To solve the problem of adding the mixed number [tex]\( 1 \frac{1}{7} \)[/tex] and the simple fraction [tex]\( \frac{1}{4} \)[/tex], follow these steps:
### Step 1: Convert the Mixed Number to an Improper Fraction
A mixed number consists of a whole part and a fraction. The mixed number here is [tex]\( 1 \frac{1}{7} \)[/tex]. To convert it to an improper fraction, follow these steps:
1. Multiply the whole number part by the denominator of the fractional part:
[tex]\[
1 \times 7 = 7
\][/tex]
2. Add the numerator of the fractional part to the result from step 1:
[tex]\[
7 + 1 = 8
\][/tex]
3. Place the result over the original denominator:
[tex]\[
1 \frac{1}{7} = \frac{8}{7}
\][/tex]
### Step 2: Ensure Both Fractions Have a Common Denominator
To add [tex]\(\frac{8}{7}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex], we need a common denominator. The denominators are 7 and 4. The least common multiple (LCM) of 7 and 4 is 28. Convert both fractions to have this common denominator.
1. Convert [tex]\(\frac{8}{7}\)[/tex] to a fraction with a denominator of 28:
[tex]\[
\frac{8}{7} = \frac{8 \times 4}{7 \times 4} = \frac{32}{28}
\][/tex]
2. Convert [tex]\(\frac{1}{4}\)[/tex] to a fraction with a denominator of 28:
[tex]\[
\frac{1}{4} = \frac{1 \times 7}{4 \times 7} = \frac{7}{28}
\][/tex]
### Step 3: Add the Fractions
With both fractions having the same denominator, we can easily add them:
[tex]\[
\frac{32}{28} + \frac{7}{28} = \frac{32 + 7}{28} = \frac{39}{28}
\][/tex]
### Step 4: Simplify the Fraction
The fraction [tex]\(\frac{39}{28}\)[/tex] is already in its simplest form because 39 and 28 have no common factors other than 1.
### Step 5: Convert to Decimal Form
To understand the result better, we can convert the simplified fraction to a decimal:
[tex]\[
\frac{39}{28} \approx 1.3928571428571428
\][/tex]
Therefore, the sum of [tex]\( 1 \frac{1}{7} \)[/tex] and [tex]\( \frac{1}{4} \)[/tex] is [tex]\( \frac{39}{28} \)[/tex], which is approximately [tex]\( 1.3928571428571428 \)[/tex] in decimal form.
