A bakery owner purchased 8 kg of premium wheat flour along with a certain quantity of standard wheat flour. The premium variety costs double the price per kilogram of the standard variety. However, due to a mix-up during dispatch, the ordered quantities of the two flours were interchanged, resulting in a 35% increase in the total cost. Determine the ratio of the originally ordered quantity of premium wheat flour to that of standard wheat flour.

The correct option is 13 : 34.

Step 1: Define the variables for price and quantity
Let the price per kilogram of standard wheat flour be $P$.
Since the price of premium wheat flour is double that of standard flour, the price per kilogram of premium wheat flour is $2P$.
Let the originally ordered quantity of standard flour be $x$ kg.
The originally ordered quantity of premium flour is given as $8\text{kg}$.

Step 2: Calculate the original total cost
The original total cost is the sum of the cost of 8 kg of premium flour and $x$ kg of standard flour:

$\text{Original Cost}=(8\times 2P)+(x\times P)=16P+xP=(16+x)P$

Step 3: Calculate the new total cost after the mix-up
Due to the mix-up, the quantities were interchanged. The bakery received $x$ kg of premium flour and 8 kg of standard flour.
The new total cost is calculated as follows:

$\text{New Cost}=(x\times 2P)+(8\times P)=2xP+8P=(2x+8)P$

Step 4: Set up the equation based on the cost increase
The interchanged quantities resulted in a 35% increase in total cost. Therefore, the new cost is 135% (or 1.35 times) of the original cost:

$\text{New Cost}=1.35\times \text{Original Cost}$
Substituting the expressions for New Cost and Original Cost:

$(2x+8)P=1.35\times (16+x)P$

Step 5: Solve for $x$
Since $P$ is non-zero, we can divide both sides of the equation by $P$:

$2x+8=1.35(16+\mathrm{x})$
Expanding the right-hand side:

$2x+8=21.6+1.35x$
Rearranging terms to isolate $x$:

$2x-1.35x=21.6-8$

$0.65x=13.6$

$x=\frac{13.6}{0.65}=\frac{1360}{65}=\frac{272}{13}$

Step 6: Determine the ratio of ordered quantities
We need to find the ratio of the originally ordered quantity of premium wheat flour to standard wheat flour, which is $8:x$:

$\text{Ratio}=8:\frac{272}{13}$
Multiply both terms by 13 to clear the fraction:

$\text{Ratio}=(8\times 13):272=104:272$
Simplify the ratio by dividing both terms by 8:

$\text{Ratio}=\frac{104}{8}:\frac{272}{8}=13:34$

Thus, the ratio of the originally ordered quantity of premium wheat flour to standard wheat flour is 13 : 34.