1. Bala had a total of 420 red and green apples in the ratio \( 4: 3 \). After he sold an equal number of red and green apples, the number of red and green apples left was in the ratio \( 8: 5 \). How many apples did Bala sell altogether?

Let the number of red apples be \( 4x \) and the number of green apples be \( 3x \). According to the problem, the total number of apples is given by: \[ 4x + 3x = 420 \] This simplifies to: \[ 7x = 420 \] Solving for \( x \): \[ x = \frac{420}{7} = 60 \] Thus, the number of red apples is: \[ 4x = 4 \times 60 = 240 \] And the number of green apples is: \[ 3x = 3 \times 60 = 180 \] Let \( y \) be the number of red and green apples that Bala sold. After selling \( y \) apples of each type, the number of red apples left is: \[ 240 - y \] And the number of green apples left is: \[ 180 - y \] According to the problem, the ratio of the remaining red apples to the remaining green apples is \( 8:5 \). Therefore, we can set up the equation: \[ \frac{240 - y}{180 - y} = \frac{8}{5} \] Cross-multiplying gives: \[ 5(240 - y) = 8(180 - y) \] Expanding both sides: \[ 1200 - 5y = 1440 - 8y \] Rearranging the equation to isolate \( y \): \[ 1200 + 8y = 1440 + 5y \] \[ 8y - 5y = 1440 - 1200 \] \[ 3y = 240 \] Dividing by 3: \[ y = 80 \] Thus, the total number of apples Bala sold altogether is: \[ 2y = 2 \times 80 = 160 \] Therefore, the total number of apples Bala sold is: \[ \boxed{160} \]