Question

State University is planning a tuition raise. The Dean of Admissions suggests that they increase tuition by $10\%$ each year for the next two years. The Provost suggests that each year they instead increase the tuition by $30\%$ of the amount of the tuition above $\$10{,}000$. So, for example, if the tuition now is $\$25{,}000$, the Dean of Admissions is suggesting a $\$2{,}500$ increase in the first year. Meanwhile, the Provost is suggesting a $\$4{,}500$ increase (since the current tuition is $\$15{,}000$ more than $\$10{,}000$, and $30\%$ of $\$15{,}000$ is $\$4{,}500$). The Treasurer points out that the tuition two years from now (after two annual increases) will be the same under either the Dean's plan or the Provost's plan. What is the tuition right now? (Assume the tuition is not $0$; State University is not free!)

Answer 1

14,375

Explanation:

Answer 2

The actual tuition is $14,375

Explanation:

Percentages

Let's call X the original tuition, and let's assume it's greater than 10,000 so far. We'll analyze the first option:

Dean of Admissions suggests increasing tuition by 10% each year for the next two years. The first year the tuition would be

`\displaystyle X+(10)/(100)X=1.1X`

The second-year the tuition would increase another 10% of the last amount, so it would end being

`\displaystyle 1.1X+(10)/(100)(1.1X)=1.21X`

Now for the second option

The Provost suggests that each year they increase the tuition by 30% of the amount of the tuition above $10,000. The first increase would be

`\displaystyle (30)/(100)(X-10,000)=0.3X-3,000`

And the new tuition at the end of the first year would be

`\displaystyle X+0.3X-3,000=1.3X-3,000`

This new amount would be increased by applying the same rule, so the second-year increase would be

`\displaystyle (30)/(100)(1.3X-3,000-10,000)=0.39X-3,900`

And the final tuition would be

`1.3X-3,000+0.39X-3,900=1.69X-6,900`

The Treasurer points out that the tuition will be the same under any scheme, so

`1.69X-6,900=1.21X`

Rearranging

`1.69X-1.21X=6,900`

`0.48X=6,900`

`\displaystyle X=(6,900)/(0.48)`

`\boxed{X=14,375}`

The actual tuition is $14,375