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A rectangular box of laundry detergent was redesigned so that its height was increased by \(10\%\), its length was decreased by \(10\%\), and its width was increased by \(1\%\). What was the resulting percentage change in the volume of the box?
Let's break down what the problem is asking in everyday terms. We have a rectangular box - think of a cereal box or detergent container. This box has three dimensions: length, width, and height. The company decided to redesign the box by changing all three dimensions:
We need to figure out how these changes affect the total volume (how much stuff the box can hold). Since \(\mathrm{volume = length \times width \times height}\), when we change all three dimensions, we need to see the combined effect.
Process Skill: TRANSLATE - Converting the percentage changes into mathematical understandingLet's use simple variables to represent our starting point. Say the original box has:
The original volume is: \(\mathrm{L \times W \times H}\)
Now, after the redesign, each dimension changes by a specific percentage. When something increases by 10%, we multiply by 1.10. When something decreases by 10%, we multiply by 0.90. When something increases by 1%, we multiply by 1.01.
So our new dimensions are:
The new volume is: \(\mathrm{(L \times 0.90) \times (W \times 1.01) \times (H \times 1.10)}\)
Here's where it gets interesting. Since volume depends on all three dimensions multiplied together, the percentage changes also get multiplied together.
New volume = \(\mathrm{L \times W \times H \times (0.90 \times 1.01 \times 1.10)}\)
The original volume was \(\mathrm{L \times W \times H}\), so the volume change factor is:
Change factor = \(\mathrm{0.90 \times 1.01 \times 1.10}\)
Let's calculate this step by step to keep the arithmetic simple:
First: \(\mathrm{0.90 \times 1.01 = 0.909}\)
Then: \(\mathrm{0.909 \times 1.10 = 0.9999}\)
So the new volume is 0.9999 times the original volume.
Process Skill: SIMPLIFY - Breaking down the calculation into manageable stepsWhen we have a change factor of 0.9999, this means the new volume is 99.99% of the original volume.
To find the percentage change:
Percentage change = \(\mathrm{\frac{New\ value - Original\ value}{Original\ value} \times 100\%}\)
Since the change factor is 0.9999:
Percentage change = \(\mathrm{(0.9999 - 1) \times 100\% = -0.0001 \times 100\% = -0.01\%}\)
The negative sign means it's a decrease, so the volume decreased by 0.01%.
The volume of the box decreased by 0.01%. Looking at our answer choices, this matches option (B) 0.01% less.
This makes intuitive sense: the 10% decrease in length had a bigger impact than the small increases in width (1%) and height (10%), resulting in a tiny overall decrease in volume.
1. Misinterpreting percentage changes: Students often confuse which direction each change goes. They might read "decreased by 10%" and incorrectly multiply by 1.10 instead of 0.90, or read "increased by 10%" and multiply by 0.10 instead of 1.10. This fundamental error in converting percentages to multiplication factors will lead to completely wrong calculations.
2. Treating percentage changes as additive instead of multiplicative: A common mistake is thinking that since height increases 10%, length decreases 10%, and width increases 1%, the net change is 10% - 10% + 1% = 1% increase. Students fail to recognize that volume changes require multiplying the individual change factors together, not adding the percentage changes.
3. Setting up the wrong formula: Some students might try to calculate each dimension's new volume separately instead of recognizing that \(\mathrm{volume = length \times width \times height}\), so the new volume equals original volume \(\mathrm{\times}\) (all change factors multiplied together).
1. Arithmetic mistakes in decimal multiplication: The calculation \(\mathrm{0.90 \times 1.01 \times 1.10}\) involves careful decimal arithmetic. Students commonly make errors like getting \(\mathrm{0.90 \times 1.01 = 0.91}\) instead of 0.909, or miscalculating the final product as 1.0099 instead of 0.9999.
2. Rounding errors or premature rounding: Students might round intermediate steps (like rounding 0.909 to 0.91) which can significantly affect the final answer in a problem where the correct answer is a very small percentage change of 0.01%.
1. Sign confusion in percentage change: After correctly calculating that the change factor is 0.9999, students might forget that this represents a decrease (since it's less than 1) and incorrectly select "0.01% greater" instead of "0.01% less".
2. Misinterpreting the magnitude: Students who get a change factor of 0.9999 might incorrectly think this means a 0.9999% change or a 99.99% change, rather than correctly calculating it as a 0.01% decrease from the original volume.
For this volume problem, we can use smart numbers by choosing convenient initial dimensions that make calculations easy.
Let's set the original box dimensions to values that will make percentage calculations clean:
These numbers are chosen because they make percentage calculations straightforward (10% of 100 = 10, 1% of 100 = 1).
Apply the given percentage changes:
Volume change = \(\mathrm{New \ volume - Original \ volume = 999,900 - 1,000,000 = -100}\) cubic units
Percentage change = \(\mathrm{\frac{Volume \ change}{Original \ volume} \times 100\%}\)
Percentage change = \(\mathrm{\frac{-100}{1,000,000} \times 100\% = -0.01\%}\)
The negative sign indicates a decrease, so the volume decreased by 0.01%.
Therefore, the answer is (B) 0.01% less.
Why these smart numbers work: By choosing 100 as each dimension, we make percentage calculations trivial while maintaining the mathematical relationships. The choice is logical because it simplifies arithmetic without affecting the underlying percentage relationships that govern the problem.