Problem 9 - Elementary and Intermediate Algebra

Question

Snacks. In a sample of 25 bags of potato chips, 2 were found to be underweight. Complete the following proportion that could be used to find the number of underweight bags that would be expected in a shipment of \(1,000\) bags of potato chips.

Step-by-Step Solution

Answer

Expected 80 underweight bags in 1,000 bags.

Step 1: Define the Known Proportion

We know from the problem that in a sample of 25 bags, 2 were found to be underweight. This gives us a proportion of underweight bags: \( \frac{2}{25} \).

Step 2: Set Up the Proportion Equation

We want to find the number of underweight bags in a shipment of 1,000 bags, represented as \( x \). The proportion of underweight bags should remain the same, so we set up the equation: \( \frac{2}{25} = \frac{x}{1000} \).

Step 3: Solve the Proportion Equation for x

Cross-multiply to solve for \( x \): \( 2 \times 1000 = 25 \times x \). This simplifies to \( 2000 = 25x \).

Step 4: Isolate x

Divide both sides of the equation by 25 to solve for \( x \): \( x = \frac{2000}{25} = 80 \).

Key Concepts

Solving Proportion Equations

Solving proportion equations involves finding an unknown value within a given ratio. Often, these equations appear when you need to predict an outcome from a smaller sample, based on known proportions. To solve a proportion equation, you essentially equate two ratios. In mathematical terms, a ratio is usually represented as a fraction. For instance, in the potato chips problem, you start by recognizing the known proportion:

• Sample of 25 bags, with 2 found underweight, gives the ratio: \( \frac{2}{25} \).
• The task is to apply this ratio to a larger group of 1,000 bags, represented by the equation \( \frac{x}{1000} \).

By solving the equation \( \frac{2}{25} = \frac{x}{1000} \), you're effectively scaling up the known small sample to predict the outcome for the larger shipment. This method ensures consistency and accuracy when extrapolating data, making proportion equations an invaluable tool in real-world scenarios.

Cross-Multiplication

Cross-multiplication is an essential technique for solving equations that involve proportions. It allows you to easily manipulate the equation to isolate the unknown variable. When dealing with a proportion like \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other:

• First, multiply \( a \) by \( d \).
• Multiply \( b \) by \( c \).

The cross-products are then set equal to each other, as shown in the equation: \[a \times d = b \times c\]This approach simplifies the problem to a linear equation. For the original problem, cross-multiplying the proportions \( \frac{2}{25} = \frac{x}{1000} \) leads to:

• Calculating \( 2 \times 1000 = 2000 \).
• Calculating \( 25 \times x \).

Which results in the new equation: \( 2000 = 25x \).This is a straightforward way to move closer to solving for \( x \), which represents the anticipated number of underweight bags.

Ratio and Proportion Concepts

Understanding ratio and proportion concepts is key to making sense of how quantities relate to each other in a consistent manner. A ratio compares two quantities, revealing their relative sizes. For example, the ratio \( \frac{2}{25} \) in the potato chips context tells us how many bags are underweight in our sample. Proportion, on the other hand, involves two equal ratios and is used to solve for unknown variables.Key points to remember about these concepts include:

• A ratio expresses a part-to-part or part-to-whole relationship, using a fraction.
• Proportion indicates that two ratios are equivalent.
• By setting up a proportion, you can solve problems where you scale a known quantity to a larger unknown quantity.

These concepts form the basis of many real-world and mathematical problems, where you're often required to maintain consistency across different scales. Proper understanding aids in accurate predictions, such as determining the expected number of underweight bags in larger samples based on observed data in smaller samples. This ensures decisions are data-driven and reliable.