Question

During a sale, a store sells 20 percent of its remaining stock each day, without replenishment. After four days, what fraction of its original stock has it sold? A. $$\frac{1}{625}$$ B. $$\frac{256}{625}$$ C. $$\frac{61}{125}$$ D. $$\frac{64}{125}$$ E. $$\frac{369}{625}$$

Short answer

E: \[ \frac{369}{625} \].

Step-by-step solution

Calculate daily remaining stock

If the store sells 20% of its remaining stock each day, then 80% of the stock remains. This can be represented as a fraction: \(1 - 0.2 = 0.8\).

Determine remaining stock after 4 days

To find the remaining stock after four days, raise 0.8 (the remaining stock fraction after one day) to the fourth power: \[ 0.8^4 = (0.8 \times 0.8 \times 0.8 \times 0.8 )= 0.4096 \]. So, 0.4096 or 40.96% of the stock is left after four days.

Find sold stock fraction

If 40.96% of the stock remains, then the fraction of the stock sold is \[1 - 0.4096 = 0.5904\].

Convert fraction to exact form

To find the exact fraction form of 0.5904, observe that \[0.5904 = \frac{5904}{10000}\]. Simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 16 in this case: \[ \frac{5904 \div 16}{10000 \div 16} = \frac{369}{625} \].

Conclude with correct option

The fraction of the original stock sold is \[ \frac{369}{625} \]. So, the correct answer is E.

Key concepts

percentages

Percentages are a way to express a number as a fraction of 100. They are very useful in many real-life contexts, especially in finance and sales.
In this example, the store sells 20 percent of its stock each day. This means that out of every 100 units of stock, 20 units are sold each day.
To convert percentages into a more usable form, you can divide by 100. For instance, 20 percent becomes 0.2 in decimal form. Similarly, 80 percent becomes 0.8.
Understanding this conversion is crucial for solving the problem since it allows for easy multiplication and further calculations.

fractions

Fractions represent a part of a whole. They are composed of a numerator (top number) and a denominator (bottom number).
In our problem, we deal with fractions when we calculate the remaining stock each day and convert percentages into fractions.
Initially, we start with the whole stock. After selling 20%, 80% or \( \frac{80}{100} \) of the stock remains. This simplifies to \( \frac{4}{5} \).
As days pass, the same fraction of the stock keeps getting removed, so understanding and managing fractions is key to solving the problem.

exponents

Exponents are a shorthand way to express repeated multiplication of the same number. For example, \( 0.8^4 \) means multiplying 0.8 by itself four times. This is crucial in the given problem to find out how much stock remains after four days.
Specifically, we use exponents to calculate the remaining stock: \( 0.8^4 = 0.4096 \). This step shows the power of exponents in simplifying complex multiplicative processes.

problem-solving steps

Solving complex problems in math can be broken down into clear steps. This GMAT problem can be solved by following a series of logical steps:

• Step 1: Identify and understand the percentage sold each day (20%).
• Step 2: Translate this percentage into a decimal (0.2) and understand the remaining stock (0.8).
• Step 3: Use exponents to find the remaining stock after several days, \( 0.8^4 = 0.4096 \).
• Step 4: Calculate the fraction sold by subtracting the remaining stock from 1 ( \(1 - 0.4096 = 0.5904 \)).
• Step 5: Convert the decimal back into a simplified fraction, \( \frac{369}{625} \).

Following these problem-solving steps ensures a clear, logical progression from understanding the problem to arriving at the solution.