Problem 49 - Prealgebra

Question

Of all the cars sold in the United States in \(2003, \frac{2}{5}\) were imported from Japan and 0.26 were imported from Germany. Are more Japanese or German cars sold in the United States? Explain.

Step-by-Step Solution

Answer

More Japanese cars were sold in the U.S. than German cars.

Step 1: Understanding the Problem

We are given the fractions of cars sold in the U.S. that were imported from Japan and Germany in 2003. We need to compare these fractions to determine which country exported more cars to the U.S.

Step 2: Converting the Fractions

The problem provides "fraction of cars imported from Germany" as a decimal: 0.26. We need to compare this with the fraction of cars from Japan: \(\frac{2}{5}\). To compare, both fractions should be in the same form: decimal or common fraction.

Step 3: Convert Japanese Fraction to Decimal

Convert \(\frac{2}{5}\) into decimal form. Divide 2 by 5 to find the decimal equivalent: \(\frac{2}{5} = 0.4\).

Step 4: Comparison

Now we compare the two decimals: 0.4 and 0.26. Since 0.4 is greater than 0.26, more cars were imported from Japan than from Germany.

Key Concepts

Decimal Conversion

Converting a fraction to a decimal is a useful skill in comparing quantities. It allows us to easily visualize and compare percentages or values. Here’s how you can do it:

• Identify the fraction: In our problem, we have the fraction \(\frac{2}{5}\).
• Perform the division: To convert \(\frac{2}{5}\) to a decimal, divide 2 by 5, yielding 0.4.
• Use the decimal: Now you have a decimal you can easily compare with other decimals like 0.26.

Converters like calculators make these conversions simple, and understanding them deepens your math skills, especially in problem-solving scenarios.

Fractions

Fractions represent parts of a whole, useful in expressing quantities that are less than one. In our exercise, \(\frac{2}{5}\) implies 2 parts of a whole that is divided into 5 equal parts.

• Numerator and Denominator: The top number (numerator) shows how many parts we have. The bottom number (denominator) shows how many parts make up a whole.
• Comparison: Though comparing fractions directly is possible using a common denominator or other methods, conversion to decimals is often easier.
• Usage: Fractions are used in various applications like splitting bills, measuring ingredients, and analyzing data.

Understanding how fractions work is fundamental in many aspects of math problem-solving.

Math Problem Solving

Math problem solving involves several steps that help in reaching a solution efficiently. Breaking down complex scenarios into simpler, understandable chunks is crucial.

• Understand the problem: Clearly identify what is being asked. In our case, we need to determine which country exported more cars to the U.S.
• Convert and compare: Use conversion techniques, such as converting fractions to decimals, to make comparison straightforward.
• Logical reasoning: Once you have comparable values, apply logical reasoning to determine which is greater or more appropriate.
• Check your answer: After solving the problem, re-evaluate the steps to ensure accuracy.

These steps streamline math problem-solving and make seemingly difficult problems approachable.