Understanding Production Costs from Company Data

The question asks for the ratio of the cost of production of item I by company A to the cost of production of item I by company D. We are given the total cost of production for both items by all seven companies is Rs. 25 crores. We also have a table showing each company's percentage of this total production and the ratio of production between Item I and Item II for each company.

Analyzing the Provided Data Table

Let's first look at the relevant data from the table for companies A and D:

Total production cost for all companies = Rs. 25 crores.

Calculating Production Cost for Item I by Company A

• Company A's total production is 15% of the total production by all companies.
• So, Company A's total production cost = $\frac{15}{100} \times 25$ crores.
• The ratio of production between Item I and Item II for Company A is 2 : 3.
• This means for every 2 units of Item I, 3 units of Item II are produced. The total parts in the ratio are $2 + 3 = 5$.
• The fraction of production cost specifically for Item I by Company A is $\frac{2}{5}$.
• Therefore, the production cost of Item I by Company A = $\frac{2}{5} \times (\text{Company A's total production cost})$
• Production cost of Item I by Company A = $\frac{2}{5} \times \left(\frac{15}{100} \times 25\right)$ crores.

Calculating Production Cost for Item I by Company D

• Company D's total production is 8% of the total production by all companies.
• So, Company D's total production cost = $\frac{8}{100} \times 25$ crores.
• The ratio of production between Item I and Item II for Company D is 3 : 5.
• This means for every 3 units of Item I, 5 units of Item II are produced. The total parts in the ratio are $3 + 5 = 8$.
• The fraction of production cost specifically for Item I by Company D is $\frac{3}{8}$.
• Therefore, the production cost of Item I by Company D = $\frac{3}{8} \times (\text{Company D's total production cost})$
• Production cost of Item I by Company D = $\frac{3}{8} \times \left(\frac{8}{100} \times 25\right)$ crores.

Determining the Ratio of Production Costs

We need to find the ratio of (Production cost of Item I by Company A) : (Production cost of Item I by Company D).

Ratio = $\left(\frac{2}{5} \times \frac{15}{100} \times 25\right) : \left(\frac{3}{8} \times \frac{8}{100} \times 25\right)$

Notice that the term $\left(\frac{1}{100} \times 25\right)$ is common on both sides of the ratio and will cancel out. Let's simplify the calculation:

Ratio = $\left(\frac{2}{5} \times 15\right) : \left(\frac{3}{8} \times 8\right)$

Simplify the terms:

• For the left side (Company A): $\frac{2}{5} \times 15 = 2 \times \frac{15}{5} = 2 \times 3 = 6$.
• For the right side (Company D): $\frac{3}{8} \times 8 = 3 \times \frac{8}{8} = 3 \times 1 = 3$.

So the ratio is $6 : 3$.

This ratio can be simplified by dividing both sides by the greatest common divisor, which is 3.

Simplified Ratio = $\frac{6}{3} : \frac{3}{3} = 2 : 1$.

Conclusion on Production Ratio

The ratio of the production of item I by company A to the cost of production of item I by company D is 2 : 1.

Revision Table: Key Calculations

Ratio of Item I cost (A:D) = 6 : 3 = 2 : 1.

Additional Information: Understanding Ratios in Production Data

When analyzing production data presented in percentages and ratios, it's crucial to break down the total figures correctly. The percentage of total production gives a company's share of the overall pie. The item ratio (like Item I : Item II) tells us how that company's production is distributed between the specific items. To find the amount for a single item, you use the ratio to determine the fraction of the company's total production allocated to that item. For example, a 2:3 ratio means Item I is 2 parts out of a total of 5 parts (2+3). When comparing the production of a specific item across different companies, as done in this question, you calculate the specific item's production amount for each company and then form the ratio. Remember that the 'total production' base value (Rs. 25 crores in this case) often cancels out when calculating ratios between parts derived from the same total.