a,b and c start a partnership the capitals of a and b are in the ratio of 4:5 the periods of b and c are in the ratio 5:3 the profit shares of a and c are in the ratio of 4:3 the capitals of b and c are in the ratio of ? (a) 5:1 (b) 5:2 (c) 5:3 (d) cannot be determined

To solve the problem, let's break down the information provided and use it to find the ratio of the capitals of B and C.

### Step 1: Define the Variables
- Let the capital of A be \( a \) and the capital of B be \( b \).
- The capitals of A and B are in the ratio \( 4:5 \), so we can write:
  \[
  \frac{a}{b} = \frac{4}{5} \quad \text{or} \quad a = \frac{4}{5}b
  \]
- Let the time periods for which B and C invested their capitals be \( t_b \) and \( t_c \) respectively.
- The periods of B and C are in the ratio \( 5:3 \), so we have:
  \[
  \frac{t_b}{t_c} = \frac{5}{3}
  \]
- Let the profit shares of A and C be \( P_a \) and \( P_c \) respectively.
- The profit shares of A and C are in the ratio \( 4:3 \), so:
  \[
  \frac{P_a}{P_c} = \frac{4}{3}
  \]

### Step 2: Use the Profit Share Relationship
The profit share is proportional to the product of capital and time. Therefore, for A and C, we have:
\[
\frac{P_a}{P_c} = \frac{a \cdot t_a}{c \cdot t_c}
\]
Substitute the given ratio:
\[
\frac{4}{3} = \frac{a \cdot t_a}{c \cdot t_c}
\]
Since we don't have the exact values of \( t_a \), assume \( t_a = 1 \) for simplicity. So:
\[
\frac{4}{3} = \frac{a}{c \cdot t_c}
\]

### Step 3: Relate to the Capitals of B and C
Using \( a = \frac{4}{5}b \) and substituting it in the equation:
\[
\frac{4}{3} = \frac{\frac{4}{5}b}{c \cdot \frac{3}{5}}
\]
Simplifying:
\[
\frac{4}{3} = \frac{4b}{5c}
\]
Cross-multiply to solve for the ratio \( \frac{b}{c} \):
\[
4c \times 3 = 4b \times 5
\]
\[
12c = 20b
\]
\[
\frac{b}{c} = \frac{12}{20} = \frac{3}{5}
\]
So, the ratio of the capitals of B and C is \( 5:3 \).

### Conclusion
The correct answer is:
\[
\boxed{(c) \, 5:3}
\]

Would you like more details on any part of the solution or have any questions?

Here are some related questions for practice:
1. If A's capital was increased by 25%, how would the profit shares change?
2. What happens to the capital ratios if C's investment time was doubled?
3. How would the ratio change if B and C invested for the same period?
4. What if the profit share ratio of A and B is also given?
5. How does varying the time periods affect the capital ratio?

**Tip:** Remember that in partnership problems, profit is always divided based on the product of capital and time.

Explore how to solve partnership problems involving capital and profit shares ratios. This problem example examines ratios of capitals and profit shares, applying mathematical concepts such as ratio analysis and proportionality. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation