A sells his goods 30% cheaper than B and 

30% dearer than C. By what percentage is the cost of C’s 

goods cheaper than B’s goods.

Let's denote:

- \( P_B \) as the price of B's goods,
- \( P_A \) as the price of A's goods,
- \( P_C \) as the price of C's goods.

### Step 1: Relationship between A and B
A sells his goods 30% cheaper than B. This means:

\[
P_A = P_B \times (1 - 0.30) = 0.70 \times P_B
\]

### Step 2: Relationship between A and C
A sells his goods 30% dearer than C. This means:

\[
P_A = P_C \times (1 + 0.30) = 1.30 \times P_C
\]

### Step 3: Expressing \( P_C \) in terms of \( P_B \)
From the first equation, we know \( P_A = 0.70 \times P_B \), and from the second equation \( P_A = 1.30 \times P_C \). Setting these two equal gives:

\[
0.70 \times P_B = 1.30 \times P_C
\]

Solving for \( P_C \):

\[
P_C = \frac{0.70 \times P_B}{1.30}
\]

Simplifying the fraction:

\[
P_C = \frac{7}{13} \times P_B
\]

### Step 4: Percentage by which C's goods are cheaper than B's
The percentage difference between B's and C's prices can be calculated as:

\[
\text{Percentage difference} = \frac{P_B - P_C}{P_B} \times 100
\]

Substitute \( P_C = \frac{7}{13} \times P_B \) into this equation:

\[
\text{Percentage difference} = \frac{P_B - \frac{7}{13} \times P_B}{P_B} \times 100
\]

Simplifying:

\[
\text{Percentage difference} = \left(1 - \frac{7}{13}\right) \times 100 = \frac{6}{13} \times 100 \approx 46.15\%
\]

### Final Answer:
C’s goods are approximately **46.15%** cheaper than B's goods.

Would you like further details or clarifications? Here are some related questions:

1. How do percentage increases and decreases relate to cost calculations?
2. Can we generalize this approach to different percentages?
3. What happens if the percentage difference between A and C changes?
4. How do we interpret percentage differences in real-world scenarios?
5. Can the relationship between A, B, and C be modeled using algebraic expressions?

**Tip:** When comparing percentage differences, always subtract smaller values from larger ones to avoid negative percentages.

A sells his goods 30% cheaper than B and 30% dearer than C. By what percentage is the cost of C’s goods cheaper than B’s goods?

Find out how to calculate the percentage by which C's goods are cheaper than B's. This problem covers algebraic concepts, including proportional relationships and percentage calculations. Suitable for students in Grades 9-12.

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