$A, B$ and $C$ enter into a partnership. $B$ contributes $\frac{1}{3}$ of the capital, while $A$ contributes as much as $B$ and $C$ together contribute. Find the ratio of their capitals.

> [!TLDR] > $A=B+C=1-A\Rightarrow A=\tfrac12$; $B=\tfrac13$; $C=\tfrac16$; clearing fractions gives $A:B:C=$ **3 : 2 : 1**. > [!FORMULA] > Let the total capital be 1. Given $B=\tfrac13$ of the total and $A=B+C$; use $A+B+C=1$. 1. Let total capital $=1$. Then $B=\dfrac13$. 2. Since $A=B+C$, and $A+B+C=1$, substitute $B+C=A$: $A+A=1\Rightarrow A=\dfrac12$. 3. $C=1-A-B=1-\dfrac12-\dfrac13=\dfrac{6-3-2}{6}=\dfrac16$. 4. Ratio $A:B:C=\dfrac12:\dfrac13:\dfrac16$. Multiply each by 6: $3:2:1$. 5. **Check:** $A=B+C\Rightarrow 3=2+1$ $\checkmark$ and $B=\tfrac13$ of $(3+2+1)=6$, i.e. $2$. $\checkmark$ > [!ANSWER] > The ratio of their capitals is $A:B:C=$ **3 : 2 : 1**.

A, B and C enter into a partnership. B contributes (1)/(3) of the capital, while A contributes as much as B an

Exercise 7 · Q1

A, B

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Concept understanding — Investment Ratio Calculation

Investment Ratio Calculation — A First Look

Imagine you and a friend decide to start a small business together. You put in ₹50,000, and your friend puts in ₹30,000. At the end of the year, you make a profit of ₹16,000. How should you split it?

The fair answer is not "half each" — you put in more money, so you should get a larger share. That's the entire intuition behind investment ratios: profit is shared in the same proportion as the money invested.

The Core Idea

An investment ratio is simply a comparison of the amounts different people (or groups) put into a venture. If you invest ₹50,000 and your friend invests ₹30,000, the ratio of your investment to theirs is:

50,000 : 30,000

Like any ratio, you simplify it by dividing both sides by their common factor (here, 10,000):

5 : 3

That's your investment ratio. It tells you that for every ₹5 you put in, your friend put in ₹3.

From Ratio to Profit Share

Now, how does this ratio translate into actual money? The total investment is ₹80,000. Your share of the total is:

\frac{50,000}{80,000} = \frac{5}{8}

Your friend's share is:

\frac{30,000}{80,000} = \frac{3}{8}

So the profit of ₹16,000 is split as:

You: $\frac{5}{8} \times 16,000 = ₹10,000$
Friend: $\frac{3}{8} \times 16,000 = ₹6,000$

Notice that $10,000 : 6,000$ simplifies to $5 : 3$ — the same as the investment ratio. Profit is always divided in the ratio of investment.

\text{Share of profit} = \frac{\text{Individual investment}}{\text{Total investment}} \times \text{Total profit}

When Investments Are for Different Durations

Here's where it gets slightly more interesting. What if you invested ₹50,000 for the full year, but your friend invested ₹30,000 only for the last 6 months? Is it still fair to split profit 5:3?

No — because your money was at work for twice as long. The solution is to convert each investment into an equivalent "money × time" unit, called the product.

Your product: $50,000 \times 12 = 6,00,000$ (rupee-months)
Friend's product: $30,000 \times 6 = 1,80,000$ (rupee-months)

Now compare these products:

6,00,000 : 1,80,000 = 20 : 6 = 10 : 3

That's the effective investment ratio. Profit is now split in the ratio 10 : 3, not 5 : 3.

Watch out

Never ignore time when investments are made at different points. The ratio of amounts alone is only valid when everyone invests for the same duration.

The Precise Statement

For any partnership (or any situation where returns are proportional to contribution):

Profit (or loss) is shared in the ratio of the product of each partner's investment and the time period for which it was invested.

If all time periods are equal, this reduces to the ratio of the investments themselves.

A Quick Example to Lock It In

Three partners start a business:

A invests ₹2,00,000 for 12 months
B invests ₹1,50,000 for 8 months
C invests ₹1,00,000 for 6 months

Compute the products:

A: $2,00,000 \times 12 = 24,00,000$
B: $1,50,000 \times 8 = 12,00,000$
C: $1,00,000 \times 6 = 6,00,000$

Ratio: $24 : 12 : 6 = 4 : 2 : 1$

If the profit is ₹70,000, then:

A gets $\frac{4}{7} \times 70,000 = ₹40,000$
B gets $\frac{2}{7} \times 70,000 = ₹20,000$
C gets $\frac{1}{7} \times 70,000 = ₹10,000$

Tip

Always simplify the ratio at the product stage — it makes the final calculation cleaner and less error-prone.

The One Thing to Remember

Investment ratio is not about who put in more money — it's about who contributed more value, where value = money × time. That single idea handles every variation you'll ever see in exams.

$A=B+C=1-A\Rightarrow A=\tfrac12$ ; $B=\tfrac13$ ; $C=\tfrac16$ ; clearing fractions gives $A:B:C=$ 3 : 2 : 1.

B=\tfrac13

Let total capital $=1$ . Then $B=\dfrac13$ .
Since $A=B+C$ , and $A+B+C=1$ , substitute $B+C=A$ : $A+A=1\Rightarrow A=\dfrac12$ .
$C=1-A-B=1-\dfrac12-\dfrac13=\dfrac{6-3-2}{6}=\dfrac16$ .
Ratio $A:B:C=\dfrac12:\dfrac13:\dfrac16$ . Multiply each by 6: $3:2:1$ .
Check: $A=B+C\Rightarrow 3=2+1$ $\checkmark$ and $B=\tfrac13$ of $(3+2+1)=6$ , i.e. $2$ . $\checkmark$

✓Final answer

A:B:C=

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Q.A,BA, BA,B and CCC enter into a partnership. BBB contributes 13\frac{1}{3}31​ of the capital, while AAA contributes as much as BBB and CCC together contribute. Find the ratio of their capitals.

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Q. $A, B$ and $C$ enter into a partnership. $B$ contributes $\frac{1}{3}$ of the capital, while $A$ contributes as much as $B$ and $C$ together contribute. Find the ratio of their capitals.

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