Understanding the Impact of Price Changes on Sugar Consumption

In the world of consumer economics, the fluctuations in commodity prices, such as sugar, can significantly affect the quantity of goods that consumers can procure. This article delves into the mathematics behind such scenarios, providing a clear understanding of how a price change impacts the quantity of sugar one can purchase. By analyzing the key factors, we will explore the potential benefits and drawbacks of price adjustments in the sugar market.

The Mathematical Analysis

Let's begin with the problem at hand: If the price of sugar falls by 10%, a consumer can buy 22 kg more than before had the price been increased by 10%. Our task is to determine the original quantity of sugar that could have been bought for the same sum.

Step-by-Step Solution

Let the original price of sugar be P per kg. When the price falls by 10%, the new price becomes:

[ text{New price} P - 0.1P 0.9P ]

Let the quantity of sugar that can be bought at the original price P be Q kg. Therefore, the total cost for Q kg is:

[ text{Total cost} Q times P ]

At the new price, the quantity of sugar that can be bought is:

[ text{New quantity} frac{Q times P}{0.9P} frac{Q}{0.9} approx 1.111Q ]

When the price increases by 10%, the new price becomes:

[ text{New price} P 0.1P 1.1P ]

The quantity of sugar that can be bought at the increased price is:

[ text{Quantity at increased price} frac{Q times P}{1.1P} frac{Q}{1.1} approx 0.909Q ]

According to the problem, the difference in quantity bought when the price falls by 10% and when it increases by 10% is 22 kg:

[ frac{Q}{0.9} - frac{Q}{1.1} 22 ]

To solve for Q, we first find a common denominator:

[ frac{Q cdot 1.1 - Q cdot 0.9}{0.9 cdot 1.1} 22 ] [ frac{0.2Q}{0.99} 22 ]

Now multiply both sides by 0.99:

[ 0.2Q 22 times 0.99 ] [ 0.2Q 21.78 ]

Now solve for Q:

[ Q frac{21.78}{0.2} 108.9 ]

Therefore, the original quantity of sugar that could have been bought for the same sum is approximately 109 kg.

Alternative Approach

Another method involves setting up the equations based on the inverse proportionality of quantity to price:

Let p0 and p1 be the original price and the decreased price respectively, and let the corresponding quantities be q0 and q1.

Quantity is inversely proportional to the price:

[ frac{q1}{q0} frac{p0}{p1} frac{100}{90} frac{10}{9} ]

q1 - q0/q0 1/9 ...eq1

In the second case, the price is increased by 10%, and the quantity decreases to q2.

[ frac{q2}{q0} frac{p0}{p2} frac{10}{11} ]

From eq1 and eq2:

[ frac{q2}{q1 - q0} frac{9}{11} times frac{10}{9} frac{90}{11} ] [ q2 frac{90}{11} (q1 - q0) ]

Given q1 - q0 18 kg:

[ q2 frac{90}{11} times 18 text{ kg} 147.27 text{ kg} ]

The key takeaway is that a decrease in sugar price can significantly increase the quantity of sugar that can be purchased, and vice versa.

Real-World Implications

The fluctuation in sugar prices can have real-world implications for consumers and the market. If sugar prices fall, as we have seen in the calculations, consumers can buy more sugar with the same amount of money, leading to increased consumption. Conversely, if the price increases, the quantity of sugar that can be bought decreases, which can have a significant impact on the overall expenditure of consumers.

For instance, if the price of sugar decreases by 12.5%, or 1/8, the consumer can obtain 1/7 more sugar, which equals 9 kg. If the original quantity of sugar was 63 kg, the new quantity would be 72 kg. At a new price of Rs. 9/4 per kg, the total cost in Rs. 126 would allow the consumer to purchase 56 kg of sugar, resulting in a 7 kg reduction in quantity.

Understanding these principles can help consumers make informed decisions and better manage their budgets in response to market fluctuations.