Problem 24 - Prealgebra

Question

The White House covers an area of 0.028 square mile. What fraction of a square mile is this?

Step-by-Step Solution

Answer

The area of the White House is \(\frac{7}{250}\) of a square mile.

Step 1: Understanding the Problem

We are given that the White House covers an area of 0.028 square mile. We need to express this area as a fraction of a square mile.

Step 2: Representation as a Fraction

Any decimal can be represented as a fraction. Here, 0.028 can be represented as \(\frac{28}{1000}\) because 0.028 means 28 thousandths.

Step 3: Simplifying the Fraction

Simplify the fraction \(\frac{28}{1000}\) by finding the greatest common factor of 28 and 1000, which is 4. Divide both the numerator and the denominator by 4 to simplify: \(\frac{28 \div 4}{1000 \div 4} = \frac{7}{250}\).

Step 4: Final Fraction

The simplified fraction that represents the area of the White House relative to a square mile is \(\frac{7}{250}\).

Key Concepts

Decimal to Fraction Conversion

Converting a decimal to a fraction might seem challenging at first, but it's a straightforward process once you understand the steps involved. A decimal number like 0.028 can be transformed into a fraction by recognizing its place value.

Each decimal digit represents a specific place value:

• The first digit after the decimal point is the tenths place.
• The second is the hundredths place.
• The third is the thousandths place.

In our example, 0.028 has three digits after the decimal, placing the number 28 in the thousandths position. This means that 0.028 can be expressed as a fraction with 28 as the numerator and 1000 as the denominator, or \(\frac{28}{1000}\).

Understanding how each decimal converts into a fraction lays a firm foundation for further simplification.

Simplifying Fractions

After converting a decimal to a fraction, the next step is to simplify it. Simplifying a fraction means reducing it to its simplest form. Simplifying fractions makes them easier to read and work with in calculations or comparisons.

To simplify \(\frac{28}{1000}\), we need to divide both the numerator (28) and the denominator (1000) by their Greatest Common Factor (GCF). This process is known as reducing.

The GCF is the largest integer that divides both numbers without leaving a remainder. By simplifying, we ensure that the fraction represents the same quantity but with smaller and more manageable numbers.

Greatest Common Factor

The Greatest Common Factor (GCF) is crucial when simplifying fractions. It helps find the highest factor that two numbers share, allowing for the reduction of the fraction to its simplest form.

To determine the GCF of 28 and 1000, factor each number:

• 28 can be factored as 1, 2, 4, 7, 14, 28.
• 1000 can be factored as 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000.

The largest number that appears in both lists is 4, which means the GCF is 4. By dividing both 28 and 1000 by 4, the fraction \(\frac{28}{1000}\) simplifies to \(\frac{7}{250}\).

Learning to find the GCF is an essential skill in simplifying fractions and improving number sense.