Zoe is trying to find fractions that satisfy some of the following conditions:
The denominator of the fraction is divisible by \(3\).
All the digits in the fraction are even.
The decimal value of the fraction is greater than \(1\) but less than \(1.5\).
The denominator of the fraction is a multiple of \(2\).
The decimal value of the fraction is between \(0.25\) and \(0.6\).
The decimal value of the fraction does not include a \(9\).
The numerator of the fraction is even.
The decimal value of the fraction is closer to \(0.75\) than to \(0.55\).
Find a fraction that satisfies at least three of the conditions.
Can you find a fraction that satisfies all of these conditions? If yes, give such a fraction. If no, explain why no such fraction exists.
What is the least number of fractions Zoe needs to find such that together they meet all eight conditions?
Answers will vary. Some examples:
\(\dfrac{2}{10}\) satisfies conditions (4), (6), and (7).
\(\dfrac{4}{9}\) satisfies conditions (1), (5), (6), and (7).
\(\dfrac{7}{9}\) satisfies conditions (1), (6), and (8).
No such fraction exists. This is because conditions (3) and (5) are contradictory, as a number cannot be both between \(0.25\) and \(0.6\) and be greater than \(1\). Similarly, conditions (5) and (8) are also contradictory.
From part (b), we know that we will need at least \(2\) fractions. It turns out that it is possible to satisfy all eight conditions together with just two fractions. Consider the fractions \(\dfrac{1}{2}\) and \(\dfrac{8}{6}\).
The fraction \(\dfrac{1}{2} = 0.5\) satisfies conditions (4), (5), and (6).
The fraction \(\dfrac{8}{6}=1.333\dots\) satisfies conditions (1), (2), (3), (4), (6), (7), (8).
Answers will vary. Many other fraction pairs exist.