In a certain community, the total amount of electricity used by all of the households during a 24-hour period was...

1. Translate the problem requirements: Clarify what 'total electricity used by all households during 24 hours' means versus 'average electricity used per household per hour', and identify what we're solving for (number of households)
2. Set up the relationship using basic multiplication logic: Use the fundamental principle that total = average × count × time to connect all given information
3. Isolate the target variable: Rearrange the equation to solve for the number of households using algebraic manipulation
4. Verify using dimensional analysis: Check that our answer makes logical sense by ensuring the units work out correctly

Execution of Strategic Approach

1. Translate the problem requirements

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

• Total electricity used by all households during 24 hours = \(\mathrm{x}\) kilowatts
  This means if we add up all the electricity that every single household used over the entire day, we get \(\mathrm{x}\) kilowatts.
• Average electricity used per household per hour = \(\mathrm{y}\) kilowatts
  This means if we take any one household and look at how much electricity it uses in just one hour, on average it's \(\mathrm{y}\) kilowatts.
• What we need to find: Number of households in the community
  We want to know how many households there are total.

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

2. Set up the relationship using basic multiplication logic

Let's think about this step by step using everyday reasoning:

Imagine you have some households, let's call the number of households '\(\mathrm{H}\)'.

• Each household uses \(\mathrm{y}\) kilowatts per hour on average
• There are 24 hours in the day
• So each household uses \(\mathrm{y \times 24}\) kilowatts for the whole day

• If we have \(\mathrm{H}\) households, and each uses \(\mathrm{(y \times 24)}\) kilowatts per day
• Then total electricity used by all households = \(\mathrm{H \times (y \times 24) = H \times 24y}\) kilowatts

But we're told the total electricity used is \(\mathrm{x}\) kilowatts.

So we have: \(\mathrm{x = H \times 24y}\)

This is our fundamental relationship connecting all the given information.

3. Isolate the target variable

Now we solve for \(\mathrm{H}\) (the number of households) from our equation:

\(\mathrm{x = H \times 24y}\)

To get \(\mathrm{H}\) by itself, we divide both sides by \(\mathrm{24y}\):

\(\mathrm{H = x \div (24y)}\)

Written as a fraction: \(\mathrm{H = \frac{x}{24y}}\)

Process Skill: MANIPULATE - Algebraic rearrangement to isolate our target variable

4. Verify using dimensional analysis

Let's check that our answer makes sense by looking at the units:

• \(\mathrm{x}\) is in kilowatts (total electricity for 24 hours)
• \(\mathrm{y}\) is in kilowatts per household per hour
• 24 is in hours

So our expression \(\mathrm{\frac{x}{24y}}\) has units:

kilowatts ÷ (hours × kilowatts per household per hour) = kilowatts ÷ (kilowatts per household) = households

Perfect! Our answer gives us the right units (number of households).

Also, logically: if the average usage per household increases (\(\mathrm{y}\) gets bigger), we'd expect fewer households for the same total usage. Our formula \(\mathrm{\frac{x}{24y}}\) shows that as \(\mathrm{y}\) increases, the number of households decreases, which makes sense.

Final Answer

The number of households in the community is \(\mathrm{\frac{x}{24y}}\).

Looking at our answer choices, this matches choice E. \(\mathrm{\frac{x}{24y}}\).

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the time period in the given rates
Students often miss that the average usage is given "per household per hour" while the total usage is for the entire "24-hour period." They might set up the equation as \(\mathrm{x = H \times y}\), forgetting to account for the 24-hour difference, leading to the incorrect answer \(\mathrm{\frac{x}{y}}\) (choice A).

2. Confusing what represents the "total" versus "average"
Students may mix up which variable represents the total consumption and which represents the average rate. They might incorrectly think \(\mathrm{y}\) represents total usage per household for 24 hours, rather than the hourly average, leading to wrong relationship setups.

3. Setting up the proportion backwards
When thinking about the relationship between total usage, households, and individual usage, students might incorrectly reason that more households means less total usage, leading them to set up equations like \(\mathrm{x = \frac{H}{24y}}\) instead of \(\mathrm{x = H \times 24y}\).

Errors while executing the approach

1. Algebraic manipulation errors when isolating H
Even with the correct equation \(\mathrm{x = H \times 24y}\), students might make errors when solving for \(\mathrm{H}\). Common mistakes include dividing by only one factor (getting \(\mathrm{H = \frac{x}{24}}\) or \(\mathrm{H = \frac{x}{y}}\)) instead of dividing by the complete product \(\mathrm{24y}\).

2. Arithmetic errors with the 24-hour conversion
Students might correctly identify that they need to convert between hourly and daily rates but make calculation errors, such as multiplying by 24 instead of dividing, or placing 24 in the wrong position in their final expression.

Errors while selecting the answer

1. Choosing a dimensionally incorrect answer without verification
Students might arrive at expressions like \(\mathrm{\frac{y}{x}}\) or \(\mathrm{\frac{24y}{x}}\) but fail to check that these give the wrong units. For instance, \(\mathrm{\frac{y}{x}}\) would give "kilowatts per hour per household divided by total kilowatts" which doesn't yield a count of households.

Alternate Solutions

Smart Numbers Approach

Instead of working with abstract variables, we can assign concrete values that satisfy the given relationships and verify our answer.

Step 1: Choose smart numbers for the variables

Let's choose convenient values:

• Average electricity per household per hour: \(\mathrm{y = 2}\) kilowatts
• Number of households: Let's say 5 households (this is what we'll solve for)

Step 2: Calculate total electricity usage

If each household uses 2 kilowatts per hour, then in 24 hours each household uses:

\(\mathrm{2 \times 24 = 48}\) kilowatts per household

With 5 households, total electricity usage is:

\(\mathrm{x = 5 \times 48 = 240}\) kilowatts

Step 3: Test each answer choice with our concrete values

We have: \(\mathrm{x = 240}\), \(\mathrm{y = 2}\), and we expect the answer to be 5 households

1. Choice A: \(\mathrm{\frac{x}{y} = \frac{240}{2} = 120 \neq 5}\)
2. Choice B: \(\mathrm{\frac{y}{x} = \frac{2}{240} = \frac{1}{120} \neq 5}\)
3. Choice C: \(\mathrm{\frac{24y}{x} = \frac{24 \times 2}{240} = \frac{48}{240} = \frac{1}{5} \neq 5}\)
4. Choice D: \(\mathrm{\frac{y}{24x} = \frac{2}{24 \times 240} = \frac{2}{5760} \neq 5}\)
5. Choice E: \(\mathrm{\frac{x}{24y} = \frac{240}{24 \times 2} = \frac{240}{48} = 5}\) ✓

Step 4: Verify the logic

Our smart numbers confirm that choice E gives us the correct number of households. The formula \(\mathrm{\frac{x}{24y}}\) makes sense because:

• \(\mathrm{x}\) is total kilowatts used by all households in 24 hours
• \(\mathrm{24y}\) is kilowatts used by one household in 24 hours
• Dividing total usage by per-household usage gives the number of households