You're on a roll — b) Step 1: Solve the equation $2^x \times 8 = 64$. Divide both sides by 8. $$ 2^x = \frac{64}{8} $$ $$ 2^x = 8 $$ Step 2: Express 8 as a power of 2. $$ 8 = 2^3 $$ Step 3: Substitute this back into the equation. $$ 2^x = 2^3 $$ Step 4: Since the bases are the same, equate the exponents. $$ x = 3 $$ The solution is $\boxed{x=3}$. c) i) The total time for the journey Step 1: Calculate the time for the first part of the journey. Distance $= 180$ km Speed $= 60$ km/h Time $= \frac{\text{Distance}}{\text{Speed}}$ $$ \text{Time}_1 = \frac{180 \text{ km}}{60 \text{ km/h}} = 3 \text{ hours} $$ Step 2: Calculate the time for the second part of the journey. Distance $= 150$ km Speed $= 50$ km/h $$ \text{Time}_2 = \frac{150 \text{ km}}{50 \text{ km/h}} = 3 \text{ hours} $$ Step 3: Calculate the total time for the journey. $$ \text{Total Time} = \text{Time}_1 + \text{Time}_2 = 3 \text{ hours} + 3 \text{ hours} = 6 \text{ hours} $$ The total time for the journey is $\boxed{6 \text{ hours}}$. ii) The average speed for the entire journey Step 1: Calculate the total distance for the journey. $$ \text{Total Distance} = 180 \text{ km} + 150 \text{ km} = 330 \text{ km} $$ Step 2: Calculate the average speed. $$ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} $$ $$ \text{Average Speed} = \frac{330 \text{ km}}{6 \text{ hours}} = 55 \text{ km/h} $$ The average speed for the entire journey is $\boxed{55 \text{ km/h}}$. a) Step 1: Find the least common multiple (LCM) of the denominators 6, 12, and 4. The LCM of 6, 12, and 4 is 12. Step 2: Convert each fraction to an equivalent fraction with a denominator of 12. $$ \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} $$ $$ \frac{7}{12} $$ $$ \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12} $$ Step 3: Perform the addition and subtraction. $$ \frac{10}{12} + \frac{7}{12} - \frac{3}{12} = \frac{10 + 7 - 3}{12} = \frac{17 - 3}{12} = \frac{14}{12} $$ Step 4: Simplify the fraction. $$ \frac{14}{12} = \frac{14 \div 2}{12 \div 2} = \frac{7}{6} $$ The simplified expression is $\boxed{\frac{7}{6}}$. b) Step 1: Determine the total number of ratio parts. Ratio of boys to girls is $7:8$. Total ratio parts $= 7 + 8 = 15$. Step 2: Calculate the value of one ratio part. Total learners $= 60$. Value of one part $= \frac{\text{Total learners}}{\text{Total ratio parts}} = \frac{60}{15} = 4$. Step 3: Calculate the number of boys. Number of boys $= 7 \times 4 = 28$. Step 4: Calculate the number of girls. Number of girls $= 8 \times 4 = 32$. There are $\boxed{28 \text{ boys}}$ and $\boxed{32 \text{ girls}}$. c) Given: radius $r = 3$ m, height $h = 5$ m, $\pi = 3.142$. i) The volume of the tank Step 1: Use the formula for the volume of a cylinder, $V = \pi r^2 h$. $$ V = 3.142 \times (3 \text{ m})^2 \times 5 \text{ m} $$ $$ V = 3.142 \times 9 \text{ m}^2 \times 5 \text{ m} $$ $$ V = 3.142 \times 45 \text{ m}^3 $$ $$ V = 141.39 \text{ m}^3 $$ The volume of the tank is $\boxed{141.39 \text{ m}^3}$. ii) The surface area of the tank Step 1: Use the formula for the surface area of a closed cylinder, $A = 2\pi r (r+h)$. $$ A = 2 \times 3.142 \times 3 \text{ m} \times (3 \text{ m} + 5 \text{ m}) $$ $$ A = 2 \times 3.142 \times 3 \text{ m} \times 8 \text{ m} $$ $$ A = 6.284 \times 24 \text{ m}^2 $$ $$ A = 150.816 \text{ m}^2 $$ The surface area of the tank is $\boxed{150.816 \text{ m}^2}$. Got more? Send 'em!