Problem 53 - Prealgebra

Question

In Exercises 53-60, simplify the given expression by first converting the decimal into a fraction. \(\frac{7}{6}-2.9\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{-26}{15}\).

Step 1: Convert the Decimal to a Fraction

The decimal \(2.9\) needs to be converted into a fraction. To do this, recognize that \(2.9 = 2 + 0.9\). The decimal \(0.9\) is \(\frac{9}{10}\) because it represents the tenths place. Therefore, \(2.9 = 2 + \frac{9}{10}\). Convert 2 as a fraction with the same denominator to have \(\frac{20}{10}\). Thus, \(2.9 = \frac{20}{10} + \frac{9}{10} = \frac{29}{10}\).

Step 2: Rewrite the Entire Expression

Now that we have the fraction representation of \(2.9\), rewrite the given expression. It becomes: \(\frac{7}{6} - \frac{29}{10}\).

Step 3: Find a Common Denominator

To perform the subtraction, we need a common denominator. The denominators are 6 and 10. The least common multiple of 6 and 10 is 30. Rewrite each fraction with the denominator of 30. \(\frac{7}{6} = \frac{7 \times 5}{6 \times 5} = \frac{35}{30}\) and \(\frac{29}{10} = \frac{29 \times 3}{10 \times 3} = \frac{87}{30}\).

Step 4: Subtract the Fractions

Now that both fractions have the same denominator, subtract their numerators: \(\frac{35}{30} - \frac{87}{30} = \frac{35 - 87}{30} = \frac{-52}{30}\).

Step 5: Simplify the Resulting Fraction

Simplify \(\frac{-52}{30}\) by finding the greatest common divisor (GCD) of 52 and 30, which is 2. Divide the numerator and the denominator by 2: \(\frac{-52 \div 2}{30 \div 2} = \frac{-26}{15}\).

Key Concepts

Converting Decimals to Fractions

Converting decimals into fractions is a fundamental skill that simplifies handling numbers and expressions, especially in arithmetic operations. When you have a decimal, the first step is to identify the place value of the digits involved. For example, with the decimal 2.9:

• The number 2 represents the whole number part.
• The decimal 0.9 represents nine-tenths because it is in the tenths place.

To express 2.9 as a fraction, we break it down:

• 2 is converted to a fraction as \(\frac{20}{10}\) because any whole number "n" can be expressed as \(\frac{n \times 10}{10}\).
• Similarly, 0.9 becomes \(\frac{9}{10}\).

Now, combine these fractions to represent the entire decimal: \(\frac{20}{10} + \frac{9}{10} = \frac{29}{10}\). Thus, 2.9 is equivalent to \(\frac{29}{10}\). Understanding this conversion process makes it easier to work with more complex expressions involving decimals.

Finding Common Denominators

In order to subtract or add fractions, it's crucial first to ensure they have a common denominator. This common denominator allows the fractions to be expressed with the same base, making addition or subtraction possible. Consider two fractions, such as \(\frac{7}{6}\) and \(\frac{29}{10}\). Their denominators are 6 and 10, respectively. To find a common denominator, we determine the least common multiple (LCM) of these denominators.The LCM of 6 and 10 is 30 because:

• Factors of 6: 2 and 3
• Factors of 10: 2 and 5

Thus, the smallest number that both denominators can evenly divide into is 30. Adjust the original fractions to have this new denominator:

• Transform \(\frac{7}{6}\) to \(\frac{35}{30}\) by multiplying both the numerator and denominator by 5.
• Convert \(\frac{29}{10}\) to \(\frac{87}{30}\) by multiplying both by 3.

Now each fraction is expressed with a denominator of 30, ready for subtraction!

Simplifying Fractions

After converting and adjusting fractions to have common denominators, you often end up with a new fraction that can be simplified. Simplification involves reducing the fraction to its smallest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).For example, after subtracting in our exercise, we achieved \(\frac{-52}{30}\). To simplify:

• First, find the GCD of 52 and 30, which is 2.
• Divide both the numerator and the denominator by their GCD: \(\frac{-52 \div 2}{30 \div 2} = \frac{-26}{15}\).

Simplifying fractions not only makes the result more aesthetically pleasing but often easier to interpret. It's a crucial final step in ensuring your expressions are presented in the clearest, most concise form possible.