Verified Answer :
- Identify the multiplier for each target denominator by dividing it by the original denominator \(11\).
- Multiply both the numerator \(8\) and the denominator \(11\) by the identified multiplier.
- The equivalent fractions are:
- \(\frac{8}{11} = \frac{\boxed{72}}{99}\)
- \(\frac{8}{11} = \frac{\boxed{16}}{22}\)
- \(\frac{8}{11} = \frac{\boxed{32}}{44}\)
Explanation
- Understanding Equivalent Fractions
We are given the fraction \(\frac{8}{11}\) and asked to find three equivalent fractions with denominators \(99\), \(22\), and \(44\). Two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\) are equivalent if one can be obtained from the other by multiplying the numerator and denominator by the same non-zero integer. Our task is to determine the unknown numerators \(N_1, N_2,\) and \(N_3\) such that \(\frac{8}{11} = \frac{N_1}{99} = \frac{N_2}{22} = \frac{N_3}{44}\).
- Finding the First Equivalent Fraction
To find the equivalent fraction with a denominator of \(99\), we determine the factor by which the denominator \(11\) must be multiplied to reach \(99\). Since \(11 \times 9 = 99\), we multiply both the numerator and denominator of \(\frac{8}{11}\) by \(9\):
\[\frac{8 \times 9}{11 \times 9} = \frac{72}{99}\]
Thus, the numerator is \(72\).
- Finding the Second Equivalent Fraction
To find the equivalent fraction with a denominator of \(22\), we determine the factor by which the denominator \(11\) must be multiplied to reach \(22\). Since \(11 \times 2 = 22\), we multiply both the numerator and denominator of \(\frac{8}{11}\) by \(2\):
\[\frac{8 \times 2}{11 \times 2} = \frac{16}{22}\]
Thus, the numerator is \(16\).
- Finding the Third Equivalent Fraction
To find the equivalent fraction with a denominator of \(44\), we determine the factor by which the denominator \(11\) must be multiplied to reach \(44\). Since \(11 \times 4 = 44\), we multiply both the numerator and denominator of \(\frac{8}{11}\) by \(4\):
\[\frac{8 \times 4}{11 \times 4} = \frac{32}{44}\]
Thus, the numerator is \(32\).
- Final Results
We have found the equivalent fractions:
\[\frac{8}{11} = \frac{72}{99}, \quad \frac{8}{11} = \frac{16}{22}, \quad \frac{8}{11} = \frac{32}{44}\]
The corresponding numerators are \(\boxed{72}\), \(\boxed{16}\), and \(\boxed{32}\).
Examples
Equivalent fractions are used frequently in daily life when scaling recipes or measurements. For example, if a recipe calls for \(\frac{8}{11}\) of a cup of flour, but you only have measuring cups that measure in 22nds, you would need to calculate that \(\frac{8}{11} = \frac{16}{22}\) of a cup. Understanding how to find equivalent fractions ensures that you can adjust quantities correctly while maintaining the correct ratio of ingredients.