M06-36

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Hi Banuel - why did we not account for the '6' in the denominator?

The '6' in the denominator acts as a constant factor in the original equation \(\frac{MT}{6}=W\). It remains the same, regardless of any changes in the individual commissions of Mike, Tom, and Walt. Since it serves as a scaling factor, it does not impact the relationship between the agents' commissions.

When calculating the new product of their commissions for the current month, there is no need to account for the '6' in the denominator. It remains a constant factor and does not affect the calculation. We only need to focus on the changes in Mike's and Tom's individual commissions and how these changes influence the overall product of the three commissions.

If you try plugging numbers into the equation, you will see how the '6' in the denominator does not impact the relationship between the agents' commissions. Let's assume the following values for M, T, and W in the last month. Using M = 30, T = 10, and W = 50, the original equation holds true. After adjusting the commissions:

Mike's commission: 1.6M = 48
Tom's commission: 0.5T = 5

The product of their commissions (240) decreased by 20%. To maintain the relationship, we decrease Walt's commission by 20%:

New W = 0.8(50) = 40

The updated equation holds true: \(\frac{(48)(5)}{6} = 40\).

Hope it's clear.

For "lovely" questions like these, that fry your brain in 2 mins, I usually don't trust myself to do the logic/calculation since I often miss steps under pressure, so I prefer to plugin numbers and just handle it that way. It makes things easier to digest. This can also be a way to resolve your concern about the 6 in the denominator.

We can take M=10, T=30 and W will be 50 since MT/6 = W

Then after the changes
M = 16
T= 15
MT/6 = 40

This means W decreased by 20% (from 50 to 40).
Does six matter? Maybe, I don’t know, I already moved on to the next question 😎 (good to ask these things in practice and make sure you understand things but he also need to pick your battles)

P.S. While perhaps this is not the way that "real GMAT giants" solve questions, it is very reliable as long as you don't fumble with the numbers but that's something you will have to watch out for with all questions so that's a diff story.