\( 4.1 .1 \quad \frac { 9 } { 15 } + \frac { 2 } { 3 } - 2 \frac { 5 } { 5 } \)

Let's break this down step by step for clarity and fun! First off, we have the fractions. To make our life easier, let's convert everything to have a common denominator. Start with \( \frac{9}{15} \) which is already in proper form, while \( \frac{2}{3} \) can be converted to \( \frac{10}{15} \) (by multiplying both the numerator and the denominator by 5). Now tackling \( 2 \frac{5}{5} \), remember that \( \frac{5}{5} \) is just 1, so \( 2 \frac{5}{5} = 2 \). Putting it all together, you have: \[ \frac{9}{15} + \frac{10}{15} - 2 \] That simplifies to \( \frac{19}{15} - 2 \). Since 2 can be expressed as \( \frac{30}{15} \), now it’s easy! You simply subtract: \[ \frac{19}{15} - \frac{30}{15} = \frac{19 - 30}{15} = \frac{-11}{15} \] And there you have it, your final answer is \( \frac{-11}{15} \). Mathematics may be serious business, but breaking down problems can make it a whole lot more fun! Keep adding and subtracting with joy!