Question

Add the following fractions and mixed numbers. Reduce to lowest terms. \(6 \frac{1}{4}+\frac{2}{9}+\frac{1}{36}=\) ______

Short answer

The sum is \(6 \frac{1}{2}\).

Step-by-step solution

Convert Mixed Number to Improper Fraction

Convert the mixed number \(6 \frac{1}{4}\) to an improper fraction. The mixed number can be rewritten as \((6 \times 4 + 1)/4 = 25/4\). Thus, \(6 \frac{1}{4} = \frac{25}{4}\).

Find a Common Denominator

Identify a common denominator for \(\frac{25}{4}, \frac{2}{9}, \) and \(\frac{1}{36}\). The denominators are 4, 9, and 36. The least common denominator (LCD) is 36.

Adjust Fractions to the Common Denominator

Convert each fraction to have the denominator of 36. - For \(\frac{25}{4}\), multiply both numerator and denominator by 9: \(\frac{25 \times 9}{4 \times 9} = \frac{225}{36}\).- \(\frac{2}{9}\), multiply both by 4: \(\frac{2 \times 4}{9 \times 4} = \frac{8}{36}\).- \(\frac{1}{36}\) is already set with the common denominator.

Add the Fractions

Add the converted fractions:\(\frac{225}{36} + \frac{8}{36} + \frac{1}{36} = \frac{225+8+1}{36} = \frac{234}{36}\).

Simplify the Fraction

Simplify \(\frac{234}{36}\) by finding the greatest common divisor (GCD). The GCD of 234 and 36 is 18. Divide both the numerator and the denominator by 18:\(\frac{234 \div 18}{36 \div 18} = \frac{13}{2}\).

Express as a Mixed Number

Convert \(\frac{13}{2}\) into a mixed number by dividing. The result is 6 with a remainder of 1: Thus, \(\frac{13}{2} = 6 \frac{1}{2}\).

Key concepts

mixed numbers

Mixed numbers are a combination of a whole number and a fraction. They can represent quantities that are larger than a simple fraction can express. For example, the mixed number \(6 \frac{1}{4}\) means you have six whole units plus another quarter of a unit.

To work with mixed numbers in arithmetic operations like addition or subtraction, it's often necessary to convert them into improper fractions first. An improper fraction has a numerator larger than its denominator, making it easier to combine with other fractions. For \(6 \frac{1}{4}\), this conversion involves multiplying the whole number by the denominator of the fraction, and then adding the numerator: \((6 \times 4) + 1 = 25\), giving you the improper fraction \(\frac{25}{4}\).

improper fractions

An improper fraction is one where the numerator, which is the top number, is greater than or equal to the denominator, the bottom number. This type of fraction represents a value greater than or equal to one whole unit.

Although improper fractions may look a bit intimidating at first, they are extremely useful when it comes to calculations because they are easier to add, subtract, multiply, or divide than mixed numbers.

To convert a mixed number to an improper fraction, like turning \(6 \frac{1}{4}\) into \(\frac{25}{4}\), you follow a simple mathematical conversion process: Multiply the whole number by the denominator and add the numerator. This creates a seamless way to integrate these into various arithmetic operations.

least common denominator

The least common denominator (LCD) is crucial when you wish to add or subtract fractions because it allows you to combine them by giving them a common denomination. The LCD is the smallest number that each of the denominators can divide into evenly.

For the fractions \(\frac{25}{4}\), \(\frac{2}{9}\), and \(\frac{1}{36}\), their denominators are 4, 9, and 36. The least number they all divide evenly into is 36.

• For \(\frac{25}{4}\), converting to have a denominator of 36 involves multiplying the numerator and denominator by 9, resulting in \(\frac{225}{36}\).
• \(\frac{2}{9}\) is multiplied by 4, yielding \(\frac{8}{36}\).
• \(\frac{1}{36}\) already has 36 as its denominator.

These conversions permit the addition or subtraction of the fractions, which would otherwise be impossible due to differing denominators.

simplifying fractions

Simplifying fractions involves reducing them to their smallest possible form while keeping the same overall value. This is sometimes necessary to reach final answers that are easy to understand and interpret. To do so, identify the greatest common divisor (GCD) of the numerator and the denominator and then divide both by this number.

For example, when summing the fractions \(\frac{225}{36} + \frac{8}{36} + \frac{1}{36}\), you get \(\frac{234}{36}\). The GCD of 234 and 36 is 18. Dividing both by 18 simplifies the fraction to \(\frac{13}{2}\).

An additional step can convert this improper fraction back to a mixed number. Divide 13 by 2 to get 6 with a remainder of 1, which results in the mixed number \(6 \frac{1}{2}\). Simplifying fractions ensures that your answers are neat and consistent with standard mathematical practice.