Question

The cost price of 19 articles is same as the selling price of 29 articles. What is the loss \(\% ?\) (a) \(35 \%\) (b) \(34.48 \%\) (c) \(52.63 \%\) (d) none of these

Short answer

Based on the solution provided, complete the short answer: In this problem, we are given that the cost price of 19 articles is equal to the selling price of 29 articles. To find the loss percentage, we did the following steps: 1. We defined the variables C (cost price of one article) and S (selling price of one article) and created an equation, \(19C = 29S\). 2. We solved for S in terms of C, resulting in \(S = \frac{19C}{29}\). 3. We calculated the loss, which is equal to the cost price minus the selling price, resulting in \(C - \frac{19C}{29}\). 4. We used the loss percentage formula, which gave us the expression: \(\frac{(C - \frac{19C}{29})}{C} \times 100\). 5. We simplified the expression and calculated the loss percentage as \(34.48 \% \). Thus, the loss percentage in this problem is approximately \(34.48 \%\).

Step-by-step solution

Define the variables

Let's assume the cost price of one article is C and selling price of one article is S. The cost price of 19 articles is the same as the selling price of 29 articles. Therefore, we can write the equation: \(19C = 29S\)

Solve for C or S

We can solve for one variable in terms of the other. Let’s solve for S: \(S = \frac{19C}{29}\)

Calculate the loss

As per the loss percentage formula we defined in the analysis, the loss is: Loss \( = \text{Cost Price} - \text{Selling Price} = C - S\) Substitute the value of S from step 2: Loss \( = C - \frac{19C}{29}\)

Calculate the loss percentage

Next, we need to find the loss percentage using the formula: Loss \(\% = \frac{(\text{Cost Price} - \text{Selling Price})}{\text{Cost Price}} \times 100\) Substitute the value of C and S in the formula: Loss \(\% = \frac{(C - \frac{19C}{29})}{C} \times 100\)

Simplify and find the answer

Now, we can simplify and find the loss percentage: Loss \(\% = \frac{(\frac{29C}{29} - \frac{19C}{29})}{C} \times 100\) Loss \(\% = \frac{10C}{29C} \times 100\) Loss \(\% = \frac{10}{29} \times 100 = 34.48\%\) Comparing with the given options, we find that the loss percentage is: (B) \(34.48 \%\)

Key concepts

Understanding Cost Price

To get a complete grasp of loss percentage, we first need to understand the concept of cost price. The cost price (CP) is the amount of money that a seller spends to acquire an article before selling it. This price is fundamental as it determines the baseline for any potential profit or loss.

- **Key Point:** Cost price does not include the selling strategies or market fluctuations but purely the amount spent on the article itself.- **Example:** If you buy a book for \(10, then the cost price is \)10.

In mathematical problems, we often set the cost price as a variable, such as \(C\), to facilitate calculations and equations. This helps in determining relationships between different financial factors such as selling price and loss or profit percentages.

Understanding Selling Price

Next, let us explore the selling price (SP). The selling price is the amount for which an article is sold to a buyer. It is crucial as it determines the revenue generated from selling the article.

- **Formula:** Selling price can be depicted as \(S = \frac{19C}{29}\) in this problem because the cost price of 19 articles equals the selling price of 29 articles.- **Example:** If you sell a book for \(12, then the selling price is \)12.

The selling price plays an essential role in calculating profit or loss, as it directly affects the revenue. By contrasting the selling price against the cost price, we can determine whether there has been a financial gain or a shortfall.

Calculating Article Cost and Loss Percentage

To determine the loss percentage in the given exercise, we begin with the article cost calculations. This involves the cost price and selling price of the articles as calculated in previous sections.

- **Calculation:** The loss percentage is determined by finding the difference between the cost price (CP) and the selling price (SP) expressed as a percentage of the CP.- **Formula:** \[ \text{Loss} \% = \frac{(\text{Cost Price} - \text{Selling Price})}{\text{Cost Price}} \times 100 \]

- **Example Use:** In the problem, we calculated this as: \[ \text{Loss} \% = \frac{(C - \frac{19C}{29})}{C} \times 100 \]

By simplifying, the result is \(34.48\%\), matching option (B) from the given choices. This percentage illustrates the shortfall between the cost price and selling price, representing a monetary loss for the seller when 19 articles have the same cumulative cost as 29 would generate from sales.