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Let the initial prices of sugar be p, the initial household consumption be x and let it be decreased by y% so that the net expenditure remains the same.

Initial expenditure $=prices\times consumption=px$

New prices = p + 25% of p = $p+\dfrac{25}{100}\times p=\text{ }p\text{ }+\text{ }0.25p\text{ }=\text{ }1.25p$

New consumption = x – y% of x $=x-\dfrac{y}{100}x=x-\dfrac{xy}{100}$

New expenditure = prices $\times $ consumption = $1.25p\times \left[ x-\dfrac{xy}{100} \right]=1.25px\left[ 1-\dfrac{y}{100} \right]$

Since new expenditure = initial expenditure we have $1.25px\left[ 1-\dfrac{y}{100} \right]=px$

Dividing both sides by px we get

$1.25\left[ 1-\dfrac{y}{100} \right]=1$

Dividing both sides by 1.25 we get

$1-\dfrac{y}{100}=\dfrac{1}{1.25}=\dfrac{100}{125}=\dfrac{4}{5}$

Subtracting 1 on both sides we get

$1-\dfrac{y}{100}-1=\dfrac{4}{5}-1$

$\Rightarrow -\dfrac{y}{100}=\dfrac{-1}{5}$

Multiplying both sides by -1 we get

$\dfrac{y}{100}=\dfrac{1}{5}$

Cross multiplying, we get

$5y=100$

Dividing both sides by 5, we get

$\dfrac{5y}{5}=\dfrac{100}{5}$

i.e. y = 20

Hence the consumption should be decreased by 20% so that there is no net change in the expenditure on sugar.

We know that net expenditure $=prices\times consumption$

So that $prices\propto \dfrac{1}{consumption}$

$prices=\dfrac{k}{consumption}$

So we have $\dfrac{price{{s}_{f}}-price{{s}_{i}}}{price{{s}_{i}}}=0.25$ where $price{{s}_{i}}$ are the initial prices and $price{{s}_{f}}$ are the final prices.

Hence, we have

$\begin{align}

& \dfrac{\dfrac{k}{consumptio{{n}_{f}}}-\dfrac{k}{consumptio{{n}_{i}}}}{\dfrac{k}{consumptio{{n}_{i}}}}=\dfrac{25}{100} \\

& \Rightarrow \dfrac{consumptio{{n}_{i}}-consumptio{{n}_{f}}}{consumptio{{n}_{f}}}=\dfrac{25}{100} \\

\end{align}$

Now we know that if $\dfrac{a}{b}=\dfrac{c}{d}$ then $\dfrac{a}{b+a}=\dfrac{c}{d+c}$

Using the above property, we get

$\begin{align}

& \dfrac{consumptio{{n}_{i}}-consumptio{{n}_{f}}}{consumptio{{n}_{f}}+\left( consumptio{{n}_{i}}-consumptio{{n}_{f}} \right)}=\dfrac{25}{100+25} \\

& \Rightarrow \dfrac{consumptio{{n}_{i}}-consumptio{{n}_{f}}}{consumptio{{n}_{i}}}=\dfrac{25}{125}=\dfrac{1}{5}=\dfrac{100}{5}\%=20\% \\

\end{align}$

Hence the consumption should be decreased by 20%.

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