answer will be: Event Y - $3,000 and Event Z - $2,000 .