Question

The price of the sugar goes up by 30%. By what percent should a house wife reduce her consumption so that the expenditure does not increase?

Answer 1

Hint: Assume the original price of sugar per kg is Rs. x and the expenditure of the wife is Rs y. In case \[{{1}^{st}}\], we have the cost of sugar per kg equal to Rs. x and the expenditure of the wife is equal to Rs y. Now, using a unitary method calculate the quantity of sugar that can be bought in case \[{{1}^{st}}\]. In case \[{{2}^{nd}}\] , we have the cost of sugar per kg is equal to Rs. (x + 30% of x). The expenditure is the same as in case \[{{1}^{st}}\] . Now, calculate the quantity of sugar that can be bought in case \[{{2}^{nd}}\]. Then, calculate the difference between the quantity of the sugar in case \[{{1}^{st}}\] and case \[{{2}^{nd}}\] . We have the formula for calculating the percentage of reduction in the quantity of sugar, \[\text{Percentage}\,\text{of}\,\text{reduction=}\dfrac{\text{Reduction in the quantity}}{\text{Original consumption of sugar}}\times 100\] . Now, use this formula and get the percentage of reduction.

Complete step by step answer:
First of all, let us assume that the original price of sugar per kg is Rs. x and the expenditure of the wife is Rs y.
In case \[{{1}^{st}}\] , we have
The cost of sugar per kg = Rs. x …………………………..(1)
The expenditure of the wife = Rs y …………………………….(2)
Using a unitary method, we can get the quantity of sugar that can be bought.
The quantity of the sugar that can be bought in case \[{{1}^{st}}\] = \[\dfrac{y}{x}\] kg …………………………..(3)
Now, it is given that the price of the sugar goes up by 30%. It means that there is an increment of 30% in the cost price of sugar. It is also given that we don’t have to increase the expenditure. The expenditure of the wife remains the same.
Now, in case \[{{2}^{nd}}\] , we have
The cost of sugar per kg = Rs. (x + 30% of x) = Rs. \[\left( x+\dfrac{30}{100}x \right)\] = Rs. \[\dfrac{130}{100}x\] = Rs. \[\dfrac{13x}{10}\] ……………………(4)
The expenditure of the wife = Rs y …………………………….(5)
Using unitary method, we can get the quantity of sugar that can be bought.
The quantity of the sugar that can be bought in case \[{{2}^{nd}}\] = \[\dfrac{y}{\dfrac{13x}{10}}=\dfrac{10y}{13x}\] kg …………………………..(6)
The reduction in the consumption in the quantity of sugar is the difference between the quantity of the sugar that can be bought in case \[{{1}^{st}}\] and the quantity of the sugar that can be bought in case \[{{2}^{nd}}\] .
From equation (4) and equation (6), we have the quantity of the sugar that can be bought in case \[{{1}^{st}}\] and the quantity of the sugar that can be bought in case \[{{2}^{nd}}\] respectively.
The reduction in the quantity of the sugar = \[\dfrac{y}{x}-\dfrac{10y}{13x}=\dfrac{13y-10y}{13x}=\dfrac{3y}{13x}\] .
Now, we have to calculate the percentage of reduction in the quantity of the sugar.
We have the formula, \[\text{Percentage}\,\text{of}\,\text{reduction=}\dfrac{\text{Reduction in the quantity}}{\text{Original consumption of sugar}}\times 100\] .
The percentage in reduction of the quantity of the sugar = \[\dfrac{\dfrac{3y}{13x}}{\dfrac{y}{x}}\times 100=\dfrac{3}{13}\times 100=23.07\] .

Therefore, the percentage in reduction of the quantity of the sugar is 23.07.

Note: We can also solve this question by another method.
Let us assume that the original price of sugar per kg is Rs. 100 and the initial quantity of the consumption is 5 kg.
The initial quantity of sugar that can be bought for Rs. 500 = 5 kg.
The expenditure of the wife = Rs. \[100\times 5\] = Rs. 500.
Now, Now, it is given that the price of the sugar goes up by 30%. It means that there is an increment of 30% in the cost price of sugar. It is also given that we don’t have to increase the expenditure. The expenditure of the wife remains same.
The expenditure of the wife = Rs. 500.
The cost of the sugar per kg = Rs. (100+ 30% of 100) = Rs. \[\left( 100+\dfrac{30}{100}\times 100 \right)\] = Rs. 130.
The final quantity of sugar that can be bought for Rs. 500 = \[\dfrac{500}{130}\] kg = \[\dfrac{50}{13}\] kg.
The reduction in the consumption in the quantity of sugar is the difference between the initial quantity of the sugar that can be bought for Rs. 500 and the final quantity of the sugar that can be bought for Rs. 500.
The reduction in the quantity of the sugar = \[5-\dfrac{50}{13}=\dfrac{65-50}{13}=\dfrac{15}{13}\] .
We have the formula, \[\text{Percentage}\,\text{of}\,\text{reduction=}\dfrac{\text{Reduction in the quantity}}{\text{Original consumption of sugar}}\times 100\]
The percentage in reduction of the quantity of the sugar = \[\dfrac{\dfrac{15}{13}}{5}\times 100=\dfrac{15}{13\times 5}\times 100=\dfrac{300}{13}=23.07\] .
Therefore, the percentage in the reduction of the quantity of sugar is 23.07.