Problem 62 - Prealgebra

Question

Simplify the given expression. \(\frac{5}{3}+0.55\)

Step-by-Step Solution

Answer

\(\frac{5}{3} + 0.55 = \frac{133}{60}\).

Step 1: Convert Decimal to Fraction

Start by converting the decimal number 0.55 into a fraction. To do so, recognize that 0.55 can be expressed as \(\frac{55}{100}\). Simplify this fraction by finding the greatest common divisor (GCD) of 55 and 100, which is 5. Dividing both the numerator and the denominator by 5 gives \(\frac{11}{20}\).

Step 2: Find a Common Denominator

The next step is to add the fractions \(\frac{5}{3}\) and \(\frac{11}{20}\). To add these fractions, you need a common denominator. The denominators 3 and 20 have a least common multiple (LCM) of 60.

Step 3: Adjust Fractions to the Common Denominator

Convert \(\frac{5}{3}\) to have the common denominator 60. Multiply both the numerator and the denominator by 20 to get \(\frac{100}{60}\). Convert \(\frac{11}{20}\) by multiplying both the numerator and the denominator by 3 to get \(\frac{33}{60}\).

Step 4: Add the Fractions

Now that both fractions have the same denominator, add them together: \(\frac{100}{60} + \frac{33}{60} = \frac{133}{60}\).

Step 5: Simplify the Result

Check if \(\frac{133}{60}\) can be simplified. Since 133 and 60 do not have any common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Decimal to Fraction Conversion

Converting a decimal to a fraction is an essential skill in math. It involves expressing a decimal as a fraction in its simplest form. Let's take the example of 0.55. This number can be directly translated into a fraction by using the place value of the decimal.
- In 0.55, the number 55 is in the hundredths place, meaning it can be written as \( \frac{55}{100} \).- To simplify \( \frac{55}{100} \), you need to find the greatest common divisor (GCD) of 55 and 100. Once simplified using the GCD, you'll arrive at the simplest form, which is \( \frac{11}{20} \). Convert any decimal to a fraction using similar steps: convert the decimal to a fraction over 10, 100, 1000, etc., and then simplify if possible.

Common Denominator

When adding fractions with different denominators, it's important to find a common denominator. This process makes it possible to perform the addition.To find a common denominator:
- Look for the least common multiple (LCM) of the denominators. For example, between 3 and 20, the LCM is 60.- Convert each original fraction to an equivalent fraction with the common denominator. This step involves multiplying both the numerator and the denominator by the necessary factor to reach the LCM.Using the fractions from our exercise: \( \frac{5}{3} \) becomes \( \frac{100}{60} \) and \( \frac{11}{20} \) becomes \( \frac{33}{60} \). With this uniformity, the fractions can be added easily.

Simplifying Fractions

After performing arithmetic operations with fractions, it's always good to simplify them. Simplifying a fraction means reducing it to its smallest possible form.
- Simplified fractions have numerators and denominators that can't be divided by any number other than one.In our problem, when we added \( \frac{100}{60} \) and \( \frac{33}{60} \), we got \( \frac{133}{60} \). To simplify further, check the greatest common divisor between 133 and 60:
- Since 133 and 60 do not share any common factors besides 1, \( \frac{133}{60} \) is already in its simplest form.Always ensure that the final fraction is simplified for clarity and to meet mathematical standards.

Greatest Common Divisor

Understanding the greatest common divisor (GCD) is crucial when working with fractions. The GCD of two numbers is the largest number that divides both without leaving a remainder.
- To simplify a fraction, divide the numerator and the denominator by their GCD.In our exercise, the GCD of 55 and 100 is 5. Dividing \( \frac{55}{100} \) by 5 gives us \( \frac{11}{20} \).Finding the GCD helps reduce fractions efficiently, making calculations easier. You can find the GCD through techniques such as listing factors or using the Euclidean algorithm for larger numbers.