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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.

 Aug 17, 2015

Best Answer 

 #1
avatar+26364 
+5

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.

 

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

 

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

 

 Aug 17, 2015
 #1
avatar+26364 
+5
Best Answer

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.

 

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

 

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

 

heureka Aug 17, 2015

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