Last year Country X generated 3 times as many kilowatt-hours of electricity as Country Y, but both countries generated the...

1. Translate the problem requirements: Country X generates \(3\) times the total electricity of Country Y, but both generate equal amounts from nuclear power. We need to find what percent of Country Y's electricity comes from nuclear power, given that \(\mathrm{k}\) percent of Country X's electricity comes from nuclear power.
2. Set up the relationship using concrete variables: Assign simple variables to represent the total electricity generation for each country to establish clear proportional relationships.
3. Express nuclear power generation for both countries: Use the given information that both countries generate the same amount of nuclear power to create an equation.
4. Solve for Country Y's nuclear percentage: Use the equal nuclear power generation to find what percentage this represents of Country Y's total electricity generation.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what the problem is telling us in plain English:

• Country X generates \(3\) times as much total electricity as Country Y
• Both countries generate the exact same amount of electricity from nuclear power plants
• Country X gets \(\mathrm{k}\) percent of its electricity from nuclear power
• We need to find what percent of Country Y's electricity comes from nuclear power

Think of it this way: if Country X has a much bigger "electricity pie" than Country Y, but they both have the same size "nuclear slice," then the nuclear slice must represent a bigger percentage of Country Y's smaller pie.

Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships

2. Set up the relationship using concrete variables

Let's use simple variables to make this concrete:

Let's say Country Y generates Y kilowatt-hours of electricity total.
Then Country X generates \(\mathrm{3Y}\) kilowatt-hours of electricity total (since it's \(3\) times as much).

This gives us a clear picture: if Country Y's electricity generation is like a small circle, then Country X's is like a circle three times as big.

3. Express nuclear power generation for both countries

Now let's figure out how much nuclear power each country generates:

Country X generates \(\mathrm{k}\) percent of its electricity from nuclear power.
So Country X's nuclear generation = \(\mathrm{k\% \text{ of } 3Y = \frac{k}{100} \times 3Y}\)

Let's call the percentage we're looking for "p" - this is the percent of Country Y's electricity that comes from nuclear power.
So Country Y's nuclear generation = \(\mathrm{p\% \text{ of } Y = \frac{p}{100} \times Y}\)

Since both countries generate the same amount of nuclear power:
\(\mathrm{\frac{k}{100} \times 3Y = \frac{p}{100} \times Y}\)

4. Solve for Country Y's nuclear percentage

Now we solve for p using our equation:

\(\mathrm{\frac{k}{100} \times 3Y = \frac{p}{100} \times Y}\)

Divide both sides by Y:
\(\mathrm{\frac{k}{100} \times 3 = \frac{p}{100}}\)

Multiply both sides by \(100\):
\(\mathrm{3k = p}\)

Therefore, Country Y generates \(\mathrm{3k}\) percent of its electricity from nuclear power.

This makes intuitive sense: Country Y has a much smaller total electricity generation, so the same amount of nuclear power represents a much larger percentage of its total.

4. Final Answer

Country Y generates \(\mathrm{3k}\) percent of its electricity from nuclear power plants.

Looking at our answer choices, this matches choice (E) \(\mathrm{3k}\).

Quick verification: If Country X generates \(3\) times as much total electricity but the same nuclear amount, then nuclear power must represent \(3\) times the percentage for the smaller country (Country Y). Since Country X has \(\mathrm{k\%}\), Country Y must have \(\mathrm{3k\%}\).

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the relationship between countries
Students often confuse which country generates more electricity. They might incorrectly assume Country Y generates \(3\) times as much as Country X, reversing the given relationship. This fundamental misreading leads to setting up equations with the wrong multiplier.

2. Overlooking the "same nuclear generation" constraint
Many students focus only on the total electricity relationship and miss the critical constraint that both countries generate the same absolute amount of nuclear power. They might incorrectly assume Country X also generates \(3\) times as much nuclear power, leading to a completely wrong approach.

3. Confusing percentages with absolute amounts
Students frequently struggle to distinguish between percentage of electricity from nuclear power versus absolute amount of nuclear electricity generated. They might try to directly compare the percentages (\(\mathrm{k\%}\) vs the unknown percentage) without accounting for the different total electricity generation levels.

Errors while executing the approach

1. Algebraic manipulation errors
When solving the equation \(\mathrm{\frac{k}{100} \times 3Y = \frac{p}{100} \times Y}\), students often make mistakes when canceling terms or dividing both sides by Y. They might forget to cancel Y properly or make errors when multiplying/dividing by \(100\).

2. Incorrect variable substitution
Students may correctly set up variables but then substitute them incorrectly in equations. For example, they might write Country X's nuclear generation as \(\mathrm{\frac{k}{100} \times Y}\) instead of \(\mathrm{\frac{k}{100} \times 3Y}\), forgetting that Country X generates \(\mathrm{3Y}\) total electricity.

Errors while selecting the answer

1. Selecting the reciprocal relationship
After correctly calculating that the answer is \(\mathrm{3k}\), some students second-guess themselves and think "Country Y is smaller, so its percentage should be smaller too." This leads them to incorrectly choose (A) \(\mathrm{\frac{k}{3}}\) instead of (E) \(\mathrm{3k}\), missing the key insight that a smaller denominator actually creates a larger percentage.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose convenient values for total electricity generation

Since Country X generates \(3\) times as much electricity as Country Y, let's assign:

• Country Y's total electricity = \(100\) kilowatt-hours
• Country X's total electricity = \(3 \times 100 = 300\) kilowatt-hours

These numbers are chosen because they make the \(3:1\) ratio clear and percentages easy to calculate.

Step 2: Choose a specific value for k

Let's say \(\mathrm{k = 20}\), meaning Country X generates \(20\%\) of its electricity from nuclear power.

Step 3: Calculate nuclear power generation for Country X

Country X's nuclear power = \(20\% \text{ of } 300 = 0.20 \times 300 = 60\) kilowatt-hours

Step 4: Use the fact that both countries generate equal nuclear power

Since both countries generate the same amount from nuclear power:

Country Y's nuclear power = \(60\) kilowatt-hours (same as Country X)

Step 5: Calculate what percentage this represents for Country Y

Country Y's nuclear percentage = \(\mathrm{\frac{60}{100} \times 100\% = 60\%}\)

Step 6: Express the result in terms of k

We found that Country Y generates \(60\%\) from nuclear power, and we used \(\mathrm{k = 20}\).

Notice that \(60 = 3 \times 20 = 3k\)

Therefore, Country Y generates \(\mathrm{3k}\) percent of its electricity from nuclear power.

Verification with different k value:

Let's verify with \(\mathrm{k = 10}\):

• Country X's nuclear power = \(10\% \text{ of } 300 = 30\) kilowatt-hours
• Country Y's nuclear power = \(30\) kilowatt-hours (equal to X)
• Country Y's nuclear percentage = \(\mathrm{\frac{30}{100} \times 100\% = 30\%}\)
• Indeed, \(30\% = 3 \times 10\% = 3k\)

The answer is (E) \(\mathrm{3k}\).