Calculating AC Price Change Over Alternate Years

This problem involves calculating the net percentage change in the price of an AC over a period where the price fluctuates annually. We need to determine the price difference between 2007 and 2004, considering the alternating increase and decrease pattern.

Understanding the Percentage Change

First, let's convert the given percentage increase/decrease into a decimal format for easier calculations.

• The price changes by $1\frac{3}{5}\%$.
• To convert the mixed fraction to an improper fraction: $1\frac{3}{5} = \frac{1 \times 5 + 3}{5} = \frac{8}{5}$.
• So, the percentage is $\frac{8}{5}\%$.
• Converting this to a decimal: $\frac{8}{5} = 1.6$. Therefore, the change is $1.6\%$.
• As a decimal multiplier, an increase of $1.6\%$ corresponds to multiplying by $(1 + \frac{1.6}{100}) = 1.016$.
• As a decimal multiplier, a decrease of $1.6\%$ corresponds to multiplying by $(1 - \frac{1.6}{100}) = 0.984$.

Tracking Price Fluctuations Year-by-Year

The problem states that the price increase/decrease happens in alternate years, starting with an increase in 2004. We need to compare the price in 2007 to the price in 2004. A common interpretation for such problems is to compare the price levels at the beginning of the respective years.

Let the initial price of the AC at the beginning of 2004 be represented by $P$.

Here's how the price changes:

• Start of 2004: The price is $P$.
• During 2004: The price increased by $1.6\%$.
• Start of 2005: The price becomes $P \times (1 + 0.016) = P \times 1.016$.
• During 2005: Since 2004 had an increase, 2005 (an alternate year) has a decrease of $1.6\%$.
• Start of 2006: The price becomes $(P \times 1.016) \times (1 - 0.016) = P \times 1.016 \times 0.984$.
• During 2006: Since 2005 had a decrease, 2006 (an alternate year) has an increase of $1.6\%$.
• Start of 2007: The price becomes $(P \times 1.016 \times 0.984) \times (1 + 0.016) = P \times 1.016 \times 0.984 \times 1.016$.

We are comparing the price in 2007 (which is the price at the start of 2007) to the price in 2004 (which is the price at the start of 2004).

Summary of Price Changes (Relative to Initial Price P)

Time Point	Change Applied	Calculation Factor	Price
Start of 2004	Base Price	-	$P$
Start of 2005	Increase ($+1.6\%$)	$(1 + 0.016)$	$P \times (1.016)$
Start of 2006	Decrease ($-1.6\%$)	$(1 - 0.016)$	$P \times 1.016 \times 0.984$
Start of 2007	Increase ($+1.6\%$)	$(1 + 0.016)$	$P \times 1.016 \times 0.984 \times 1.016$

Determining the Net Percentage Change

Now, let's calculate the final price at the start of 2007 relative to the initial price $P$.

The price at the start of 2007 is:

$$ P_{2007} = P \times (1.016) \times (0.984) \times (1.016) $$

We can group the terms:

$$ P_{2007} = P \times [(1.016) \times (0.984)] \times (1.016) $$

First, calculate the product of the increase and decrease factors:

$$ (1.016) \times (0.984) = (1 + 0.016) \times (1 - 0.016) $$

This is in the form $(a+b)(a-b) = a^2 - b^2$. Here, $a=1$ and $b=0.016$.

$$ = 1^2 - (0.016)^2 $$ $$ = 1 - 0.000256 $$ $$ = 0.999744 $$

Now substitute this back into the equation for $P_{2007}$:

$$ P_{2007} = P \times (0.999744) \times (1.016) $$

Perform the final multiplication:

$$ P_{2007} = P \times 1.015737984 $$

To find the percentage change compared to the price in 2004 ($P$), we use the formula:

$$ \text{Percentage Change} = \frac{P_{2007} - P}{P} \times 100 $$ $$ \text{Percentage Change} = \left( \frac{P_{2007}}{P} - 1 \right) \times 100 $$ $$ \text{Percentage Change} = (1.015737984 - 1) \times 100 $$ $$ \text{Percentage Change} = 0.015737984 \times 100 $$ $$ \text{Percentage Change} = 1.5737984\% $$

Rounding this to two decimal places, we get $1.57\%$. Since the value is positive, it represents an increase.

Therefore, the price of the AC in 2007, as compared to that in 2004, increased by approximately $1.57\%$.