Question

The original selling price of an item at a store is 40 percent more than the cost of the item to the retailer. If the retailer reduces the price of the item by 15 percent of the original selling price, then the difference between the reduced price and the cost of the item to the retailer is what percent of the cost of the item to the retailer?

Short answer

The difference between the reduced price and the cost to the retailer is 19% of the cost of the item to the retailer.

Step-by-step solution

Calculation of the Original Selling Price

Let's denote the original cost to the retailer with $C$. Therefore, the original selling price ($P$) which is 40% more, would be: $P=C+C\times 0.4=1.4C$

Calculation of the Reduced Price

The price was then reduced by 15% of the original selling price, so the new reduced price ($R$) would be: $R=P-P\times 0.15=1.4C-0.15\times 1.4C=1.4C\times 0.85$

Calculation of the Difference (Reduced Price - Cost)

To find the difference ($D$) between the reduced price and the cost of the item to the retailer, subtract the cost from the reduced price: $D=R-C=1.4C\times 0.85-C$

Calculation of the Difference as Percentage of Cost

The last part of the problem requires us to find this difference as a percentage of the cost to the retailer. We solve for $D$ as a percentage of the cost, i.e., $\frac{D}{C}\times 100\mathrm{\%}$. After plugging in the formula for $D$ found in the previous step and simplifying, we find the answer

Key concepts

Percentages

Percentages are a mathematical concept used to express how much something has changed in relation to its original value. This is often seen in financial contexts, such as determining discounts or increases in prices. A percentage represents a fraction of 100 and is commonly used to demonstrate proportion.
For example, when we say an item’s selling price is 40% more than its cost, it means that the difference between the selling price and the cost is 40% of the cost. In this context, if the cost is denoted by $C$, then a 40% increase in price makes the selling price $1.4C$. The 1.4 factor comes from the sum of the whole (which is $C$ or 100%) and the additional 40%.

• To increase a number by a percentage: add the percentage value to 1, then multiply.
• To apply a reduction: subtract the percentage value from 1, then multiply the original number by this value.

Understanding percentages helps in calculating prices, interest rates, population growth, and many more applications.

Retail Price Calculations

In retail, price calculations are essential for determining how much an item should be sold for to achieve profits. Calculations take into account the cost of goods sold, desired markups, discounts, and other factors.
From this exercise, the original selling price $P$ is calculated as 40% more than the retailer's cost $C$. Given that a markup is often a percentage-based increase on the cost, the formula becomes $P=C+0.4C=1.4C$.
Furthermore, to find the reduced price after a discount, say the discount is 15% on the selling price, we calculate it by reducing $15\mathrm{\%}$ from the original selling price. The reduced price $R$ is then calculated as:

• $R=1.4C\times (1-0.15)=1.4C\times 0.85$.

This process shows the importance of understanding how percentage reductions affect retail pricing.

Word Problems in Math

Word problems are narrative-based mathematical problems that require translation from text to mathematical operations. To tackle them, breaking the problem into smaller parts is essential. The word problem in this exercise gives us a real-world scenario where we calculate selling prices, discounts, and percentage differences.
First, identify what is being asked. Here, the problem asks us to find the percentage difference between a reduced price and the original cost to the retailer. Each piece of information is turned into a math operation: calculating percentages, arithmetic operations for reductions, and comparisons between numbers.

• Understand what each number and term signifies.
• Translate those into variable expressions.
• Follow logical arithmetic steps to solve each part.

Tackling word problems can build critical thinking and problem-solving skills essential in mathematics.

Step by Step Math Solutions

Step by step solutions breakdown complex problems into manageable stages, allowing one to understand the logic behind each progression. Here, we use four steps:
1. **Calculate the Original Selling Price**: Express the selling price $P$ with the equation $P=1.4C$. This includes a 40% addition to the cost $C$.
2. **Calculate the Reduced Price**: Implementing a reduction of 15% gives $R=1.4C\times 0.85$.
3. **Determine the Difference**: Find $D=R-C$, the difference between reduced selling price and cost.
4. **Express Difference as Percentage of Cost**: Finally, compute $\frac{D}{C}\times 100\mathrm{\%}$. This shows how much the reduced price deviates from cost as a percentage.
Approaching problems step by step not only highlights the individual parts of the problem but reinforces understanding of the overall solution.