##### Question Capital One (C1) is evaluating a promotional offer on its Rent-a-Home (RH) product and wants you to size the unit economics. **Base assumptions** - C1 has 2,000,000 active cardholders. - Currently 5% use Rent-a-Home (RH) annually. - Average RH spend per RH user per year is $500. - A proposed promo gives 30% off RH spend, fully funded by C1. - Average first-year net revenue per new cardholder (after non-RH rewards, servicing, and expected credit loss) is $550. - Ignore interchange on RH transactions for simplicity. Let **x** = the number of new customers acquired because of the promo. 1. **Base break-even.** Write the break-even equation where incremental revenue from the new customers equals the promo cost across existing RH users plus new RH users, then solve for **x**. Show your work and final numbers. 2. **Sizing and realism.** Express **x** as a percentage of the 2,000,000 base, and discuss whether this target is realistic given typical acquisition funnels and approval rates. 3. **Spend-uplift extension.** Now extend the scenario and assume that during the promo: (i) existing RH users increase RH spend by 20% (from $500 to $600); (ii) new customers who join because of the promo spend twice as much on RH as existing users during the promo period (i.e., $1,000). The discount remains 30% on RH spend and average first-year net revenue per new cardholder remains $550. Write and solve the break-even equation: `550·x − 0.30·(1,000·x + 2,000,000·5%·600) = 0` 4. **Compare and explain.** Compare the new break-even **x** from part 3 to the original break-even from part 1, and explain intuitively why the required new-customer count changes despite the higher spend.