Arithmetic

Ratio, Proportion & Variation

Medium

Suppose, $C1, C2, C3, C4,$ and $C5$ are five companies. The profits made by $C1, C2,$ 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

$438$ crore

$435$ crore

$348$ crore

$345$ crore

✅ Correct Option: 1

We have two separate ratios:

C1 : C2 : C3 = 9 : 10 : 8

C2 : C4 : C5 = 18 : 19 : 20

Notice that C2 appears in both ratios - this is our connecting link!

From the first ratio, we can write:

C1 = 9k (where k is some multiplier)

C2 = 10k

C3 = 8k

From the second ratio, we can write:

C2 = 18m (where m is some multiplier)

C4 = 19m

C5 = 20m

Since C2 is the same company in both ratios:

$10k = 18m$

$k = \dfrac{18m}{10} = \dfrac{9m}{5}$

Substituting $k = \dfrac{9m}{5}$ into our first set of equations:

C1 = $9k = 9 \times \dfrac{9m}{5} = \dfrac{81m}{5}$

C2 = $10k = 10 \times \dfrac{9m}{5} = 18m$ (matches our second ratio)

C3 = $8k = 8 \times \dfrac{9m}{5} = \dfrac{72m}{5}$

C4 = $19m$

C5 = $20m$

We're told that C5 - C1 = 19 crore.

$20m - \dfrac{81m}{5} = 19$

Converting to common denominator:

$\dfrac{100m - 81m}{5} = 19$

$\dfrac{19m}{5} = 19$

$m = 5$

Now we can find each company's profit:

C1 = $\dfrac{81m}{5} = \dfrac{81(5)}{5} = 81$ crore

C2 = $18m = 18(5) = 90$ crore

C3 = $\dfrac{72m}{5} = \dfrac{72(5)}{5} = 72$ crore

C4 = $19m = 19(5) = 95$ crore

C5 = $20m = 20(5) = 100$ crore

Total profit = 81 + 90 + 72 + 95 + 100 = 438 crore

When dealing with overlapping ratios, we always identify the common element (C2 in this case) and use it to connect the ratios into a single system. This technique is essential for solving many ratio-based problems efficiently.

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