Understanding the Fraction Number Series

Let's analyze the given number series to find the pattern and determine which number replaces the question mark (?). The series is:

\( {{1} \over 2}\), \({{3} \over 4}\), ?, \({{5} \over 4}\)

We have a series of fractions. To easily identify a pattern, it's often helpful to have a common denominator for all the fractions. Let's look at the denominators we have: 2 and 4. The least common multiple of 2 and 4 is 4.

Converting Fractions to a Common Denominator

Convert the first term, \({{1} \over 2}\), to a fraction with a denominator of 4. To do this, we multiply both the numerator and the denominator by 2:

\({{1} \over 2} = {{1 \times 2} \over {2 \times 2}} = {{2} \over 4}\)

Now, let's rewrite the series with the first term converted:

\({{2} \over 4}\), \({{3} \over 4}\), ?, \({{5} \over 4}\)

Identifying the Pattern in the Numerators

With a common denominator of 4, we can now look at the sequence of numerators:

2, 3, ?, 5

Let's see if there is a simple arithmetic pattern in this sequence of numerators. From the first term (2) to the second term (3), the increase is \(3 - 2 = 1\). From the term after the question mark (5) to the second term (3), there is a difference. If the pattern is consistent addition, let's assume the unknown numerator is \(x\). The sequence of numerators would be 2, 3, \(x\), 5.

If the pattern is adding a constant value, say \(d\), then:

• \(3 = 2 + d \implies d = 1\)
• \(x = 3 + d \implies x = 3 + 1 = 4\)
• \(5 = x + d \implies 5 = 4 + 1 = 5\) (This confirms the pattern)

So, the sequence of numerators is indeed increasing by 1 each time: 2, 3, 4, 5.

The missing numerator is 4.

Forming the Missing Fraction

Since the denominator throughout the series (after converting the first term) is 4, and the missing numerator is 4, the missing fraction is:

\({{4} \over 4}\)

Simplifying the Missing Term

The fraction \({{4} \over 4}\) simplifies to 1.

\({{4} \over 4} = 1\)

Verifying the Series with the Missing Term

Let's put the missing term back into the original series (using the common denominator representation for clarity):

\({{2} \over 4}\), \({{3} \over 4}\), \({{4} \over 4}\), \({{5} \over 4}\)

This is an arithmetic progression where each term is obtained by adding \({{1} \over 4}\) to the previous term:

• \({{2} \over 4} + {{1} \over 4} = {{3} \over 4}\)
• \({{3} \over 4} + {{1} \over 4} = {{4} \over 4}\)
• \({{4} \over 4} + {{1} \over 4} = {{5} \over 4}\)

The pattern holds true. The missing number is \({{4} \over 4}\), which simplifies to 1.

Conclusion

The number that replaces the question mark (?) in the given series is 1.

Revision Table: Fraction Series Analysis

Additional Information on Number Series Patterns

Number series questions often involve identifying a specific pattern. Common patterns include:

• Arithmetic Progression: Adding a constant difference between consecutive terms (like in this problem's numerators).
• Geometric Progression: Multiplying by a constant ratio between consecutive terms.
• Difference Series: The difference between consecutive terms follows a pattern itself (e.g., increasing by a constant).
• Mixed Series: Alternating patterns or a combination of different patterns.
• Fibonacci Series: Each term is the sum of the two preceding terms.

For fraction series, it is often useful to find a common denominator first, as it helps to simplify the pattern identification process, especially if the pattern involves the numerators or the fractions as a whole being an arithmetic progression.