To add the fractions [tex]\(\frac{7}{10}\)[/tex] and [tex]\(\frac{4}{7}\)[/tex] and express the answer as a mixed number in its simplest form, follow these steps:
### Step 1: Identify the Least Common Denominator (LCD)
Determine the least common denominator for the fractions [tex]\(\frac{7}{10}\)[/tex] and [tex]\(\frac{4}{7}\)[/tex]. The denominators are 10 and 7.
### Step 2: Convert Each Fraction to Have the LCD as Denominator
The least common denominator of 10 and 7 is 70 (the smallest number that both 10 and 7 can divide into without a remainder).
Convert each fraction to have this common denominator:
- For [tex]\(\frac{7}{10}\)[/tex]: [tex]\(\frac{7 \times 7}{10 \times 7} = \frac{49}{70}\)[/tex]
- For [tex]\(\frac{4}{7}\)[/tex]: [tex]\(\frac{4 \times 10}{7 \times 10} = \frac{40}{70}\)[/tex]
### Step 3: Add the Fractions
Now that both fractions have the same denominator, add their numerators:
[tex]\[
\frac{49}{70} + \frac{40}{70} = \frac{49 + 40}{70} = \frac{89}{70}
\][/tex]
### Step 4: Convert to a Mixed Number
To convert [tex]\(\frac{89}{70}\)[/tex] to a mixed number:
1. Divide the numerator by the denominator: [tex]\( 89 \div 70 = 1 \text{ with a remainder of } 19\)[/tex]
2. This gives the integer part 1 and the fractional part [tex]\(\frac{19}{70}\)[/tex].
### Step 5: Simplify the Fraction (if possible)
Check if [tex]\(\frac{19}{70}\)[/tex] can be simplified. The greatest common divisor (GCD) of 19 and 70 is 1, meaning [tex]\(\frac{19}{70}\)[/tex] is already in its simplest form.
### Final Answer:
Thus, the sum of [tex]\(\frac{7}{10}\)[/tex] and [tex]\(\frac{4}{7}\)[/tex] is:
[tex]\[
1 \frac{19}{70}
\][/tex]
