Price of sugar rises by 20%. By how much percent should the consumption of sugar be reduced so that the expenditure does not change?
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We have a problem about sugar pricing. The price of sugar rises by 20 percent. We need to find by how much percent the consumption should be reduced so that the total expenditure remains unchanged. The key relationship here is that expenditure equals price times consumption. Our goal is to keep expenditure constant while price increases and consumption decreases.
Let's set up the problem systematically. We'll define the initial price as P and the initial consumption as C. The initial expenditure is P times C. After the price increase, the new price becomes 1.2 times P. The new consumption becomes C times 1 minus x, where x is the reduction rate we need to find. Since expenditure must remain constant, we set up the equation: new expenditure equals initial expenditure.
Now let's solve the equation step by step. Starting with 1.2P times C times 1 minus x equals P times C. We can divide both sides by PC to get 1.2 times 1 minus x equals 1. Expanding this gives us 1.2 minus 1.2x equals 1. Rearranging, we get 1.2x equals 0.2. Therefore, x equals 0.2 divided by 1.2, which equals 1 over 6, or approximately 16.67 percent.
The answer is that consumption should be reduced by 16.67 percent. Let's verify this result. Initially, expenditure equals P times C. With the new price of 1.2P and new consumption of 5C over 6, the new expenditure becomes 1.2P times 5C over 6, which equals PC. This confirms that the expenditure remains constant, so our answer is correct.
Now let's set up the mathematical framework. We define the original price as P and original consumption as C, so original expenditure is P times C. After the increase, the new price becomes 1.2P. Let the new consumption be C minus x percent of C, where x is the reduction percentage we need to find. For expenditure to remain constant, we set up the equation: original expenditure equals new expenditure.
Now let's solve the equation step by step. Starting with P times C equals 1.2P times C minus x percent of C. We can factor out C on the right side and divide both sides by PC to get 1 equals 1.2 times 1 minus x over 100. Expanding this gives us 1 equals 1.2 minus 1.2x over 100. Rearranging, we get 1.2x over 100 equals 0.2, so 1.2x equals 20, and therefore x equals 16.67 percent.
Let's verify our answer with a concrete example. Suppose the original price is 10 dollars per kilogram and consumption is 5 kilograms, giving an expenditure of 50 dollars. With a 20 percent price increase, the new price becomes 12 dollars. Reducing consumption by 16.67 percent gives us 4.167 kilograms. The new expenditure is 12 times 4.167, which equals 50 dollars. This confirms our answer is correct.