Problem Analysis: Discount and Profit Ratio

We are given the following information:

• Selling Price (SP) is 3 times the Discount offered (D): $SP = 3 \times D$
• The percentage of discount equals the percentage of profit: $Discount\% = Profit\%$
• We need to find the ratio of Discount (D) to Cost Price (CP): $D : CP$

Calculating Discount to Cost Price Ratio

Let's use the standard formulas:

• Discount Amount: $D = Marked Price (MP) - Selling Price (SP)$
• Discount Percentage: $Discount\% = \frac{D}{MP} \times 100$
• Profit Percentage: $Profit\% = \frac{SP - CP}{CP} \times 100$

Step 1: Relate Marked Price and Discount

Using the given relation $SP = 3 \times D$ and the definition of discount:

$D = MP - SP$

Substitute $SP = 3D$:

$D = MP - 3D$

Rearrange the equation to find MP in terms of D:

$MP = D + 3D = 4D$

Step 2: Use the Equality of Percentages

We are given $Discount\% = Profit\%$.

Substituting the formulas:

$\frac{D}{MP} \times 100 = \frac{SP - CP}{CP} \times 100$

Simplify by canceling $100$:

$\frac{D}{MP} = \frac{SP - CP}{CP}$

Step 3: Substitute Known Relations

Substitute $MP = 4D$ and $SP = 3D$ into the equation from Step 2:

$\frac{D}{4D} = \frac{3D - CP}{CP}$

Simplify the left side:

$\frac{1}{4} = \frac{3D - CP}{CP}$

Step 4: Solve for the Ratio D : CP

Cross-multiply:

$1 \times CP = 4 \times (3D - CP)$

$CP = 12D - 4CP$

Group the CP terms:

$CP + 4CP = 12D$

$5CP = 12D$

Rearrange to find the required ratio $D : CP$ (which is $D / CP$):

$\frac{D}{CP} = \frac{5}{12}$

Final Answer

The ratio of the discount offered to the cost price is $5 : 12$.