A man increases his consumption of sugar by 44.44% because the price of sugar is reduced by P%. Find P. (Expenditure remains constant.)

Correct option is A

Given:

A man increases consumption by 44.44%

Solution:

Let the price of sugar be Rs.100

And consumption be 100

Total expenditure = $$100 \times 100$$​ = 10000

Consumption increased by 44.44% = $$\frac{4}{9}$$​​

New consumption = 100 + 44.44% of 100 = $$\frac{1300}{9}$$​​

Let the dropped price be $$x$$​​

Then, expenditure = $$x \times \frac{1300}{9}$$ 

But the expenditure is same in both cases;

Hence,

​$$x \times \frac{1300}{9}= 10000$$​​

​$$x =\frac{10000\times 9}{1300}= \frac{900}{13}$$​​

Price dropped = $$100- \frac{900}{13}= \frac{1300-900}{13}$$​​

$$=\frac{400}{13}$$​​

Percentage dropped =$$\frac{\frac{400}{13}}{100} \times 100 = \frac{400}{13}\% =30\frac{10}{13}\%$$ 

Alternate Method:

Consumption increases by 44.44% (i.e. $$\frac{4}{9})$$​.

Expenditure is constant.

With constant expenditure: ($$\text{Price factor}) \times (\text{Quantity factor}) =$$​ 1.

Price increase by P%.

Price factor = $$1-\dfrac{P}{100}$$​; Quantity factor = 1$$+\dfrac{4}{9}=\dfrac{13}{9}$$​.

​$$\left(1-\frac{P}{100}\right)\times\left(\frac{13}{9}\right)=1$$​​
​$$1-\frac{P}{100}=\frac{9}{13}$$​

​$$\frac{P}{100}=1-\frac{9}{13}=\frac{4}{13}$$​​

​$$P=\frac{400}{13}\%=30\frac{10}{13}\%$$​​
​​