Analyzing a Fundraising Raffle with Costs and Break-Even
A school club plans a raffle. Tickets cost $5 each or 3 for $12. The grand prize costs the club $450, and printing tickets and advertising costs $180. The club must sell tickets in the 3-for-$12 bundle or singly, and most buyers choose the bundle. The club wants to raise at least $2,000 after prize and printing costs. A local business offers to cover the $450 prize if the club gives it 20% of all ticket revenue. The club is unsure whether to accept, and also wants to know how many tickets must be sold to break even.
1. Ignoring the business offer, calculate how many tickets must be sold to raise $2,000 net, assuming all tickets are sold in 3-for-$12 bundles.
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Costs = 450 + 180 = $630. To net $2,000, revenue needed = 2,000 + 630 = $2,630.
Each bundle of 3 tickets earns $12, so revenue per ticket = $12 / 3 = $4.
Number of tickets = 2,630 / 4 = 657.5, so 658 tickets. But bundles come in threes, so round up to a multiple of 3: 660 tickets (220 bundles).
Check: 220 bundles × $12 = $2,640. Net = 2,640 - 630 = $2,010, which meets the $2,000 target. At 219 bundles, revenue = $2,628, net = $1,998, just short. So 660 tickets are needed.
2. Evaluate the business offer: the business covers the $450 prize but takes 20% of ticket revenue. How many bundle tickets are needed to net $2,000 under this arrangement?
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Now costs to the club = printing $180 only (prize covered). Revenue is reduced by 20%, so the club keeps 80% of ticket revenue.
Let T = number of tickets. Revenue = 4T (bundle rate). Club keeps 0.80 × 4T = 3.2T.
Net = 3.2T - 180. Set ≥ 2,000: 3.2T ≥ 2,180, so T ≥ 681.25, meaning 684 tickets (round up to a multiple of 3).
Check: 684 tickets = 228 bundles, revenue = $2,736. Club keeps 0.80 × 2,736 = $2,188.80. Net = 2,188.80 - 180 = $2,008.80, meeting the goal. Without the offer, 660 tickets sufficed; with it, 684 are needed.
3. Recommend whether to accept the business offer, considering break-even and risk, and find the break-even ticket count under each option.
Read the worked answer
Without offer: Costs = $630, revenue per ticket = $4. Break-even: 4T = 630, T = 157.5, so 158 tickets (round up to 159 for bundles). At 159 tickets: revenue = 53 bundles × $12 = $636, net = $6.
With offer: Costs = $180, club keeps 80% of revenue. Break-even: 3.2T = 180, T = 56.25, so 57 tickets (round up to 60 for bundles). At 60 tickets: revenue = 20 bundles × $12 = $240, kept = $192, net = $12.
The offer lowers break-even dramatically (from about 159 to 60 tickets), reducing risk if sales are weak. However, to reach the $2,000 goal, the offer requires 684 tickets versus 660 without it, because 20% of revenue grows large as sales rise. Recommendation: accept the offer if the club is unsure about selling many tickets, since it sharply lowers the break-even point; decline if the club is confident of selling well above about 660 tickets, since the 20% cut then costs more than the $450 prize it saves. The two options are equal in net when 3.2T - 180 = 4T - 630, giving 0.8T = 450, T = 562.5, so above roughly 563 tickets the no-offer option is better.