Problem 9 - Prealgebra

Question

Write each number as a fraction or a mixed number. Do not reduce your answers. $$405.36$$

Step-by-Step Solution

Answer

405.36 can be written as \( 405 \frac{36}{100} \) or \( \frac{40536}{100} \).

Step 1: Understand the Components

The number given is 405.36, which consists of a whole number part and a decimal part. Here, the whole number part is 405 and the decimal part is 0.36.

Step 2: Write the Whole Number as a Fraction

Any whole number can be written as a fraction by placing it over 1. Therefore, 405 as a fraction is \( \frac{405}{1} \).

Step 3: Convert the Decimal Part to a Fraction

The decimal 0.36 is equivalent to \( \frac{36}{100} \) because it is in the hundredths place.

Step 4: Combine into a Mixed Number

A mixed number consists of a whole number and a fraction. Combine the whole number 405 with the fraction \( \frac{36}{100} \) to get the mixed number form: \( 405 \frac{36}{100} \).

Step 5: Write the Complete Fraction

Combine the whole number fraction with the decimal fraction over a common denominator: \( 405 \frac{36}{100} = \frac{40500}{100} + \frac{36}{100} = \frac{40536}{100} \).

Key Concepts

Decimal to Fraction Conversion

Converting a decimal to a fraction is a key skill in math that makes it easier to perform operations and understand numbers. When you see a decimal number like 0.36, realize that it represents a certain number of parts out of a whole, usually out of powers of ten, such as tenths, hundredths, etc. Here, 0.36 is in the "hundredths" place, which makes it equal to \( \frac{36}{100} \). This is because:

• The digit 3 is in the "tenths" place, representing 30 out of 100.
• The digit 6 is in the "hundredths" place, representing 6 out of 100.

Combine the two and you have 36 out of 100. This method allows you to transform any decimal into a fraction by calculating its equivalent in terms of tens, hundreds, and so on. This step is crucial before mixing it with a whole number to form a mixed number.

Mixed Numbers

Mixed numbers are a combination of whole numbers and fractions. They are particularly useful when dealing with quantities that have portions greater than one whole. In our example, the number 405.36 can be broken down into a whole number (405) and a fraction part (0.36). By converting 0.36 into a fraction \( \frac{36}{100} \), you can then express the entire decimal number as a mixed number: 405 \( \frac{36}{100} \).Mixed numbers are great for understanding values beyond 1 but under a larger number, just like pizza slices; one whole pizza and a partial slice can express as a mixed number. You see the entire value but can also figure out what part is extra. This improves clarity for mathematical operations that involve real-life measurements and quantities.

Mathematics Education

Teaching students to convert decimals to fractions and express them as mixed numbers enriches their mathematical comprehension. These skills build foundational understanding crucial for more complex problems and math literacy. By mastering such conversions:

• Students can better perform arithmetic calculations, as fractions are easier to add, subtract, multiply, and divide than decimals in many situations.
• It provides a clearer understanding of underlying numerical structures, useful across various STEM disciplines.
• Students learn to appreciate the versatility of numbers in different forms, fostering flexibility in problem-solving.

Through practice with exercises like translating 405.36 into a fraction or a mixed number, learners encourage cognitive development in handling numbers creatively and critically, integral to successful mathematics education.