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Natansha
Natansha
Here is my explanation (if this helps!)-
Quick Weighted Average Formula - The Distance Method:
In any weighted average problem, the weights are inversely proportional to the distances from the average.
Step 1: Identify the Components and Average
- Cars: -11% change
- Trucks: +7% change
- Overall average: +1% change
Step 2: Calculate Distances from Average
- Distance of cars from average: \(|-11 - 1| = 12\)
- Distance of trucks from average: \(|7 - 1| = 6\)
Step 3: Apply the Inverse Proportion Rule The weights are inversely proportional to distances:
- Ratio of weights = \(\frac{1}{12} : \frac{1}{6} = 1\)
Since the weights represent the proportion of revenues in 1996: Answer: Ratio of car revenue to truck revenue in 1996 = \(1\\
\)
Quick Verification Method:
To verify your weights are correct, check if they produce the given average: \(\frac{1}{3} \times (-11 \%)\) + \(\frac{2}{3} \times (7 \% )\) = \(-3.67 \% + 4.67 \% \)= \(1 \% \) ✓
Strategic Tips for Weighted Averages:
Recognition Pattern: Look for "overall/combined/total" language with different component change
Quick Mental Check: The average must lie between the extremes (-11% and +7%)
Distance Rule: Component closer to average has MORE weight
Verification Shortcut: If weights don't sum to 1 or don't produce the stated average, recalculate
Alternative Quick Method - The Alligation Cross: Draw a cross with the average (1%) in center, components on left (-11%, 7%), and find differences:
- Top difference: \(7 - 1 = 6\) (becomes weight of cars)
- Bottom difference: \(1 - (-11) = 12\) (becomes weight of trucks)
- Ratio = \(6= 1\)
Remember: In GMAT weighted averages, you're essentially finding "how much" of each component creates the final mix. The component farther from the average needs less representation to pull the average its way!
Let me know if you have any follow-up questions to this!
