3) \( 21 \frac{8}{35}-3 \frac{19}{70}=10 \frac{5}{14}+6 \frac{6}{10} \)

To solve the equation \( 21 \frac{8}{35} - 3 \frac{19}{70} = 10 \frac{5}{14} + 6 \frac{6}{10} \), we will first convert all mixed numbers to improper fractions and then perform the calculations. 1. **Convert mixed numbers to improper fractions:** - For \( 21 \frac{8}{35} \): \[ 21 \frac{8}{35} = \frac{21 \times 35 + 8}{35} = \frac{735 + 8}{35} = \frac{743}{35} \] - For \( 3 \frac{19}{70} \): \[ 3 \frac{19}{70} = \frac{3 \times 70 + 19}{70} = \frac{210 + 19}{70} = \frac{229}{70} \] - For \( 10 \frac{5}{14} \): \[ 10 \frac{5}{14} = \frac{10 \times 14 + 5}{14} = \frac{140 + 5}{14} = \frac{145}{14} \] - For \( 6 \frac{6}{10} \): \[ 6 \frac{6}{10} = \frac{6 \times 10 + 6}{10} = \frac{60 + 6}{10} = \frac{66}{10} = \frac{33}{5} \] 2. **Rewrite the equation with improper fractions:** \[ \frac{743}{35} - \frac{229}{70} = \frac{145}{14} + \frac{33}{5} \] 3. **Find a common denominator for the left side:** The least common multiple of 35 and 70 is 70. Convert \( \frac{743}{35} \) to have a denominator of 70: \[ \frac{743}{35} = \frac{743 \times 2}{35 \times 2} = \frac{1486}{70} \] Now the left side becomes: \[ \frac{1486}{70} - \frac{229}{70} = \frac{1486 - 229}{70} = \frac{1257}{70} \] 4. **Find a common denominator for the right side:** The least common multiple of 14 and 5 is 70. Convert both fractions to have a denominator of 70: \[ \frac{145}{14} = \frac{145 \times 5}{14 \times 5} = \frac{725}{70} \] \[ \frac{33}{5} = \frac{33 \times 14}{5 \times 14} = \frac{462}{70} \] Now the right side becomes: \[ \frac{725}{70} + \frac{462}{70} = \frac{725 + 462}{70} = \frac{1187}{70} \] 5. **Set the two sides equal:** \[ \frac{1257}{70} = \frac{1187}{70} \] 6. **Cross-multiply to check for equality:** Since the denominators are the same, we can compare the numerators: \[ 1257 \neq 1187 \] Thus, the equation \( 21 \frac{8}{35} - 3 \frac{19}{70} = 10 \frac{5}{14} + 6 \frac{6}{10} \) is **not true**.