Question

Subtract the following fractions and mixed numbers. Reduce to lowest terms. \(39 \frac{11}{18}-8 \frac{3}{6}=\) ______

Short answer

\(31 \frac{1}{9}\)

Step-by-step solution

Convert Mixed Numbers to Improper Fractions

Firstly, convert the mixed numbers to improper fractions. For \(39 \frac{11}{18}\), multiply 39 by 18, then add 11: \((39 \times 18) + 11 = 702 + 11 = 713\). So, \(39 \frac{11}{18} = \frac{713}{18}\).\For \(8 \frac{3}{6}\), simplify the fraction first. \(\frac{3}{6} = \frac{1}{2}\). Now, multiply 8 by 2, then add 1: \((8 \times 2) + 1 = 16 + 1 = 17\). So, \(8 \frac{1}{2} = \frac{17}{2}\).

Make Denominators the Same

The fractions are \(\frac{713}{18}\) and \(\frac{17}{2}\). We need a common denominator. The least common multiple of 18 and 2 is 18. Convert \(\frac{17}{2}\) to have a denominator of 18 by multiplying both numerator and denominator by 9: \(\frac{17 \times 9}{2 \times 9} = \frac{153}{18}\).

Subtract the Fractions

Now, subtract \(\frac{153}{18}\) from \(\frac{713}{18}\):\\[\frac{713}{18} - \frac{153}{18} = \frac{713 - 153}{18} = \frac{560}{18}\].

Simplify the Fraction

To reduce \(\frac{560}{18}\) to its lowest terms, find the greatest common divisor (GCD) of 560 and 18, which is 2. Divide both numerator and denominator by their GCD: \(\frac{560 \div 2}{18 \div 2} = \frac{280}{9}\).

Convert Improper Fraction to Mixed Number

Finally, convert \(\frac{280}{9}\) back to a mixed number. Divide 280 by 9: 280 ÷ 9 = 31 remainder 1. This gives \(31 \frac{1}{9}\).

Key concepts

Improper Fractions

An improper fraction happens when the numerator, which is the top part of the fraction, is larger than the denominator, which is the bottom part. This means you have more parts than make up a whole, indicating that the value is greater than 1. To convert a mixed number into an improper fraction, you multiply the whole number by the denominator and then add the numerator. For example, with the mixed number \(39 \frac{11}{18}\), you multiply 39 (the whole number) by 18 (the denominator). Then add the 11 (the numerator) to get \(\frac{713}{18}\). Improper fractions are useful for performing arithmetic operations like addition and subtraction because they simplify the process by dealing with only numerators and denominators, rather than whole numbers.

Mixed Numbers

Mixed numbers combine a whole number with a fraction. They can be easier to understand when visualizing quantities larger than 1 whole. For instance, \(8 \frac{1}{2}\) clearly shows that there are 8 complete portions and half of another. For mathematical operations, it is often convenient to convert mixed numbers into improper fractions. Once calculated, the results can be converted back into mixed numbers for simplicity and clarity. For example, an improper fraction like \(\frac{280}{9}\) can be divided to show how many complete groups of 9 fit into 280, resulting in a mixed number \(31 \frac{1}{9}\). Remember that mixed numbers are ideal for expressing real-world quantities but may require conversion for calculation purposes.

Common Denominator

Fractions must have the same denominator to be added or subtracted directly. This shared denominator is known as the "common denominator", and it represents quantities divided into an equal number of parts. To find a common denominator, compute the least common multiple (LCM) of the numbers. In the example of subtracting \(\frac{713}{18}\) and \(\frac{17}{2}\), the LCM of 18 and 2 is 18. Thus, \(\frac{17}{2}\) is converted into \(\frac{153}{18}\) so that both fractions have the common denominator of 18. Using a common denominator simplifies the arithmetic operations, allowing for subtraction or addition of the numerators while keeping the denominator the same for both fractions.

Simplifying Fractions

This process involves reducing a fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD). It simplifies fractions to make them easier to understand and work with. For instance, if you have the fraction \(\frac{560}{18}\), identify the GCD of 560 and 18, which in this case is 2. Divide both the numerator and the denominator by 2 to simplify the fraction to \(\frac{280}{9}\). A simplified fraction shows the most concise way to express that ratio, making it easier to interpret and use in further mathematical operations.