Question - Fraction Addition and Subtraction
Solution:
The given expression is:\[ \frac{2}{5} \cdot 3 + \frac{4}{7} \div 2 \]Let's solve this step by step.Step 1: Perform the multiplication and division within the expression.Multiplication:\[ \frac{2}{5} \cdot 3 = \frac{2 \cdot 3}{5} = \frac{6}{5} \]Division:Since dividing by a number is the same as multiplying by its reciprocal, we have:\[ \frac{4}{7} \div 2 = \frac{4}{7} \cdot \frac{1}{2} = \frac{4 \cdot 1}{7 \cdot 2} = \frac{4}{14} = \frac{2}{7} \](Simplifying the fraction by dividing the numerator and denominator by 2)Step 2: Perform the addition between the two fractions.To add fractions, we need a common denominator. The least common multiple of 5 and 7 is 35, so we adjust the fractions accordingly:\[ \frac{6}{5} = \frac{6 \cdot 7}{5 \cdot 7} = \frac{42}{35} \]\[ \frac{2}{7} = \frac{2 \cdot 5}{7 \cdot 5} = \frac{10}{35} \]Adding the fractions together gives us:\[ \frac{42}{35} + \frac{10}{35} = \frac{42 + 10}{35} = \frac{52}{35} \]We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor which is 1 (since 52 and 35 have no common divisors other than 1), so the fraction is already in its simplest form.Therefore, the result of the expression is:\[ \frac{52}{35} \] Since 52 divided by 35 gives us 1 with a remainder of 17, we can also express this as a mixed number:\[ \frac{52}{35} = 1 \frac{17}{35} \]So the final answer is:\[ 1 \frac{17}{35} \]
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