- Equation 1: \(3 \frac{2}{7} + \frac{2}{7} = 3 \frac{4}{7}
eq 3 \frac{2}{7}\), so it is false.
- Equation 2: \(6 \frac{5}{8} + \frac{6}{8} = 6 \frac{11}{8} = 7 \frac{3}{8}\), which matches the right side, so it is true.
- Equation 3: \(5 \frac{4}{9} + 1 \frac{1}{2} = (5+1) + (\frac{8}{18} + \frac{9}{18}) = 6 \frac{17}{18}\), which matches the right side, so it is true.
- Final answer: \(\boxed{2, 3}\).
Explanation
- Problem Analysis and Evaluation Strategy
We are tasked with evaluating three arithmetic equations involving mixed numbers and fractions to determine which ones are true.
The equations are:
- \(3 \frac{2}{7} + \frac{2}{7} = 3 \frac{4}{14}\)
- \(6 \frac{5}{8} + \frac{3}{4} = 7 \frac{3}{8}\)
- \(5 \frac{4}{9} + 1 \frac{1}{2} = 6 \frac{17}{18}\)
A mixed number \(a \frac{b}{c}\) is equivalent to \(a + \frac{b}{c}\). To evaluate each equation, we perform the arithmetic operations on both sides and compare the results.
- Evaluating Equation 1
To evaluate the first equation \(3 \frac{2}{7} + \frac{2}{7} = 3 \frac{4}{14}\):
- Left side: \(3 \frac{2}{7} + \frac{2}{7} = 3 + \frac{2}{7} + \frac{2}{7} = 3 + \frac{4}{7}\).
- Right side: \(3 \frac{4}{14} = 3 + \frac{4}{14} = 3 + \frac{2}{7}\).
Comparing the two sides, \(3 + \frac{4}{7}
eq 3 + \frac{2}{7}\). Therefore, the first equation is false.
- Evaluating Equation 2
To evaluate the second equation \(6 \frac{5}{8} + \frac{3}{4} = 7 \frac{3}{8}\):
- Left side: \(6 \frac{5}{8} + \frac{3}{4} = 6 + \frac{5}{8} + \frac{6}{8}\) (since \(\frac{3}{4} = \frac{6}{8}\)).
- Summing the fractions: \(\frac{5}{8} + \frac{6}{8} = \frac{11}{8} = 1 \frac{3}{8}\).
- Adding to the whole number: \(6 + 1 \frac{3}{8} = 7 \frac{3}{8}\).
- Right side: \(7 \frac{3}{8}\).
Since both sides are equal to \(7 \frac{3}{8}\), the second equation is true.
- Evaluating Equation 3
To evaluate the third equation \(5 \frac{4}{9} + 1 \frac{1}{2} = 6 \frac{17}{18}\):
- Left side: \(5 \frac{4}{9} + 1 \frac{1}{2} = (5 + 1) + (\frac{4}{9} + \frac{1}{2})\).
- Finding a common denominator for the fractions: \(\frac{4}{9} = \frac{8}{18}\) and \(\frac{1}{2} = \frac{9}{18}\).
- Summing the fractions: \(\frac{8}{18} + \frac{9}{18} = \frac{17}{18}\).
- Adding to the sum of the whole numbers: \(6 + \frac{17}{18} = 6 \frac{17}{18}\).
- Right side: \(6 \frac{17}{18}\).
Since both sides are equal to \(6 \frac{17}{18}\), the third equation is true.
- Conclusion
Based on our evaluations:
- Equation 1 is false.
- Equation 2 is true.
- Equation 3 is true.
The true equations are the second and third. Thus, the final answer is 2 and 3.
Examples
When you are cooking, you often need to adjust ingredient measurements. If a recipe calls for \(6 \frac{5}{8}\) cups of flour and you need to add an extra \(\frac{3}{4}\) cup to adjust the consistency, calculating the total correctly ensures your final dish has the right texture. Mastery of fraction addition helps you scale recipes, manage supplies, and ensure precision in everyday measurements.