Verified Answer :
- For the denominator \(117\), multiply \(-1\) by \(117/13=9\) to get the numerator \(-9\).
- For the denominator \(26\), multiply \(-1\) by \(26/13=2\) to get the numerator \(-2\).
- For the denominator \(-52\), multiply \(-1\) by \(-52/13=-4\) to get the numerator \(4\).
- The equivalent fractions are \(\boxed{-9}\), \(\boxed{-2}\), and \(\boxed{4}\).
Explanation
- Understanding Equivalent Fractions
To find a fraction equivalent to a given fraction, we multiply both the numerator and the denominator by the same non-zero integer \(k\). That is, \(\frac{a}{b} = \frac{a \times k}{b \times k}\). We are given the base fraction \(\frac{-1}{13}\) and we need to find the missing numerators for the denominators \(117\), \(26\), and \(-52\).
- Finding the First Equivalent Fraction
To find the numerator for the denominator \(117\), we determine the multiplier \(k\) such that \(13 \times k = 117\). Dividing \(117\) by \(13\), we get \(k = 9\). Multiplying the numerator \(-1\) by \(9\), we have \(-1 \times 9 = -9\). Thus, \(\frac{-1}{13} = \frac{-9}{117}\).
- Finding the Second Equivalent Fraction
To find the numerator for the denominator \(26\), we determine the multiplier \(k\) such that \(13 \times k = 26\). Dividing \(26\) by \(13\), we get \(k = 2\). Multiplying the numerator \(-1\) by \(2\), we have \(-1 \times 2 = -2\). Thus, \(\frac{-1}{13} = \frac{-2}{26}\).
- Finding the Third Equivalent Fraction
To find the numerator for the denominator \(-52\), we determine the multiplier \(k\) such that \(13 \times k = -52\). Dividing \(-52\) by \(13\), we get \(k = -4\). Multiplying the numerator \(-1\) by \(-4\), we have \(-1 \times (-4) = 4\). Thus, \(\frac{-1}{13} = \frac{4}{-52}\).
- Conclusion
The fractions equivalent to \(\frac{-1}{13}\) are \(\frac{-9}{117}\), \(\frac{-2}{26}\), and \(\frac{4}{-52}\). The missing numerators are \(-9\), \(-2\), and \(4\). The final answers are \(\boxed{-9}, \boxed{-2}, \boxed{4}\).
Examples
Equivalent fractions are essential in everyday life when measuring quantities, such as in cooking or construction. For instance, if a recipe calls for \(\frac{-1}{13}\) of a cup of a certain ingredient (using a negative value to represent a withdrawal or a specific direction), and you only have a measuring cup marked with a denominator of 117, you would need to calculate \(\frac{-9}{117}\) to get the correct amount. Understanding equivalent fractions allows us to express the same quantity in different ways to suit the tools we have available.