In the summer, water consumption is known to decrease overall by 25%. A Water Board official states that in the summer household consumption decreases by 20%, while other consumption increases by 70%. Which of the following statements is correct?

Question

Prepp Team · Accepted Answer

Water Consumption: Understanding the Problem

The question presents a scenario involving water consumption changes during the summer. We are given three pieces of information related to water consumption:

Overall water consumption is known to decrease by 25%.
Household consumption decreases by 20%.
Other consumption increases by 70%.

Our task is to determine which of the given statements is correct by checking the consistency of the official's statement regarding household and other consumption changes with the overall decrease in water consumption.

Water Consumption: Initial Setup

To analyze the situation, let's define variables for the initial water consumption before summer:

Let \( H \) represent the initial household water consumption.
Let \( O \) represent the initial other water consumption.
Let \( W \) represent the initial total water consumption.

By definition, the total water consumption is the sum of household and other consumption:

\( W = H + O \)

Water Consumption: Summer Changes

Now, let's calculate the water consumption for each category during the summer based on the percentages provided by the Water Board official:

Overall Water Consumption in Summer: The total consumption decreases by 25%.
\( W_{\text{summer}} = W - 0.25W = (1 - 0.25)W = 0.75W \)
Household Consumption in Summer: Household consumption decreases by 20%.
\( H_{\text{summer}} = H - 0.20H = (1 - 0.20)H = 0.80H \)
Other Consumption in Summer: Other consumption increases by 70%.
\( O_{\text{summer}} = O + 0.70O = (1 + 0.70)O = 1.70O \)

Consistency Check of Water Consumption Data

For the official's statement to be correct, the sum of household consumption and other consumption in summer must equal the total overall water consumption in summer. That is:

\( H_{\text{summer}} + O_{\text{summer}} = W_{\text{summer}} \)

Substitute the expressions we derived for summer consumption:

\( 0.80H + 1.70O = 0.75W \)

We know that \( W = H + O \). Substitute this into the equation:

\( 0.80H + 1.70O = 0.75(H + O) \)

Water Consumption: Solving for Consistency

Now, let's expand and rearrange the equation to check for consistency:

\( 0.80H + 1.70O = 0.75H + 0.75O \)

Gather the terms involving \( H \) on one side and terms involving \( O \) on the other side:

\( 0.80H - 0.75H = 0.75O - 1.70O \)

Perform the subtractions:

\( 0.05H = -0.95O \)

Water Consumption: Conclusion on Official's Statement

The equation \( 0.05H = -0.95O \) implies that \( H = \frac{-0.95}{0.05}O \), which simplifies to \( H = -19O \).

In real-world scenarios, water consumption values ( \( H \) and \( O \) ) must be positive. It is impossible for a positive quantity (like \( H \)) to be equal to a negative multiple of another positive quantity ( \( -19O \) ). This mathematical inconsistency indicates that the figures provided in the official's statement are contradictory and cannot simultaneously be true.

Therefore, there are errors in the official's statement.

Statement	Analysis
Overall water consumption decreases by 25%.	Given as a known fact.
Household consumption decreases by 20%.	Official's claim.
Other consumption increases by 70%.	Official's claim.
Mathematical Consistency	If \( H \) and \( O \) are initial positive consumptions, then \( 0.05H = -0.95O \) leads to \( H = -19O \), which is impossible as consumption cannot be negative.

Based on our analysis, the figures given by the official for household and other consumption changes are inconsistent with the overall 25% decrease in water consumption.