1. **State the problem:** Kate had 165 and Cheryl had 93. After both spent the same amount of money, Kate had 4 times as much money left as Cheryl. We need to find how much Kate spent. 2. **Define variables:** Let $x$ be the amount of money both Kate and Cheryl spent. 3. **Write the equation:** After spending, Kate has $165 - x$ left. Cheryl has $93 - x$ left. According to the problem, $165 - x = 4(93 - x)$. 4. **Solve the equation:** $$165 - x = 4(93 - x)$$ $$165 - x = 372 - 4x$$ 5. **Isolate $x$:** Add $4x$ to both sides: $$165 - x + 4x = 372 - 4x + 4x$$ $$165 + 3x = 372$$ 6. **Subtract 165 from both sides:** $$165 + 3x - 165 = 372 - 165$$ $$3x = 207$$ 7. **Divide both sides by 3:** $$\frac{\cancel{3}x}{\cancel{3}} = \frac{207}{3}$$ $$x = 69$$ 8. **Interpret the result:** Kate spent 69 on the shopping trip. **Final answer:** $$\boxed{69}$$