Sure, let's go through the steps to find the reciprocals of the given numbers.
### Problem 3: Finding the reciprocals of [tex]\(\frac{6}{7}\)[/tex] and [tex]\(\frac{2}{6}\)[/tex]
1. Fraction [tex]\(\frac{6}{7}\)[/tex]:
- The reciprocal of a fraction [tex]\(\frac{a}{b}\)[/tex] is given by [tex]\(\frac{b}{a}\)[/tex].
- Here, the reciprocal of [tex]\(\frac{6}{7}\)[/tex] is [tex]\(\frac{7}{6}\)[/tex].
- Converting [tex]\(\frac{7}{6}\)[/tex] to a decimal form gives approximately 1.1667.
2. Fraction [tex]\(\frac{2}{6}\)[/tex]:
- The reciprocal of [tex]\(\frac{2}{6}\)[/tex] is [tex]\(\frac{6}{2}\)[/tex].
- Simplifying [tex]\(\frac{6}{2}\)[/tex] gives 3.
- Therefore, the reciprocal of [tex]\(\frac{2}{6}\)[/tex] is 3.0.
### Problem 29: Finding the reciprocal of [tex]\(1 \frac{4}{7}\)[/tex] and [tex]\(\frac{?}{11}\)[/tex]
1. Mixed Number [tex]\(1\frac{4}{7}\)[/tex]:
- First, convert the mixed number to an improper fraction.
- A mixed number [tex]\(a \frac{b}{c}\)[/tex] can be converted to an improper fraction as [tex]\(\frac{a \cdot c + b}{c}\)[/tex].
- For [tex]\(1\frac{4}{7}\)[/tex], we calculate:
- [tex]\(1\cdot7 + 4 = 7 + 4 = 11\)[/tex]
- So, [tex]\(1\frac{4}{7}\)[/tex] becomes [tex]\(\frac{11}{7}\)[/tex].
- The reciprocal of [tex]\(\frac{11}{7}\)[/tex] is [tex]\(\frac{7}{11}\)[/tex].
- Converting [tex]\(\frac{7}{11}\)[/tex] to a decimal form gives approximately 0.6364.
2. Unknown Fraction [tex]\(\frac{?}{11}\)[/tex]:
- To find the reciprocal, we assume the numerator is 1, as it makes the calculation straightforward.
- The reciprocal of [tex]\(\frac{1}{11}\)[/tex] is [tex]\(\frac{11}{1}\)[/tex], which simplifies to 11.
- Therefore, the reciprocal of [tex]\(\frac{?}{11}\)[/tex] (assuming ? = 1) is 11.0.
In summary, the reciprocals are:
- For [tex]\(\frac{6}{7}\)[/tex]: 1.1667
- For [tex]\(\frac{2}{6}\)[/tex]: 3.0
- For [tex]\(1 \frac{4}{7}\)[/tex]: 0.6364
- For [tex]\(\frac{?}{11}\)[/tex] (with ? assumed to be 1): 11.0
These values explain all necessary reciprocals step-by-step.
