1. Problem: In a batch of 250 students, 62 are in the Sports Club, 170 are in the Dramatic Club, and 40 students are in both clubs. We want to find how many students are in either club and how many are in neither club. Step 1: Use the formula for the union of two sets: $$|A \cup B| = |A| + |B| - |A \cap B|$$ where $|A|$ is Sports Club members, $|B|$ is Dramatic Club members, and $|A \cap B|$ is members in both. Step 2: Substitute values: $$|A \cup B| = 62 + 170 - 40 = 192$$ Step 3: Total students not in either club: $$250 - 192 = 58$$ 2. Problem: The difference of two numbers is 192. One number is 9 times the other. Find the bigger number. Step 1: Let the smaller number be $x$. Then the bigger number is $9x$. Step 2: Difference equation: $$9x - x = 192$$ Step 3: Simplify and solve: $$8x = 192 \implies x = \frac{192}{8} = 24$$ Step 4: Bigger number is: $$9x = 9 \times 24 = 216$$ 3. Problem: The units digit of a two-digit number is 5 more than the tens digit. Adding the units digit to the number with its digits reversed gives 79. Find the number. Step 1: Let tens digit be $t$, units digit be $u$. Given: $$u = t + 5$$ Step 2: The number is $10t + u$, reversed number is $10u + t$. Step 3: According to the problem: $$u + (10u + t) = 79$$ Step 4: Substitute $u = t + 5$: $$ (t+5) + 10(t+5) + t = 79 $$ $$ t + 5 + 10t + 50 + t = 79 $$ $$ 12t + 55 = 79 $$ Step 5: Solve for $t$: $$ 12t = 79 - 55 = 24 \implies t = 2 $$ Step 6: Find $u$: $$ u = t + 5 = 2 + 5 = 7 $$ Step 7: The number is: $$ 10t + u = 10 \times 2 + 7 = 27 $$ 4. Problem: Find three numbers where the second is 6 less than 3 times the first, the third is 2 more than twice the first, and their sum is 68. Step 1: Let first number be $x$. Then: Second = $3x - 6$ Third = $2x + 2$ Step 2: Equation for sum: $$ x + (3x - 6) + (2x + 2) = 68 $$ Step 3: Simplify: $$ x + 3x - 6 + 2x + 2 = 68 $$ $$ 6x - 4 = 68 $$ Step 4: Solve for $x$: $$ 6x = 72 \implies x = 12 $$ Step 5: Find second and third numbers: Second = $3 \times 12 - 6 = 36 - 6 = 30$ Third = $2 \times 12 + 2 = 24 + 2 = 26$ 5. Problem: Find two consecutive integers such that two times the first plus three times the second is 78. Step 1: Let the first integer be $n$. The second consecutive integer is $n+1$. Step 2: Equation: $$ 2n + 3(n + 1) = 78 $$ Step 3: Simplify: $$ 2n + 3n + 3 = 78 $$ $$ 5n + 3 = 78 $$ Step 4: Solve for $n$: $$ 5n = 75 \implies n = 15 $$ Step 5: The integers are $15$ and $16$. Graph description problem: The rectangle has vertices at (1,1), (7,1), (7,5), and (1,5). The region is defined by inequalities: $$1 \leq x \leq 7$$ $$1 \leq y \leq 5$$