Get the most out of your questions with IDNLearn.com's extensive resources. Our experts provide timely and precise responses to help you understand and solve any issue you face.
Sagot :
To find the missing rational number in the series [tex]\(\frac{4}{9}, \frac{9}{20}, \ldots, \frac{45}{100}\)[/tex], we need to observe patterns in both the numerators and denominators.
### Step 1: Observing the numerators
The given numerators are:
1. 4 in the fraction [tex]\(\frac{4}{9}\)[/tex]
2. 9 in the fraction [tex]\(\frac{9}{20}\)[/tex]
3. ? (missing numerator)
4. 45 in the fraction [tex]\(\frac{45}{100}\)[/tex]
Let's find the missing numerator by observing a possible pattern in the given numerators.
The numerators (4, 9, ?, 45) suggest an increasing sequence. One obvious guess is to see if they fit a polynomial or a simple number sequence pattern.
Notice the pattern:
- [tex]\(4 = 2^2\)[/tex]
- [tex]\(9 = 3^2\)[/tex]
- [tex]\(45 = 45\)[/tex] (actually [tex]\(45 = 5 \times 9\)[/tex])
Upon closer inspection, it seems they do not follow a simple square pattern, but they could follow a sequence.
### Step 2: Observing the denominators
The given denominators are:
1. 9 in the fraction [tex]\(\frac{4}{9}\)[/tex]
2. 20 in the fraction [tex]\(\frac{9}{20}\)[/tex]
3. ? (missing denominator)
4. 100 in the fraction [tex]\(\frac{45}{100}\)[/tex]
The denominators (9, 20, ?, 100) also need to be examined for a pattern.
### Step 3: Infer the Patterns
To better infer the intermediate term, we can consider both numerators and denominators together:
- Look for patterns like arithmetic sequence, geometric sequence or fractional relationship changes.
One approach could be to assume linear relationships.
#### Numerator Pattern:
- The difference between the numerators are:
[tex]\(9 - 4 = 5\)[/tex],
[tex]\(45 - 9 = 36\)[/tex]
- We might interpolate the mid-point numerical value between an increasing gap.
#### Denominator Pattern:
- The difference between denominators are:
[tex]\(20 - 9 = 11\)[/tex],
[tex]\(100 - 20 = 80\)[/tex]
Another approach is the fractional mean:
- For numerators: [tex]\((4 + 45)/2 = 24.5\)[/tex]
- For denominators: [tex]\((9 + 100)/2 = 54.5\)[/tex].
However, since fractions only work in rational terms, and the series seems arithmetic-like yet large differences, other plausible interpolations are:
Let’s test with simpler middle terms. Notice potential:
- Both satisfy approximate equal splits in terms of scaling similarity.
We should refine:
- Numerator precise intermediate: [tex]\(24 \implies 17.5\)[/tex]; simplified for cleaner pairs closer to natural doubling would produce: [tex]\(((9)+ (20))/2) approx \(65\)[/tex] divisible simpler scaled thus correspondence fraction:
Intermediate suggests:
[tex]\[\boxed{\frac{21}{60}}\][/tex]
Thus: Exact rational missing term predicted is [tex]\(\frac{21}{58}?\)[/tex]
### Step 1: Observing the numerators
The given numerators are:
1. 4 in the fraction [tex]\(\frac{4}{9}\)[/tex]
2. 9 in the fraction [tex]\(\frac{9}{20}\)[/tex]
3. ? (missing numerator)
4. 45 in the fraction [tex]\(\frac{45}{100}\)[/tex]
Let's find the missing numerator by observing a possible pattern in the given numerators.
The numerators (4, 9, ?, 45) suggest an increasing sequence. One obvious guess is to see if they fit a polynomial or a simple number sequence pattern.
Notice the pattern:
- [tex]\(4 = 2^2\)[/tex]
- [tex]\(9 = 3^2\)[/tex]
- [tex]\(45 = 45\)[/tex] (actually [tex]\(45 = 5 \times 9\)[/tex])
Upon closer inspection, it seems they do not follow a simple square pattern, but they could follow a sequence.
### Step 2: Observing the denominators
The given denominators are:
1. 9 in the fraction [tex]\(\frac{4}{9}\)[/tex]
2. 20 in the fraction [tex]\(\frac{9}{20}\)[/tex]
3. ? (missing denominator)
4. 100 in the fraction [tex]\(\frac{45}{100}\)[/tex]
The denominators (9, 20, ?, 100) also need to be examined for a pattern.
### Step 3: Infer the Patterns
To better infer the intermediate term, we can consider both numerators and denominators together:
- Look for patterns like arithmetic sequence, geometric sequence or fractional relationship changes.
One approach could be to assume linear relationships.
#### Numerator Pattern:
- The difference between the numerators are:
[tex]\(9 - 4 = 5\)[/tex],
[tex]\(45 - 9 = 36\)[/tex]
- We might interpolate the mid-point numerical value between an increasing gap.
#### Denominator Pattern:
- The difference between denominators are:
[tex]\(20 - 9 = 11\)[/tex],
[tex]\(100 - 20 = 80\)[/tex]
Another approach is the fractional mean:
- For numerators: [tex]\((4 + 45)/2 = 24.5\)[/tex]
- For denominators: [tex]\((9 + 100)/2 = 54.5\)[/tex].
However, since fractions only work in rational terms, and the series seems arithmetic-like yet large differences, other plausible interpolations are:
Let’s test with simpler middle terms. Notice potential:
- Both satisfy approximate equal splits in terms of scaling similarity.
We should refine:
- Numerator precise intermediate: [tex]\(24 \implies 17.5\)[/tex]; simplified for cleaner pairs closer to natural doubling would produce: [tex]\(((9)+ (20))/2) approx \(65\)[/tex] divisible simpler scaled thus correspondence fraction:
Intermediate suggests:
[tex]\[\boxed{\frac{21}{60}}\][/tex]
Thus: Exact rational missing term predicted is [tex]\(\frac{21}{58}?\)[/tex]
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Find the answers you need at IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.
