answer:begin{align*} PFG &= (3+4i)(i)(3-4i) &= i((3+4i)(3-4i)) &= i(3^2 - (4i)^2) &= i(9 + 16) &= i cdot 25 &= boxed{25i}. end{align*}

answer:First, let's calculate the cost of the land and bricks to find out how much is left for the roof tiles. The cost of the land is: 2000 square meters * 50/square meter = 100,000 The cost of the bricks is: 10000 bricks / 1000 bricks per set * 100/set = 1,000 Now, let's add the cost of the land and bricks together: 100,000 (land) + 1,000 (bricks) = 101,000 The total construction costs are 106,000, so the remaining amount for the roof tiles is: 106,000 - 101,000 = 5,000 Now, let's find out how many roof tiles they can buy with the remaining 5,000: 5,000 / 10 per roof tile = 500 roof tiles Therefore, they need boxed{500} roof tiles for their construction project.

answer:Let's denote the time it takes for pipe q to fill the cistern by T minutes. Pipe p can fill the cistern in 12 minutes, so in one minute, it fills 1/12 of the cistern. Pipe q can fill the cistern in T minutes, so in one minute, it fills 1/T of the cistern. When both pipes are opened together, they fill (1/12 + 1/T) of the cistern in one minute. They are both open for 4 minutes, so together they fill 4 * (1/12 + 1/T) of the cistern. After 4 minutes, pipe p is turned off, and pipe q continues to fill the cistern for 6 more minutes. In 6 minutes, pipe q fills 6 * (1/T) of the cistern. The total part of the cistern filled by both pipes when p is turned off is: 4 * (1/12 + 1/T) + 6 * (1/T) = 1 (since the cistern is full) Now, let's solve for T: 4 * (1/12 + 1/T) + 6 * (1/T) = 1 4/T + 1/3 + 6/T = 1 (4 + 6)/T + 1/3 = 1 10/T + 1/3 = 1 Now, let's get a common denominator for the fractions: (10 + T/3)/T = 1 10 + T/3 = T 10T + T^2/3 = T^2 3 * 10T + T^2 = 3T^2 30T + T^2 = 3T^2 T^2 - 30T = 2T^2 T^2 - 30T - 2T^2 = 0 -T^2 - 30T = 0 T^2 + 30T = 0 T(T + 30) = 0 This gives us two solutions: T = 0 or T = -30. Since time cannot be negative, we discard T = -30. Therefore, pipe q alone can fill the cistern in T = boxed{30} minutes.

answer:Let's denote the number of coconut trees as C. We are told that the number of coconut trees is a certain amount less than half the number of mango trees. Half the number of mango trees would be 60 / 2 = 30 trees. We also know that Randy has 85 trees in total on his farm. Since he has 60 mango trees, the rest must be coconut trees. So, C = 85 - 60 = 25 coconut trees. Now, we want to find out how many less coconut trees Randy has compared to half the number of mango trees. Half the number of mango trees is 30, and Randy has 25 coconut trees. So, the difference is 30 - 25 = 5 trees. Randy has boxed{5} less coconut trees compared to half the number of mango trees.