1. **Problem:** Aunt Anna is 42 years old. Caitlin is 5 years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin? 2. **Problem:** Which of these numbers is less than its reciprocal? 3. **Problem:** How many whole numbers lie in the interval between $\frac{5}{8}$ and $2\pi$? 4. **Problem:** Given percentages of working adults working at home in Carlin City over years 1960, 1970, 1980, and 1990, identify the correct graph. 5. **Problem:** Each principal of Lincoln High School serves exactly one 3-year term. What is the maximum number of principals this school could have during an 8-year period? 6. **Problem:** Figure ABCD is a square with side length 5. Inside it, three smaller squares with side lengths 3, 1, and 1 form an L-shaped shaded region. Find the area of the shaded region. --- ### Step-by-step solutions: 1. 1. Aunt Anna's age is $42$. 2. Brianna's age is half of Aunt Anna's: $$\text{Brianna} = \frac{42}{2} = 21.$$ 3. Caitlin is 5 years younger than Brianna: $$\text{Caitlin} = 21 - 5 = 16.$$ 4. **Answer:** (B) 16 2. 1. We check each number $x$ to see if $x < \frac{1}{x}$. 2. For $x = -2$, $-2 < -\frac{1}{2}$? No, because $-2 < -0.5$ is true. 3. For $x = -1$, $-1 < -1$? No, equal. 4. For $x = 0$, reciprocal undefined. 5. For $x = 1$, $1 < 1$? No. 6. For $x = 2$, $2 < \frac{1}{2}$? No. 7. So only $-2$ satisfies $x < \frac{1}{x}$. 8. **Answer:** (A) -2 3. 1. Interval is between $\frac{5}{8} = 0.625$ and $2\pi \approx 6.283$. 2. Whole numbers strictly between are $1, 2, 3, 4, 5, 6$. 3. Count: 6 whole numbers. 4. Options max is 5, so closest is (D) 5. 5. But question says "lie in the interval between", so exclude endpoints. 6. Whole numbers strictly greater than 0.625 and less than 6.283 are $1,2,3,4,5,6$ (6 numbers). 7. Since 6 is less than 6.283, it counts. 8. So 6 whole numbers, but option not given. 9. Possibly question expects counting whole numbers strictly between, so answer is 6, but closest option is (D) 5. 4. 1. Percentages increase from 5% in 1960 to 30% in 1990 gradually. 2. Graph (A) shows gradual increase from 5% to 30% over years. 3. Other graphs do not match this trend. 4. **Answer:** (A) 5. 1. Each principal serves 3 years. 2. In 8 years, maximum number of principals is the greatest integer $n$ such that $3n \leq 8$ or $3(n-1) < 8$ if partial terms allowed. 3. $3 \times 2 = 6$ years, $3 \times 3 = 9$ years. 4. So maximum full principals is 2 if no overlap, but if a principal can start before previous ends, max is 3. 5. Since terms are exactly 3 years, and 8 years total, max principals is 3. 6. **Answer:** (B) 3 6. 1. Large square side length: $5$ units. 2. Area of large square: $$5^2 = 25.$$ 3. Smaller squares have sides 3, 1, and 1. 4. Areas of smaller squares: $$3^2 = 9, \quad 1^2 = 1, \quad 1^2 = 1.$$ 5. Total area of smaller squares: $$9 + 1 + 1 = 11.$$ 6. Shaded L-shaped area: $$25 - 11 = 14.$$ 7. **Answer:** (D) 14