CAT 2017

Slot 1

QA

Question & SolutionQues & Sol

Suppose, C1, C2, C3, C4, and C5 are five companies. The profits made by Cl, C2, and C3 are in the ratio 9 : 10 : 8 while the profits made by C2, C4, and C5 are in the ratio 18 : 19 : 20. If C5 has made a profit of Rs 19 crore more than C1, then the total profit (in Rs) made by all five companies is

1. Concept Used

• Topic: Ratio & Proportion — Merging Ratios by Equating the Common Element
• Formula: $$\text{To merge two ratios sharing a common term, scale both ratios so that the common term has the same value in both.}$$

2. Calculation

We are given two separate ratios that share C2 as a common element:

$$C1 : C2 : C3 = 9 : 10 : 8 \quad \cdots (i)$$

$$C2 : C4 : C5 = 18 : 19 : 20 \quad \cdots (ii)$$

To merge these into a single combined ratio, we need to make the value of C2 equal in both ratios. In ratio (i), C2 = 10, and in ratio (ii), C2 = 18. The LCM of 10 and 18 is 90.

Scale ratio (i) by multiplying every term by 9 (so that C2 becomes 90):

$$C1 : C2 : C3 = 81 : 90 : 72$$

Scale ratio (ii) by multiplying every term by 5 (so that C2 becomes 90):

$$C2 : C4 : C5 = 90 : 95 : 100$$

Now, since C2 = 90 in both, we can merge all five companies into one unified ratio:

$$C1 : C2 : C3 : C4 : C5 = 81 : 90 : 72 : 95 : 100$$

Let the common multiplier be x, so the actual profits are:

$$C1 = 81x, \quad C2 = 90x, \quad C3 = 72x, \quad C4 = 95x, \quad C5 = 100x$$

Using the condition that C5 has made Rs 19 crore more profit than C1:

$$C5 - C1 = 19$$

$$100x - 81x = 19$$

$$19x = 19$$

$$x = 1$$

Therefore, the total profit made by all five companies is:

$$\text{Total} = 81x + 90x + 72x + 95x + 100x = 438x$$

$$\text{Total} = 438 \times 1 = 438 \text{ crore}$$

3. Solution

Answer = Option A ✅

The total profit made by all five companies is Rs 438 crore.