Problem 28 - Elementary and Intermediate Algebra

Question

Translate each ratio into a fraction in simplest form. 5 tons to \(4,000\) pounds

Step-by-Step Solution

Answer

The fraction in simplest form is \( \frac{5}{2} \).

Step 1: Understanding the Unit Conversion

Before we can express the ratio as a fraction, we need to ensure both quantities are in the same units. Given that 1 ton equals 2,000 pounds, we need to convert tons into pounds. Thus, 5 tons is equivalent to \( 5 \times 2000 = 10000 \) pounds.

Step 2: Set Up the Ratio

Write the ratio with the same units in fraction form. The ratio given in the problem is 5 tons to 4,000 pounds. Convert this directly to pounds to get \( \frac{10000}{4000} \).

Step 3: Simplify the Fraction

Now, simplify the fraction \( \frac{10000}{4000} \). Find the greatest common divisor (GCD) of 10,000 and 4,000, which is 2,000. Divide both the numerator and the denominator by their GCD: \( \frac{10000 \div 2000}{4000 \div 2000} = \frac{5}{2} \).

Step 4: Express the Final Solution

After simplifying, we find the ratio as a fraction in its simplest form is \( \frac{5}{2} \).

Key Concepts

Ratio Conversion

When dealing with ratios, it’s vital to understand that they simply compare two quantities. Think about ratios as a way to express how much of one thing there is compared to another. In our given exercise, we began with a scenario of 5 tons compared to 4,000 pounds. To translate this into a fraction, we need to ensure that both parts of the ratio are expressed in the same units.

Here's a simple process:

• Identify the quantities being compared.
• Ensure both are in the same measurement units (more about this in the next section).
• Set them up as a fraction.

By translating the ratio into a fraction, you create a straightforward method to express the relationship numerically. For instance, after unit conversion, the ratio of 5 tons to 4,000 pounds becomes the fraction \( \frac{10000}{4000} \). This makes it both easier to understand and manipulate mathematically.

Unit Conversion

Unit conversion is a crucial step when working with ratios, especially when they involve different measurement units such as weight or length. It involves converting one unit into another to ensure a fair comparison. In the context of our problem, we needed to convert tons into pounds before creating the fraction.

Here's how to do it:

• Understand the conversion factor between the units. For this case: 1 ton = 2,000 pounds.
• Multiply the quantity you are converting by the conversion factor. Thus, 5 tons convert to \(5 \times 2000 = 10000 \) pounds.

After completing the unit conversion, you can proceed to express your ratio accurately, ensuring that you are comparing apples to apples. Proper unit conversion eliminates confusion and ensures accurate and meaningful relationships between the quantities being compared.

Simplifying Fractions

Simplifying fractions is an essential mathematical skill that helps to express fractions in their simplest form. Simplification makes it easier to work with numbers, allowing for clearer interpretation and more straightforward calculations.

To simplify:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by their GCD.

In our exercise, the fraction \( \frac{10000}{4000} \) was simplified by identifying their GCD, which is 2,000. By dividing both by 2,000, we reduced the fraction to \( \frac{5}{2} \). This new fraction communicates the same ratio as before but in a clearer and more straightforward format.