Question

A tire store arranged its tires in rows from the most expensive tires to the lowest price. A set of tires from the first row cost $420. Each row had tires that were about 10% less than the tires in row before. Which row has a set of tires that cost about $75.00 per tire?

Answer 1

1st row:   Cost = $420
2nd row: Cost = $420(0.9)
3rd row:  Cost = $420(0.9)²

By induction, the cost in the n-th row is
Cost = $420(0.9)ⁿ⁻¹

When the cost is $75, the row number, n, is given by
420(0.9)ⁿ⁻¹ = 75
(0.9)ⁿ⁻¹ = 75/420 = 0.1786

Take logs of each side.
$(n-1)ln(0.9) = ln(0.1786) \\ n-1 = \frac{ln(0.1786)}{ln(0.9)} =16.35 \\ n=17.35$
Therefore
n = 17 (nearest integer)

Answer: Row 17