In a certain apartment building, there are one-bedroom and two-bedroom apartments. The rental prices of the apartment depend on a number of factors, but on average, two-bedroom apartments have higher rental prices than do one-bedroom apartments. Let R be the average rental price for all apartments in the building. If R is $5,600 higher than the average rental price for all one-bedroom apartments, and if the average rental price for all two-bedroom apartments is $10,400 higher that R, then what percentage of apartments in the building are two-bedroom apartments?
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Is there a quick way to set this up without getting lost in variables? I always struggle with these average mixture problems.
Think of it as a weighted average: if x is the fraction of two-bedroom apartments, then the overall average R sits between the one-bedroom and two-bedroom averages. The differences from R are inversely proportional to the group sizes.
Yeah, just set the one-bedroom average as R - 5600 and two-bedroom as R + 10400. Then the weighted average equation simplifies nicely.
Got 35%? I set up (1-x)(R-5600) + x(R+10400) = R and solved, but my x came out to 0.35. Is that right?
Check your algebra: the R terms cancel, and you get 10400x = 5600(1-x). Solving that gives x = 5600/16000 = 0.35, so 35%.
Yep, that's what I got too. The math works out cleanly.
This is a classic weighted average problem, but the large dollar amounts make it look scarier than it is. Once you realize the overall average is closer to the one-bedroom average, it's clear that one-bedroom apartments must be the majority, so two-bedroom is less than 50%. That eliminates 52% right away.
Why is the answer not 42%? I keep getting confused with the direction of the differences.
The overall average R is $5,600 above the one-bedroom average, but $10,400 below the two-bedroom average. So R is closer to the one-bedroom average, meaning there are more one-bedroom apartments. So the percentage of two-bedroom must be less than 50%, and actually it's less than 39% based on the ratio.
What it tests
Your ability to compute percentage change, percentage of a whole, and to work backwards from a final value.
Common trap
Treating successive percentage changes as additive — a 10% rise then a 10% fall is not back to the start.