A rectangular box of laundry detergent was redesigned so that its height was increased by 10%, its length was decreased...

1. Translate the problem requirements: We need to find how the volume of a rectangular box changes when height increases by 10%, length decreases by 10%, and width increases by 1%. Since \(\mathrm{volume = length \times width \times height}\), we need to determine the combined effect of these three percentage changes.
2. Set up the original and new dimensions: Use simple variables to represent the original dimensions and express the new dimensions in terms of percentage multipliers.
3. Calculate the volume change factor: Multiply the three percentage change factors together to find the overall volume change, since volume depends on all three dimensions multiplicatively.
4. Convert to percentage change: Transform the volume change factor into a percentage decrease or increase to match the answer format.

Execution of Strategic Approach

1. Translate the problem requirements

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:

• Height goes UP by 10% (the box gets taller)
• Length goes DOWN by 10% (the box gets shorter from front to back)
• Width goes UP by 1% (the box gets slightly wider)

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 understanding

2. Set up the original and new dimensions

Let's use simple variables to represent our starting point. Say the original box has:

• Length = L
• Width = W
• Height = H

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:

• New length = \(\mathrm{L \times 0.90}\) (decreased by 10%)
• New width = \(\mathrm{W \times 1.01}\) (increased by 1%)
• New height = \(\mathrm{H \times 1.10}\) (increased by 10%)

The new volume is: \(\mathrm{(L \times 0.90) \times (W \times 1.01) \times (H \times 1.10)}\)

3. Calculate the volume change factor

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 steps

4. Convert to percentage change

When 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%.

5. Final Answer

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.

Common Faltering Points

Errors while devising the approach

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).

Errors while executing the approach

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%.

Errors while selecting the answer

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.

Alternate Solutions

Smart Numbers Approach

For this volume problem, we can use smart numbers by choosing convenient initial dimensions that make calculations easy.

Step 1: Choose Smart Initial Dimensions

Let's set the original box dimensions to values that will make percentage calculations clean:

• Original length = 100 units
• Original width = 100 units
• Original height = 100 units

These numbers are chosen because they make percentage calculations straightforward (10% of 100 = 10, 1% of 100 = 1).

Step 2: Calculate New Dimensions

Apply the given percentage changes:

• New height = \(\mathrm{100 + 10\% \ of \ 100 = 100 + 10 = 110}\) units
• New length = \(\mathrm{100 - 10\% \ of \ 100 = 100 - 10 = 90}\) units
• New width = \(\mathrm{100 + 1\% \ of \ 100 = 100 + 1 = 101}\) units

Step 3: Calculate Original and New Volumes

• Original volume = \(\mathrm{100 \times 100 \times 100 = 1,000,000}\) cubic units
• New volume = \(\mathrm{90 \times 101 \times 110 = 999,900}\) cubic units

Step 4: Find the Percentage Change

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 = \(\mathrm{\frac{-100}{1,000,000} \times 100\% = -0.01\%}\)

Step 5: Interpret the Result

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.