Math〈4.4a〉Alligation – clarifying confusion

The link to our Math Center is here: grecenter.org/math

I already explained how to use alligation method for solving GRE-GMAT math problems in my Bangla video in Youtube. The video is once again embedded here:

Which number to choose while multiplying fractions?

This refers to the discussion at 9.01 minute where I chose a desired value 9.25 from two initial values of 12 and 4. It looked like this:

\begin{matrix} 12 & - & 4 \\ - & 9.25 & - \\ ? & - & ? \end{matrix}[latex] If you look closely then you'll find that this can be written as: [latex]\begin{matrix} 12 & - & 4 \\ - & \frac { 925 }{ 100 } & - \\ ? & - & ? \end{matrix}[latex] Now, we need to multiply all the numbers such a way that we get integers. Remember, this multiplication is performed only in order to simplify it. You can still go ahead by subtracting 9.25 from 12 and get the same result. After multiplying we get: [latex]\begin{matrix} 1200 & - & 400 \\ - & \frac { 925 }{ 100 } \times 100 & - \\ ? & - & ? \end{matrix}[latex] , which means [latex]\begin{matrix} 1200 & - & 400 \\ - & 925 & - \\ ? & - & ? \end{matrix}[latex] Which gives us the following answer: [latex]\begin{matrix} 1200 & - & 400 \\ - & 925 & - \\ 525 & - & 275 \end{matrix}[latex] In a graphical representation: <iframe style="border: 0px;" src="https://tube.geogebra.org/material/iframe/id/878039/width/600/height/600/border/888888/rc/false/ai/false/sdz/true/smb/false/stb/false/stbh/true/ld/false/sri/true/at/auto" width="600px" height="600px" scrolling="no"> </iframe> This means, the 12 and 4 concentrations should have mixed in [latex]\frac { 525 }{ 275 }[latex] ratio, which equals [latex]\frac{21}{11}[latex] Please move to the 10.23 minute's discussion in the video and see that the desired concentration had a denominator of 2. For making all numbers integer, I multiplied all of them with 2. Got it? As a rule of thumb, if there's a situation like this: [latex]\begin{matrix} \frac { a }{ b} & - & \frac { c }{ d} \\ - & \frac { x }{ y } & - \\ ? & - & ? \end{matrix}[latex] then you choose a number which is LCM of all the three denominators, b, d and y. So, after multiplying all the fractions with [latex]b\times d\times y[latex] we get: [latex]\begin{matrix} \frac { a }{ b } \times bdy & - & \frac { c }{ d } \times bdy \\ - & \frac { x }{ y } \times bdy & - \\ ? & - & ? \end{matrix}[latex] And our formula is simplified as: [latex]\begin{matrix} ady & - & cby \\ - & xbd & - \\ ? & - & ? \end{matrix}

And we determine our numbers as:

\(\)\begin{matrix} ady & – & cby \\ – & xbd& – \\ \left| cby-xbd \right| & – & \left| ady-xbd\right| \end{matrix}\(\)

Hope I could clarify this specific question. If you still have any confusion please comment below.

Sincerely,

Mamoon Rashid, PhD
CEO of GRE Center