Water Consumption Analysis

This solution analyzes the water consumption figures provided by a Water Board official to determine if there are any inconsistencies.

Breakdown of Official's Claims

Let the initial total water consumption be represented by $C$. Let the initial household consumption be $H_0$ and the initial other consumption be $O_0$. Therefore, $C = H_0 + O_0$.

• The official states that overall water consumption decreases by 25% in the summer. The new total consumption ($C_1$) would be $C - 0.25C = 0.75C$. The total decrease amount is $0.25C$.

• Household consumption decreases by 20%. The new household consumption ($H_1$) is $H_1 = H_0 - 0.20H_0 = 0.80H_0$. The decrease in household consumption is $0.20H_0$.

• Other consumption increases by 70%. The new other consumption ($O_1$) is $O_1 = O_0 + 0.70O_0 = 1.70O_0$. The increase in other consumption is $0.70O_0$.

• The new total consumption ($C_1$) is also the sum of the new household and other consumptions: $C_1 = H_1 + O_1 = 0.80H_0 + 1.70O_0$.

• We equate the two expressions for the new total consumption:

  $0.80H_0 + 1.70O_0 = 0.75C$

  Substitute $C = H_0 + O_0$ into the equation:

  $0.80H_0 + 1.70O_0 = 0.75(H_0 + O_0)$

  Distribute the 0.75:

  $0.80H_0 + 1.70O_0 = 0.75H_0 + 0.75O_0$

• Rearrange the terms to solve for the relationship between $H_0$ and $O_0$:

  $1.70O_0 - 0.75O_0 = 0.75H_0 - 0.80H_0$

  $0.95O_0 = -0.05H_0$

  Solve for $H_0$:

  $H_0 = \frac{0.95}{-0.05} O_0$

  $H_0 = -19 O_0$

• Since initial consumptions ($H_0$ and $O_0$) must represent positive quantities, the equation $H_0 = -19 O_0$ is mathematically impossible. This contradiction indicates that the figures provided by the official are inconsistent.

Conclusion on Official's Statement

The analysis reveals a mathematical inconsistency in the Water Board official's statement. Therefore, the correct conclusion is that there are errors in the official's statement.