Problem 7 - Prealgebra

Question

Write each decimal as a fraction or mixed number in simplest form. $$3.16$$

Step-by-Step Solution

Answer

3.16 as a mixed number is \( 3 \frac{4}{25} \).

Step 1: Understand the Decimal Representation

The number given is a decimal, 3.16, which means it consists of a whole number and a fractional part. The whole number is 3.

Step 2: Convert Decimal to Fraction

The decimal 0.16 can be expressed as a fraction. Since 16 is in the hundredths place, it can be written as \( \frac{16}{100} \).

Step 3: Simplify the Fraction

Simplify the fraction \( \frac{16}{100} \) by finding the greatest common divisor (GCD) of 16 and 100, which is 4. This gives \( \frac{16 \div 4}{100 \div 4} = \frac{4}{25} \).

Step 4: Form the Mixed Number

Combine the whole number with the simplified fraction to express the entire number as a mixed number: \( 3 + \frac{4}{25} \), which is simply \( 3 \frac{4}{25} \).

Key Concepts

Simplifying Fractions

Fractions can seem tricky at first, but simplifying them makes them much easier to work with. Simplifying a fraction means reducing it to its simplest form. To do this, we need to divide both the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor (GCD). For example, let's simplify the fraction \( \frac{16}{100} \). First, we need to find the GCD of 16 and 100. A handy shortcut is to list the factors for each number:

• Factors of 16: 1, 2, 4, 8, 16
• Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100

The greatest number in both lists is 4. So, the GCD is 4. Now, divide both the numerator and the denominator by 4: \( \frac{16 \div 4}{100 \div 4} = \frac{4}{25} \). Now, the fraction is simplified! Simplifying helps us see the fraction in its most basic form, making calculations and comparisons easier.

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. This format is quite common and very useful for representing quantities that are not whole, like \(3 \frac{4}{25} \). Understanding mixed numbers helps when dealing with everyday situations, such as measurements in cooking or distance in miles and yards.
To convert a decimal to a mixed number, follow these easy steps:

• Identify the whole number part of the decimal. For 3.16, the whole number part is 3.
• The decimal part (0.16) can be converted into a fraction.

Once you have the whole number and the fraction, simply combine them. That's it! Mixed numbers allow us to express quantities in a friendly, easily understandable format.

Decimal Place Value

Understanding decimal place value is important for converting decimals to fractions. Each digit in a decimal number has a place value, which tells us how much that digit really represents. Decimal numbers have parts smaller than 1, separately counted by tenths, hundredths, thousandths, and so on.Let's break down 3.16:

• The '3' is in the units (or ones) place, representing 3 whole units.
• The '1' after the decimal point is in the tenths place, representing 0.1.
• The '6' is in the hundredths place, representing 0.06.

Adding the decimal part together, 0.1 (one tenth) plus 0.06 (six hundredths), we arrive at 0.16.
This knowledge helps us write the decimal as a fraction, such as \(\frac{16}{100}\), for 0.16. By simplifying this fraction, we refine it further to the simplest form, \(\frac{4}{25}\). This understanding is vital as it bridges decimals and fractions through clear place values.