1. **Problem 22: Percent Change in Undergraduate Population** The problem states that in 2013, the undergraduate population was 5517, and in 2014 it was 6220. We need to find the percent change from 2013 to 2014, rounded to the nearest percent. 2. **Formula for Percent Change:** $$\text{Percent Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\%$$ 3. **Calculate the difference:** $$6220 - 5517 = 703$$ 4. **Divide by the old value:** $$\frac{703}{5517} \approx 0.1275$$ 5. **Convert to percent:** $$0.1275 \times 100\% = 12.75\%$$ 6. **Round to nearest percent:** $$13\%$$ --- 7. **Problem 23: Find the missing dimension $x$ in similar triangles** Given triangles $\triangle QRS \sim \triangle QPT$, corresponding sides are proportional. 8. **Identify corresponding sides:** - $RS$ corresponds to $PT$ - $QR$ corresponds to $QP$ - $QS$ corresponds to $QT$ Given: - $PT = 12$ cm - $RS = 8$ cm - $ST = 20$ cm (side in large triangle) We want to find $x = QR$ (side opposite $RS$ in smaller triangle). 9. **Set up proportion using corresponding sides:** $$\frac{RS}{PT} = \frac{QR}{QP}$$ We know $RS = 8$, $PT = 12$, and $QR = x$. We need $QP$. 10. **Find $QP$ using $ST$ and $QT$:** Since $ST = 20$ cm and $PT = 12$ cm, and $QPT$ is a triangle, $QP$ is a side of the large triangle. However, $QP$ is not given directly. But since $QRS \sim QPT$, and $RS$ corresponds to $PT$, the scale factor is: $$k = \frac{RS}{PT} = \frac{8}{12} = \frac{2}{3}$$ 11. **Use scale factor to find $x$:** Since $QR$ corresponds to $QP$, and the smaller triangle is scaled by $\frac{2}{3}$, then: $$x = QR = k \times QP = \frac{2}{3} \times QP$$ 12. **Find $QP$ using $ST$ and $QS$:** Note that $ST$ is a side of the large triangle, and $QS$ is a side of the smaller triangle. Since $QS$ corresponds to $QT$, and $ST$ corresponds to $PT$, but $ST$ is given as 20 cm, and $PT$ is 12 cm, this suggests $ST$ is not a side of the large triangle but part of the figure. Without additional information about $QP$ or $QT$, we cannot find $x$ numerically. **Assuming $QP = 30$ cm (hypothetical for calculation):** $$x = \frac{2}{3} \times 30 = 20$$ **Final answers:** - Percent change: $13\%$ - Missing dimension $x$: $\frac{2}{3} \times QP$ (depends on $QP$ value)