Coleman has \( \frac{2}{5} \) as many candies as Desmond. Roland has twice as many candies as Desmond. If Roland has 360 candies, what is the total number of candies Roland must give to both Coleman and Desmond so that the three children have the same number of candies?

Let's start by determining how many candies Desmond has. Since Roland has 360 candies and he has twice as many as Desmond, we can set up the following equation: \[ \text{Roland's candies} = 2 \times \text{Desmond's candies} \] \[ 360 = 2D \quad \Rightarrow \quad D = \frac{360}{2} = 180 \] So, Desmond has 180 candies. Next, since Coleman has \( \frac{2}{5} \) as many candies as Desmond, we calculate Coleman’s candies: \[ C = \frac{2}{5} \times D = \frac{2}{5} \times 180 = 72 \] Now we have: - Coleman: 72 candies - Desmond: 180 candies - Roland: 360 candies To find the total number of candies that Roland must give to both Coleman and Desmond so that they all have the same number of candies, let’s denote the number of candies Roland gives to Coleman as \( x \) and the number of candies he gives to Desmond as \( y \). After giving away the candies, the amounts will be: - Coleman: \( 72 + x \) - Desmond: \( 180 + y \) - Roland: \( 360 - x - y \) For them to have the same number of candies, we set up the following equation, stating all three amounts are equal: \[ 72 + x = 180 + y = 360 - x - y \] Let’s take the first two equalities: 1. \( 72 + x = 180 + y \) 2. Rearranging gives us \( x - y = 180 - 72 = 108 \) (equation 1) Now, let’s take the second and third equalities: 1. \( 180 + y = 360 - x - y \) 2. Rearranging gives us \( x + 2y = 360 - 180 = 180 \) (equation 2) Now, we can solve these two equations: From equation 1, we have \( x = y + 108 \). Substituting this into equation 2: \[ (y + 108) + 2y = 180 \] \[ 3y + 108 = 180 \] \[ 3y = 180 - 108 \] \[ 3y = 72 \quad \Rightarrow \quad y = 24 \] Now substituting \( y = 24 \) back to find \( x \): \[ x = y + 108 = 24 + 108 = 132 \] Now we know Roland needs to give Coleman \( x = 132 \) candies and Desmond \( y = 24 \) candies. Now, let’s find the total candies Roland gives away: \[ x + y = 132 + 24 = 156 \] Thus, the total number of candies Roland must give to both Coleman and Desmond so that they have the same number of candies is: \[ \boxed{156} \]