Problem 5 - Algebra 2

Question

STANDARDIZED TEST PRACTICE The weight of a bar of soap decreases by 2.5\(\%\) each time it is used. If the bar weighs 95 grams when it is new, what is its weight to the nearest gram after 15 uses? $$\begin{array}{llll}{\text { A } 57.5 \mathrm{g}} & {\text { B } 59.4 \mathrm{g}} & {\text { C } 65 \mathrm{g}} & {\text { D } 93 \mathrm{g}}\end{array}$$

Step-by-Step Solution

Answer

B: 59.4 grams

Step 1: Understand the Problem

We have a bar of soap that weighs 95 grams initially and loses 2.5% of its weight each time it is used. We need to calculate its weight after 15 uses.

Step 2: Formulate the Weight Loss Equation

The weight loss formula for a compounding percentage decrease is given by \( W = W_0 \times (1 - r)^n \), where \( W_0 \) is the initial weight, \( r \) is the rate of decrease per use (2.5% or 0.025), and \( n \) is the number of uses (15).

Step 3: Substitute the Values in the Equation

Let's substitute the given values into the formula: \( W = 95 \times (1 - 0.025)^{15} \).

Step 4: Calculate the Value Inside the Parenthesis

Compute \( 1 - 0.025 = 0.975 \). This gives the retained weight after each use.

Step 5: Raise the Result to the Power of 15

Calculate \( 0.975^{15} \). This calculation helps determine the cumulative effect of 15 uses on the soap’s weight.

Step 6: Calculate the Final Weight

Multiply the initial weight by the calculated power: \( W = 95 \times 0.975^{15} \). Compute this to find the remaining weight of the soap.

Step 7: Round to the Nearest Gram

After computing, round the answer to the nearest whole number to find the final weight of the soap after 15 uses.

Key Concepts

Compounding Percentage Decrease

When items, like our bar of soap, decrease in weight or value by a certain percentage each time they are used, we encounter a concept called compounding percentage decrease. This is different from a simple percentage decrease where the same fixed amount is subtracted every time. In compounding, the decrease after every use is based on the remaining amount.

• If you start with a 95g bar of soap and it loses 2.5% weight diminishes to 97.5% of its previous weight after each use.
• Each time the soap is used, it will weigh 97.5% of its last weight.

To calculate this, we use the formula: \[ W = W_0 \times (1 - r)^n \]where:

• \(W\) is the final weight after all uses,
• \(W_0\) is the starting weight of the soap,
• \(r\) is the decimal fraction of the percentage decrease (2.5% becomes 0.025),
• \(n\) is the total number of times the soap has been used.

Weight Loss Equation

The weight loss equation helps us predict the future weight of an object subjected to repeated percentage decrease, like our soap bar. This equation is particularly useful in scenarios involving depreciation of asset value or biologically changing weights (e.g., weight loss in dieting). For our soap, plugging in the numbers looks like:\[ W = 95 \times (1 - 0.025)^{15} \]

• The initial weight, \(95\), represents the soap's original state before use.
• The term \((1 - 0.025)\) or 0.975, represents the remaining weight after one use.
• The exponent, \(15\), represents the compounded use cycle.

By calculating the expression \(0.975^{15}\), you can determine how much of the soap remains after being used 15 times.

Rounding

After calculating the new weight using the weight loss equation, it's essential to round the final figure. Rounding is important because it simplifies numbers, making them easier to interpret and use practically. For instance, you may calculate the soap's weight to be a value like 59.467 grams. In everyday scenarios, especially when dealing with weight measurements, it's typical to round to the nearest whole number.

• To round, look at the first decimal place.
• If the decimal is 5 or above, round up by adding 1 to the whole number.
• If it's 4 or below, round down by keeping the whole number as is.

Thus, 59.467 rounds to 59 grams as our final answer. This rounded value is more practical for everyday use without sacrificing accuracy significantly.