If x and y are positive numbers and z = xy^2, a 50 percent increase in x and a 20...

1. Translate the problem requirements: We need to find how \(\mathrm{z = xy^2}\) changes when x increases by 50% and y decreases by 20%. This means finding the percent change in the new value of z compared to the original.
2. Express the changes in terms of the original variables: Convert the percentage changes into algebraic expressions - new x becomes \(\mathrm{1.5x}\) and new y becomes \(\mathrm{0.8y}\).
3. Calculate the new value of z: Substitute the changed values into the original formula \(\mathrm{z = xy^2}\) to find the new expression.
4. Determine the percent change: Compare the new z value to the original z value to find the percentage change.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're being asked to find. We have a formula \(\mathrm{z = xy^2}\) where x and y are positive numbers. We need to see what happens to z when:

• x increases by 50%
• y decreases by 20%

Think of this like adjusting the settings on a machine - we're changing two inputs and want to see how the output changes. The key question is: will z go up or down, and by how much?

2. Express the changes in terms of the original variables

Let's convert these percentage changes into simple mathematical terms:

When x increases by 50%, the new value becomes the original plus 50% more. In everyday terms, if you had $100 and got a 50% raise, you'd have $150. So:

• New x = \(\mathrm{x + 0.50x = 1.5x}\)

When y decreases by 20%, the new value becomes the original minus 20%. If you had $100 and lost 20%, you'd have $80. So:

• New y = \(\mathrm{y - 0.20y = 0.8y}\)

Now we can work with these concrete expressions: x becomes \(\mathrm{1.5x}\) and y becomes \(\mathrm{0.8y}\)

3. Calculate the new value of z

Originally, we had \(\mathrm{z = xy^2}\). Now let's substitute our new values:

New z = \(\mathrm{(new\,x) \times (new\,y)^2}\)

New z = \(\mathrm{(1.5x) \times (0.8y)^2}\)

Let's calculate \(\mathrm{(0.8y)^2}\) step by step:

\(\mathrm{(0.8y)^2 = 0.8^2 \times y^2 = 0.64y^2}\)

So: New z = \(\mathrm{1.5x \times 0.64y^2}\)

New z = \(\mathrm{1.5 \times 0.64 \times xy^2}\)

New z = \(\mathrm{0.96 \times xy^2}\)

Since the original \(\mathrm{z = xy^2}\), we can write:

New z = \(\mathrm{0.96z}\)

4. Determine the percent change

Now we compare the new value to the original value. We found that:

• Original z = z
• New z = \(\mathrm{0.96z}\)

This means the new value is 96% of the original value. Since 96% is less than 100%, z has decreased.

The decrease is: 100% - 96% = 4%

We can verify this using the percent change formula:

Percent change = \(\mathrm{\frac{New\,value - Original\,value}{Original\,value} \times 100\%}\)

Percent change = \(\mathrm{\frac{0.96z - z}{z} \times 100\% = \frac{-0.04z}{z} \times 100\% = -4\%}\)

The negative sign indicates a decrease of 4%.

5. Final Answer

The changes result in a decrease of 4% in z.

Looking at our answer choices, this matches option (A) A decrease of 4%.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting percentage changes
Students often confuse what "50% increase" and "20% decrease" mean mathematically. They might think a 50% increase means multiplying by 0.5 instead of 1.5, or a 20% decrease means multiplying by 0.2 instead of 0.8. This fundamental misunderstanding of percentage changes will lead to completely wrong calculations.

2. Not recognizing the compound effect
Students may try to simply add the percentage changes (50% - 20% = 30% increase) without realizing that changes to different variables in a formula like \(\mathrm{z = xy^2}\) don't combine linearly. They might miss that y appears squared in the formula, making its impact on z more significant than x's impact.

Errors while executing the approach

1. Arithmetic errors when squaring
When calculating \(\mathrm{(0.8y)^2}\), students commonly make the error of getting \(\mathrm{0.8y^2}\) instead of \(\mathrm{0.64y^2}\). They forget to square the coefficient 0.8, treating \(\mathrm{(0.8y)^2}\) as \(\mathrm{0.8 \times y^2}\) rather than \(\mathrm{0.8^2 \times y^2 = 0.64y^2}\).

2. Multiplication errors with decimals
When multiplying \(\mathrm{1.5 \times 0.64}\), students might incorrectly calculate this as 0.94 or 1.04 instead of the correct 0.96. These decimal multiplication errors are common under time pressure and lead to selecting the wrong final answer.

Errors while selecting the answer

1. Sign confusion in percent change
After correctly calculating that the new \(\mathrm{z = 0.96z}\), students might interpret this as a 96% increase rather than recognizing it represents 96% of the original value, which means a 4% decrease. This leads them to select "increase of 4%" instead of "decrease of 4%."

Alternate Solutions

Smart Numbers Approach

Step 1: Choose convenient values for x and y

Let's select \(\mathrm{x = 10}\) and \(\mathrm{y = 10}\). These are simple, positive numbers that make percentage calculations straightforward.

Step 2: Calculate the original value of z

\(\mathrm{z = xy^2 = 10 \times (10)^2 = 10 \times 100 = 1,000}\)

Step 3: Apply the percentage changes

• New x after 50% increase: \(\mathrm{x_{new} = 10 + (0.50 \times 10) = 10 + 5 = 15}\)
• New y after 20% decrease: \(\mathrm{y_{new} = 10 - (0.20 \times 10) = 10 - 2 = 8}\)

Step 4: Calculate the new value of z

\(\mathrm{z_{new} = x_{new} \times (y_{new})^2 = 15 \times (8)^2 = 15 \times 64 = 960}\)

Step 5: Find the percent change in z

Percent change = \(\mathrm{\frac{(z_{new} - z_{original})}{z_{original}} \times 100\%}\)

Percent change = \(\mathrm{\frac{(960 - 1,000)}{1,000} \times 100\% = \frac{-40}{1,000} \times 100\% = -4\%}\)

Conclusion: This represents a decrease of 4%.

Why this approach works: The beauty of this smart numbers method is that percentage changes are independent of the specific values we choose for x and y. As long as we pick positive numbers and apply the same percentage changes, we'll always get the same final percentage change in z.