In the formula R = x/y^2, x and y are both positive. If x is decreased by 30 percent and...

1. Translate the problem requirements: We need to find how the value of R changes when x decreases by 30% and y increases by 10% in the formula \(\mathrm{R = \frac{x}{y^2}}\)
2. Set up concrete values for easy calculation: Choose simple initial values for x and y that make the arithmetic straightforward
3. Calculate the original R value: Find the baseline value of R using our chosen numbers
4. Apply the percentage changes and find new R: Calculate the new x and y values after the changes, then find the new R
5. Determine the percentage decrease in R: Compare the original and new R values to find the percentage change

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what this problem is asking us to do. We have a formula \(\mathrm{R = \frac{x}{y^2}}\), and we need to see what happens to R when:

• x decreases by 30%
• y increases by 10%

The key insight here is that we're looking for the percentage change in R after these changes occur. This is a classic problem about how changes in input variables affect the output of a formula.

Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding is crucial here

2. Set up concrete values for easy calculation

Instead of working with abstract variables, let's choose specific numbers that make our arithmetic simple. This is much easier than trying to work with percentages algebraically right from the start.

Let's choose:

• x = 100 (makes percentage calculations easy)
• y = 10 (gives us y² = 100, which is also easy to work with)

These values are completely arbitrary - we could choose any positive numbers and get the same percentage result. The beauty of percentage problems is that the specific starting values don't matter for the final percentage change.

3. Calculate the original R value

Using our chosen values in the original formula:

\(\mathrm{R = \frac{x}{y^2}}\)
\(\mathrm{R = \frac{100}{10^2}}\)
\(\mathrm{R = \frac{100}{100}}\)
\(\mathrm{R = 1}\)

So our original value of R is 1. This nice round number will make it easy to calculate percentage changes later.

4. Apply the percentage changes and find new R

Now let's apply the changes described in the problem:

New value of x:
x decreases by 30%, so the new x = 100 - (30% of 100) = 100 - 30 = 70

New value of y:
y increases by 10%, so the new y = 10 + (10% of 10) = 10 + 1 = 11

New value of R:
\(\mathrm{R_{new} = \frac{\text{new x}}{(\text{new y})^2}}\)
\(\mathrm{R_{new} = \frac{70}{11^2}}\)
\(\mathrm{R_{new} = \frac{70}{121}}\)

To make this easier to work with, let's convert this to a decimal:
\(\mathrm{R_{new} = \frac{70}{121} \approx 0.578}\)

Process Skill: SIMPLIFY - Using concrete numbers instead of algebraic expressions makes the calculation much more manageable

5. Determine the percentage decrease in R

Now we can find the percentage change in R:

Original R = 1
New R ≈ 0.578

The decrease in R = 1 - 0.578 = 0.422

Percentage decrease = (Decrease/Original value) × 100%
Percentage decrease = (0.422/1) × 100% = 42.2%

Since we're looking for an approximate answer, this rounds to 42%.

4. Final Answer

The value of R decreases by approximately 42%.

Looking at our answer choices, this matches choice B. 42%.

We can verify this makes sense: when x decreases significantly (30%) and y increases (making y² increase even more due to the squaring), we expect \(\mathrm{R = \frac{x}{y^2}}\) to decrease substantially. A 42% decrease is reasonable given these changes.

Common Faltering Points

Errors while devising the approach

• Misunderstanding the direction of change: Students might get confused about whether they're calculating an increase or decrease in R. Since both x decreases and y increases, and y is in the denominator with a square, students may incorrectly think R could increase rather than recognizing it will definitely decrease.
• Not recognizing the compound effect: Students may try to simply combine the percentage changes (30% decrease + 10% increase = 20% net effect) without understanding that changes in both numerator and denominator create a compound effect, especially with y being squared.
• Overlooking the squaring effect: Students might forget that y appears as y² in the denominator, so a 10% increase in y actually creates a \(\mathrm{(1.1)^2 = 1.21}\) factor, which is a 21% increase in the denominator, not just 10%.

Errors while executing the approach

• Arithmetic errors with fractions: When calculating \(\mathrm{\frac{70}{121}}\), students often make computational errors. They might incorrectly calculate this as \(\mathrm{\frac{70}{121} \approx 0.6}\) instead of the correct 0.578, leading to a wrong percentage decrease.
• Percentage calculation mistakes: Students frequently confuse the percentage decrease formula. They might calculate (New - Original)/New instead of (Original - New)/Original, or forget to multiply by 100 to convert to percentage form.
• Rounding errors: Students might round intermediate steps too aggressively (for example, rounding \(\mathrm{\frac{70}{121}}\) to 0.6 early in the process) rather than keeping more decimal places until the final answer, leading to significant errors in the final percentage.

Errors while selecting the answer

• Selecting based on rough estimation: Students might do a quick mental calculation like "30% down, 10% up, so maybe around 36%" and choose answer A without doing the proper calculation that accounts for the squaring effect.
• Sign confusion: Some students might calculate the percentage change correctly but then select an answer that represents an increase rather than a decrease, or vice versa, especially if they lost track of whether R was increasing or decreasing.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose convenient values for x and y

Let's select x = 100 and y = 10. These numbers are chosen because:

• They make percentage calculations straightforward (30% of 100 = 30)
• They result in clean arithmetic when calculating y² and the formula \(\mathrm{R = \frac{x}{y^2}}\)
• The values are simple enough to minimize calculation errors

Step 2: Calculate the original value of R

\(\mathrm{R = \frac{x}{y^2} = \frac{100}{10^2} = \frac{100}{100} = 1}\)

Step 3: Apply the percentage changes to find new values

• New x: Decreased by 30% → x_new = 100 - (30% of 100) = 100 - 30 = 70
• New y: Increased by 10% → y_new = 10 + (10% of 10) = 10 + 1 = 11

Step 4: Calculate the new value of R

\(\mathrm{R_{new} = \frac{x_{new}}{(y_{new})^2} = \frac{70}{11^2} = \frac{70}{121}}\)

Step 5: Find the percentage decrease in R

Original R = 1 = \(\mathrm{\frac{121}{121}}\)

New R = \(\mathrm{\frac{70}{121}}\)

Decrease in R = \(\mathrm{\frac{121}{121} - \frac{70}{121} = \frac{51}{121}}\)

Percentage decrease = \(\mathrm{\frac{51}{121} \div 1 \times 100\% = \frac{51}{121} \times 100\%}\)

Since \(\mathrm{\frac{51}{121} \approx 0.42}\), the percentage decrease ≈ 42%

Answer: B. 42%