June had twice as many apples as oranges. Ken had 1/4 as many oranges as apples. The number of oranges Ken had was 2/9 as many as the number

June had twice as many apples as oranges. Ken had 1/4 as many oranges as apples. The number of oranges Ken had was 2/9 as many as the number of apples June had. Ken had 120 apples and oranges.

(a) express the number of oranges June had as a fraction of the number of apples Ken had. Give you answers in its SIMPLEST FORM.

$$\small{ \begin{array}{lrcl} & O_1 &=& \text{ oranges June }\\ & A_1 &=& \text{ apples June }\\\\ & O_2 &=& \text{ oranges Ken }\\ & A_2 &=& \text{ apples Ken }\\\\ (1) & A_1 &=& 2\cdot O_1\\ (2) & O_2 &=& \frac14 \cdot A_2 \\\\ (3) & O_2 &=& \frac29\cdot A_1 \\\\ (4) & 120 &=& O_2 + A_2\\\\\\ (1), (2), (3) & \frac14 \cdot A_2 &=& \frac29 \cdot A_1 \quad |\quad A_1 = 2\cdot O_1 \\\\ & \frac14 \cdot A_2 &=& \frac29 \cdot 2 \cdot O_1 \\\\ & \frac14 \cdot A_2 &=& \frac49 \cdot O_1 \\\\ & O_1 &=& \frac14 \cdot \frac94 \cdot A_2 \\\\ & \mathbf{O_1} & \mathbf{=} & \mathbf{ \frac9{16} \cdot A_2 }\\\\ \end{array} }$$

The number of oranges June had is 9/16 of the number of apples Ken had.

$$\small{ \begin{array}{lrcl} (A_2?) & 120 &=& O_2 + A_2 \qquad | \qquad O_2 = \frac14 \cdot A_2\\\\ & 120 &=& \frac14 \cdot A_2 + A_2\\\\ &120 &=& \frac54 \cdot A_2\\\\ & A_2 &=& 120 \cdot \frac45 \\\\ & \mathbf{A_2} & \mathbf{=} & \mathbf{96 }\\\\\\ (O_1?) &O_1 &=& \frac9{16}\cdot A_2 \\\\ & O_1 &=& \frac9{16} \cdot 96 \\\\ & \mathbf{O_1} & \mathbf{=} & \mathbf{54 }\\\\\\ (A_1?) &A_1 &=& 2\cdot O_1\\\\ & A_1 &=&2 \cdot 54\\\\ & \mathbf{A_1} & \mathbf{=} & \mathbf{ 108 }\\\\\\ (O_2?) &O_2 &=& \frac14\cdot A_2 \\\\ & O_2 &=& \frac14 \cdot 96\\\\ & \mathbf{O_2} & \mathbf{=} & \mathbf{ 24 }\\\\\\ \end{array} }$$

$$\small{ \begin{array}{|r|r|r|} \hline \text {oranges} & \text{apples} & \\ \hline 54 & 108 & \text{June} \\ \hline 24 & 96 &\text{Ken}\\ \hline \end{array} }$$