(4) There are blue, yellow and red marbles in a box. \( \frac{3}{8} \) of the marbles are blue and \( \frac{9}{10} \) of the remaining marbles are yellow. (a) What fraction of the marbles are yellow? (b) What fraction of the marbles are red?

To solve this problem, let \( x \) represent the total number of marbles in the box. First, we know that \( \frac{3}{8} \) of the marbles are blue. This means the number of blue marbles is \( \frac{3}{8}x \). The remaining marbles are \( x - \frac{3}{8}x = \frac{5}{8}x \). Now, we are told that \( \frac{9}{10} \) of the remaining marbles are yellow. So the number of yellow marbles is: \[ \frac{9}{10} \cdot \frac{5}{8}x = \frac{9 \times 5}{10 \times 8}x = \frac{45}{80}x = \frac{9}{16}x \] To find the fraction of the total marbles that are yellow: \[ \text{Fraction of yellow marbles} = \frac{\text{Number of yellow marbles}}{x} = \frac{9}{16} \] Next, to find the fraction of marbles that are red, we first determine the total number of yellow and blue marbles: - Number of blue marbles: \( \frac{3}{8}x \) - Number of yellow marbles: \( \frac{9}{16}x \) To find the number of red marbles, we subtract the blue and yellow marbles from the total \( x \): First, we convert \( \frac{3}{8} \) into a fraction with a denominator of 16 (the same as \( \frac{9}{16} \)): \[ \frac{3}{8} = \frac{6}{16} \] Now we add the fractions of blue and yellow marbles: \[ \text{Total of blue and yellow marbles} = \frac{6}{16}x + \frac{9}{16}x = \frac{15}{16}x \] Thus, the remaining marbles that are red are: \[ \text{Number of red marbles} = x - \frac{15}{16}x = \frac{1}{16}x \] To find the fraction of the total marbles that are red: \[ \text{Fraction of red marbles} = \frac{\text{Number of red marbles}}{x} = \frac{1}{16} \] So, in summary: (a) The fraction of the marbles that are yellow is \( \frac{9}{16} \). (b) The fraction of the marbles that are red is \( \frac{1}{16} \).