Question

Find each sum or difference, showing each step of your work. Give your answers in lowest terms. If an answer is greater than 1 , write it as a mixed number. $$3 \frac{5}{6}+\frac{6}{7}$$

Short answer

4 \frac{29}{42}

Step-by-step solution

- Convert Mixed Number to Improper Fraction

Convert the mixed number to an improper fraction. For the number \(3 \frac{5}{6}\), multiply the whole number part (3) by the denominator (6) and add the numerator (5): \[ 3 \frac{5}{6} = \frac{3 \times 6 + 5}{6} = \frac{18 + 5}{6} = \frac{23}{6} \]

- Find Common Denominator

The fractions \(\frac{23}{6}\) and \(\frac{6}{7}\) must have the same denominator to be added. The least common denominator (LCD) for 6 and 7 is 42. Convert each fraction to have this common denominator: \[ \frac{23}{6} = \frac{23 \times 7}{6 \times 7} = \frac{161}{42} \] \[ \frac{6}{7} = \frac{6 \times 6}{7 \times 6} = \frac{36}{42} \]

- Add the Fractions

Add the fractions with the common denominator: \[ \frac{161}{42} + \frac{36}{42} = \frac{197}{42} \]

- Convert to Mixed Number

If the resulting fraction is greater than 1, convert it to a mixed number. Divide 197 by 42: \[ 197 \div 42 = 4 \text{ remainder } 29 \] So, \( \frac{197}{42} = 4 \frac{29}{42} \)

- Ensure Lowest Terms

Make sure the fraction part of the mixed number is in its lowest terms. The fraction \(\frac{29}{42}\) is already in its lowest terms because 29 and 42 have no common factors.

Key concepts

improper fractions

Improper fractions are fractions where the numerator (the top number) is larger than the denominator (the bottom number). These fractions represent a value greater than or equal to 1.
For example, let's take the fraction \(\frac{23}{6}\). Here, 23 is larger than 6, so it's an improper fraction.
To convert a mixed number to an improper fraction, follow these steps:

• Multiply the whole number part by the denominator of the fractional part.
• Add the product to the numerator of the fractional part.
• Place the result over the original denominator.

Let's see an example with the mixed number \(\text{3} \frac{5}{6}\):

• Multiply the whole number (3) by the denominator (6): \(\text{3} \times \text{6} = \text{18}\).
• Add the numerator of the fraction (5): \(\text{18} + \text{5} = \text{23}\).
• Place 23 over the denominator, giving us \(\frac{23}{6}\).

least common denominator

When adding or subtracting fractions, it's important to have a common denominator. If the denominators are different, we need to find the least common denominator (LCD).
The LCD is the smallest number that both denominators can evenly divide into.
For example, with the fractions \(\frac{23}{6}\) and \(\frac{6}{7}\):
Find the multiples of each denominator:

• Multiples of 6: 6, 12, 18, 24, 30, 36, 42...
• Multiples of 7: 7, 14, 21, 28, 35, 42...

From these lists, we see that 42 is the smallest common multiple.
To convert the fractions to have the LCD:

• Multiply the numerator and denominator of \(\frac{23}{6}\) by 7: \(\frac{23 \times 7}{6 \times 7} = \frac{161}{42}\).
• Multiply the numerator and denominator of \(\frac{6}{7}\) by 6: \(\frac{6 \times 6}{7 \times 6} = \frac{36}{42}\).

Now both fractions have the same denominator and can be added together.

mixed numbers

A mixed number consists of a whole number and a fraction. For instance, 4 \(\frac{29}{42}\) is a mixed number.
To convert an improper fraction back to a mixed number, divide the numerator by the denominator.
For example, with the fraction \(\frac{197}{42}\):

• Divide 197 by 42, getting a whole number quotient of 4 and a remainder of 29.

Thus, \(\frac{197}{42}\) can be expressed as 4 \(\frac{29}{42}\).
Make sure the fraction part of the mixed number is in its lowest terms. To do this, check if the numerator and the denominator share any common factors. If they don't, the fraction is already in its simplest form.
For \(\frac{29}{42}\), there are no common factors, so it's in its lowest terms.
Mixed numbers make it easier to interpret values that are greater than 1, offering a clear representation of whole parts and fractional parts.