Problem 56 - Prealgebra

Question

Simplify the given expression by first converting the decimal into a fraction. \(\frac{11}{6}-0.375\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{35}{24} \).

Step 1: Convert the Decimal to a Fraction

First, we need to convert the decimal number 0.375 into a fraction. Because 0.375 has three decimal places, we can write it as: \[ 0.375 = \frac{375}{1000} \]Now, simplify this fraction. Both the numerator and the denominator can be divided by 125:\[ \frac{375 \div 125}{1000 \div 125} = \frac{3}{8} \]So, 0.375 can be expressed as \( \frac{3}{8} \).

Step 2: Find a Common Denominator

Now we need to perform the subtraction \( \frac{11}{6} - \frac{3}{8} \) using equations with common denominators. The denominators are 6 and 8. The least common multiple of 6 and 8 is 24, so we will convert both fractions to have 24 as the denominator.

Step 3: Convert Fractions to Common Denominator

Convert \( \frac{11}{6} \) to a fraction with a denominator of 24:\[ \frac{11}{6} \times \frac{4}{4} = \frac{44}{24} \]Convert \( \frac{3}{8} \) to a fraction with a denominator of 24:\[ \frac{3}{8} \times \frac{3}{3} = \frac{9}{24} \]

Step 4: Subtract the Fractions

Now that both fractions have a common denominator, we can subtract them:\[ \frac{44}{24} - \frac{9}{24} = \frac{44 - 9}{24} = \frac{35}{24} \]

Step 5: Simplify the Result if Necessary

Check if the fraction \( \frac{35}{24} \) can be simplified. Since 35 and 24 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Decimal to Fraction Conversion

Turning a decimal into a fraction may sound tricky, but it's quite straightforward when you break it down. Take the decimal 0.375. The process involves a few simple steps:

• Count the number of decimal places. Here, 0.375 has three decimal places.
• Convert the decimal into a fraction by using a denominator that is a power of ten. So, 0.375 becomes \( \frac{375}{1000} \).
• Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator. For \( \frac{375}{1000} \), the GCD is 125.
• Divide both the numerator and the denominator by this number to simplify the fraction. \( \frac{375 \div 125}{1000 \div 125} = \frac{3}{8} \).

Keep in mind that not all decimals become whole numbers in the numerator when multiplied by powers of ten. Always simplify the fraction to its lowest terms.

Least Common Denominator

Fractions need a common denominator to be added or subtracted effectively. This is because fractions are parts of a whole and need to be measured consistently. The least common denominator (LCD) is the smallest number that both denominators can divide into evenly. Finding the LCD for fractions like \( \frac{11}{6} \) and \( \frac{3}{8} \) involves:

• Listing the multiples of each denominator. For 6, it's 6, 12, 18, 24, etc., and for 8, it's 8, 16, 24, etc.
• Finding the smallest common multiple. In this exercise, both 6 and 8 have a common multiple at 24.
• Changing each fraction so that the denominator becomes the LCD. Multiply the numerator and denominator of \( \frac{11}{6} \) by 4 to get \( \frac{44}{24} \) and of \( \frac{3}{8} \) by 3 to get \( \frac{9}{24} \).

Making the denominators the same allows you to seamlessly carry out arithmetic operations between fractions.

Subtracting Fractions

Once both fractions in an expression share a least common denominator, subtracting them is a breeze. You only need to focus on the numerators since the denominators are already aligned. Here's how to subtract fractions like \( \frac{44}{24} \) and \( \frac{9}{24} \):

• Subtract the numerator of the second fraction from the numerator of the first fraction: 44 - 9 = 35.
• Keep the common denominator: 24.
• The result is \( \frac{35}{24} \).

Remember, always check if your resulting fraction can be simplified by finding factors common to the numerator and denominator. In this case, since 35 and 24 don't share any common factors other than 1, \( \frac{35}{24} \) is already in its simplest form. Simplification ensures your answer is as reduced and clean as possible, much like keeping your math workspace tidy!