Problem 87 - Algebra for College Students

Question

Solve each problem. Maritime losses. The amount paid to an insured party by the American Insurance Company is computed by using the proportion \(\frac{\text { value shipped }}{\text { amount of loss }}=\frac{\text { amount of declared premium }}{\text { amount insured party gets paid }}\) If the value shipped was \(\$ 300,000,\) the amount of loss was \(\$ 250,000,\) and the amount of declared premium was \(\$ 200,000,\) then what amount is paid to the insured party?

Step-by-Step Solution

Answer

The amount paid to the insured party is $166,666.67.

Step 1 - Understand the Proportion

The problem states that the amount paid by the insurance company follows a specific proportion: \[\frac{\text{value shipped}}{\text{amount of loss}} = \frac{\text{amount of declared premium}}{\text{amount paid to insured party}}\]

Step 2 - Identify Given Values

Identify the values provided in the problem: \[\text{value shipped} = 300,000 \ \text{amount of loss} = 250,000 \ \text{amount of declared premium} = 200,000\]

Step 3 - Set Up the Proportion

Substitute the given values into the proportion: \[\frac{300,000}{250,000} = \frac{200,000}{\text{amount paid to insured party}}\]

Step 4 - Solve for the Amount Paid

To find the amount paid by the insurance company, let’s call this amount \( P \). Substitute into the equation and solve for \( P \): \[\frac{300,000}{250,000} = \frac{200,000}{P}\] Cross-multiply to get: \[300,000 \times P = 250,000 \times 200,000 \] Solving for \( P \): \[ P = \frac{250,000 \times 200,000}{300,000} = \frac{50,000,000,000}{300,000} = 166,666.67\]

Key Concepts

solving proportions

Proportions are mathematical expressions that demonstrate that two ratios or fractions are equal. To solve a proportion, we typically find out the unknown value in one of the ratios. This is a common problem in various subjects, including algebra.
In real-life contexts, proportions can depict scenarios like monetary transactions, similar to the insurance problem we have. For example, if you know the proportion for a business deal, you can find out missing values related to cost, revenue, or losses.
Let's break down the steps to solve a proportion:

• Identify the given values and label them correctly.
• Set up the proportion with the unknown value represented by a variable, typically denoted as 'x' or, in our exercise, as 'P'.
• Use cross-multiplication to eliminate the fractions.
• Solve the equation to find the unknown value.

cross-multiplication

One of the key techniques for solving proportions is cross-multiplication. This method involves multiplying the numerator of one fraction by the denominator of the other fraction. Let's see how it works in our insurance problem:
The proportion provided is:
\(\frac{300,000}{250,000} = \frac{200,000}{P}\)
Here, we need to find the amount paid to the insured party, which is denoted as P.
To solve for P, we cross-multiply:

• Multiply 300,000 by P.
• Multiply 250,000 by 200,000.

This gives us the equation:
\(300,000 \times P = 250,000 \times 200,000\)
Now, isolate P by dividing both sides of the equation by 300,000:
\(P = \frac{250,000 \times 200,000}{300,000}\)
Simplify to get the final value of P:
\(P = 166,666.67\)
Cross-multiplication is a powerful tool because it simplifies solving for an unknown value quickly and accurately.

real-world applications of algebra

Algebra isn't just confined to textbooks and classrooms; it has numerous real-world applications. The maritime insurance problem we solved is a perfect example of this.
Insurance companies use algebraic proportions to determine payouts, but other applications include:

• Financial calculations such as loans, mortgages, and interests.
• Medicine, for dosage calculations based on patient weight and age.
• Engineering, where it's used to calculate forces, distances, and currents.
• Recipe adjustments in cooking, where ingredients proportions need to be altered to serve different numbers of people.

Understanding how to solve proportions using cross-multiplication not only helps in academic problems but also prepares one to tackle practical challenges efficiently. Hence, building a strong foundation in these algebraic techniques is essential for critical thinking and problem-solving in everyday life.