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A rectangle is divided into four small rectangles as shown. The areas of three of the four small rectangles are labeled in the diagram. What is the area of the remaining small rectangle?


 Oct 7, 2018

Okay, lets work this one through. 


The rectangle with area 3 is sharing its length with the rectangle with area 4. Let this value be \(x\)


The rectangle with area 3 is sharing its width with the rectangle with area 5. Let this value be \(y\)


The rectangle with area 4 is sharing its width with the rectangle with unknown area. Let this value be \(z\)


The rectangle with area 5 is sharing its length with the rectangle with unknown area. Let this value be \(w\)


If we set up and equation for each rectangle, we get the following:


Bottom left rectangle: \(xy = 3\)

Bottom right rectangle: \(wy = 5\)

Top left rectangle: \(xz = 4\)

Top right rectangle: \(wz = ?\)


First off, I am going to solve for the \(x\) in the first equation. We get \(x = \frac{3}{y}\).


Next, I am going to use the second equation to solve for \(y\). We get \(y = \frac{5}{w}\).


I am going to use the second variable we solved for and plug it into the first, like this: \(x = \frac{3}{\frac{5}{w}}\)

If we simplify this eqution, we get: \(x = 3\div \frac{5}{w} = 3(\frac{w}{5}) = \frac{3w}{5}\)


At this point, we know we're getting close to the answer, but we're not done yet.


If we use the value we just got for \(x\) and plug it into the third equation, look what we get:


\(xz = 4 , (\frac{3w}{5})(z) = 4 , (3w)(z) = 20 , wz = \frac{20}{3}\)


Wait...isn't \(wz\) the unknown area?


Therefore, the answer is \(\boxed {\frac{20}{3}}\), or \(\boxed {6.\overline {6}}\)

 Oct 8, 2018

Excellent, Knockout  !!!!


BTW....welcome aboard    !!!!


cool cool cool

CPhill  Oct 8, 2018

CPhill is right. This is excellent. Not your math, but your amazing skills for shoving 4 variables of bullshitt in a 1 variable bag.

If you were in Texas and wanted to go to Mexico, you first go to Russia, and then to Hong Kong, then swing over to equatorial Africa then to Mexico.  That is the scenic route.  

For the direct rout, just divide 4/3 and multiply by 5. Or divide 5/3 and multiply by 4.  

Guest Oct 8, 2018

Thank you so much Knock Out and Guest for helping me

 Oct 8, 2018

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