Due to an increase in the price of sugar by 25%,by how much percent must a householder decrease the consumption of sugar so that there is no increase in the expenditure on sugar?

(1.25x)(r) = x

1.25r = 1
r = 1/1.25 = .8 = 80%

So it must be reduced by 20%

check:
suppose the original was $100
after the increase it would be 125
after a reduction of 20% is would be
.8(125) = 100

To determine the percentage by which a householder must decrease their sugar consumption in order to offset the increase in price and maintain the same expenditure on sugar, we can follow these steps:

Step 1: Understand the problem
- Let's assume the initial price of sugar is P and the initial consumption is C.
- The initial expenditure on sugar is calculated as Expenditure = Price * Consumption.

Step 2: Calculate the new price after the increase of 25%
- The new price of sugar, after the increase of 25% to the initial price P, can be calculated as New Price = P + (P * 25%).

Step 3: Calculate the new expenditure with the increased price
- The new expenditure on sugar, with the increased price and the unchanged initial consumption, can be calculated as New Expenditure = New Price * Consumption.

Step 4: Find the percentage decrease in consumption required
- We need to find the percentage decrease in consumption that would result in the new expenditure being equal to the initial expenditure.
- Let X represent the percentage decrease in consumption, so the new consumption becomes (1 - X/100) * C.

Step 5: Set up the equation and solve for X
- Equate the initial expenditure with the new expenditure:
Expenditure = New Expenditure
Price * Consumption = New Price * New Consumption
P * C = (P + (P * 25%)) * ((1 - X/100) * C)

Simplifying the equation:
P * C = (P * (1 + 0.25)) * ((1 - X/100) * C)
P * C = P * 1.25 * (1 - X/100) * C

Cancelling out the common terms "P" and "C":
1 = 1.25 * (1 - X/100)

Simplifying further:
1 = 1.25 - 1.25 * X/100

Rearranging the equation:
1.25 * X/100 = 0.25
X/100 = 0.25 / 1.25
X/100 = 0.2

Multiplying both sides by 100:
X = 20

Therefore, a householder must decrease their sugar consumption by 20% in order to offset the 25% increase in price and maintain the same expenditure on sugar.