A firm ordered 15 premium machines and some standard machines. The price of a premium machine was 6 times that of a standard machine. By mistake, the number of premium and standard machines was interchanged in the delivery, which increased the total bill by 62.5%. What was the ratio of the number of premium machines to standard machines in the original order?

The correct answer is 1:2.

Step 1: Define the variables for prices and quantities.

Let the price of a standard machine be $p$.
Since the price of a premium machine is 6 times that of a standard machine, the price of a premium machine is $6p$.

Let the number of standard machines ordered originally be $x$.
The number of premium machines ordered originally is given as $15$.

Step 2: Calculate the original total bill.

The total cost of the original order is:
$\text{Original Bill}=(15\times 6p)+(x\times p)$
$\text{Original Bill}=90p+xp=p(90+x)$

Step 3: Calculate the interchanged (delivered) total bill.

Due to the mistake, the quantities were interchanged. So, $x$ premium machines and $15$ standard machines were delivered.

The total cost of the delivered order is:
$\text{Delivered Bill}=(x\times 6p)+(15\times p)$
$\text{Delivered Bill}=6xp+15p=p(6x+15)$

Step 4: Form the equation using the given percentage increase.

The interchanged delivery increased the total bill by 62.5%.
Note that $62.5\%=\frac{62.5}{100}=\frac{5}{8}$.

Therefore, the relationship between the delivered bill and original bill is:
$\text{Delivered Bill}=\text{Original Bill}\times (1+\frac{5}{8})$
$\text{Delivered Bill}=\text{Original Bill}\times \frac{13}{8}$

Substitute the expressions for the bills into the equation:
$p(6x+15)=p(90+x)\times \frac{13}{8}$

Divide both sides by $p$ (since $p\ne 0$):
$6x+15=\frac{13(90+x)}{8}$

Step 5: Solve for $x$.

Multiply both sides by 8:
$8(6x+15)=13(90+x)$

Expand both sides:
$48x+120=1170+13x$

Rearrange the terms to solve for $x$:
$48x-13x=1170-120$
$35x=1050$
$x=\frac{1050}{35}$
$x=30$

So, the original number of standard machines ordered was 30.

Step 6: Find the required ratio.

The question asks for the ratio of the number of premium machines to standard machines in the original order:
$\text{Ratio}=\frac{\text{Number of Premium Machines}}{\text{Number of Standard Machines}}=\frac{15}{30}=\frac{1}{2}$

Thus, the ratio is 1:2.