Question

What is a fraction equivalent to 1/2 where the numerator is less than 3 times its denominator?

Answer 1

To find a fraction equivalent to \(\frac{1}{2}\) where the numerator is less than 3 times its denominator, we can express a fraction equivalent to \(\frac{1}{2}\) as follows:

\[
\frac{a}{b} = \frac{1}{2}
\]

This means:

\[
2a = b
\]

Now, we want the numerator \(a\) to be less than \(3\) times the denominator \(b\):

\[
a < 3b
\]

Substituting \(b\) from \(2a = b\) into the inequality:

\[
a < 3(2a)
\]
\[
a < 6a
\]

This inequality is always true for \(a > 0\). Hence, we can choose various values for \(a\) and find the corresponding value of \(b\).

Let's take some examples:

1. If \(a = 1\):
\[
b = 2a = 2(1) = 2 \implies \text{Fraction} = \frac{1}{2}
\]

2. If \(a = 2\):
\[
b = 2a = 2(2) = 4 \implies \text{Fraction} = \frac{2}{4}
\]

3. If \(a = 3\):
\[
b = 2a = 2(3) = 6 \implies \text{Fraction} = \frac{3}{6}
\]

4. If \(a = 4\):
\[
b = 2a = 2(4) = 8 \implies \text{Fraction} = \frac{4}{8}
\]

5. If \(a = 5\):
\[
b = 2a = 2(5) = 10 \implies \text{Fraction} = \frac{5}{10}
\]

So far, all these values satisfy the condition \(a < 3b\):

- For \(\frac{1}{2}\): \(1 < 3 \times 2\) (True)
- For \(\frac{2}{4}\): \(2 < 3 \times 4\) (True)
- For \(\frac{3}{6}\): \(3 < 3 \times 6\) (True)
- For \(\frac{4}{8}\): \(4 < 3 \times 8\) (True)
- For \(\frac{5}{10}\): \(5 < 3 \times 10\) (True)

Now, let's pick a fraction with a numerator less than \(3\) times the denominator. Consider:

\[
\frac{2}{4}
\]

The numerator \(2\) is less than \(3\) times the denominator \(4\) (since \(3 \times 4 = 12\)).

Thus, one example of a fraction equivalent to \(\frac{1}{2}\) where the numerator is less than \(3\) times its denominator is:

\[
\frac{2}{4}
\]

This satisfies all conditions given.