Sure! Let's work through each of these problems step by step.
Problem 1: Find a rational number between 7 and 8. Write your answer as a fraction.
A rational number can be expressed as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b \neq 0 \).
To find a rational number between 7 and 8, we can take the average of 7 and 8.
\[ \text{Average} = \frac{7 + 8}{2} = \frac{15}{2} \]
Thus, a rational number between 7 and 8 is:
\[ \frac{15}{2} \]
Problem 2: Find an irrational number between \(2 \frac{1}{16}\) and \(4 \frac{1}{9}\).
First, we convert the mixed numbers to improper fractions:
\[ 2 \frac{1}{16} = \frac{2 \cdot 16 + 1}{16} = \frac{32 + 1}{16} = \frac{33}{16} \] \[ 4 \frac{1}{9} = \frac{4 \cdot 9 + 1}{9} = \frac{36 + 1}{9} = \frac{37}{9} \]
Next, we find a common denominator to compare these two fractions:
The least common multiple of 16 and 9 is 144.
Now we convert both fractions to have this common denominator:
\[ \frac{33}{16} = \frac{33 \cdot 9}{16 \cdot 9} = \frac{297}{144} \] \[ \frac{37}{9} = \frac{37 \cdot 16}{9 \cdot 16} = \frac{592}{144} \]
We need to find an irrational number between \( \frac{297}{144} \) and \( \frac{592}{144} \).
An example of an irrational number in this range is the square root of a number between 297 and 592. We can take:
\[ \sqrt{400} = 20 \]
Since \( 400 \) is clearly between \( 297 \) and \( 592 \), \( \sqrt{400} \) gives us the irrational number we need.
Thus, an irrational number between \(2 \frac{1}{16}\) and \(4 \frac{1}{9}\) is \( 20 \).
Problem 3: Find an irrational number between 7 and 8 written as a square root. Explain your answer.
A simple choice is to look for the square root of a number between 49 and 64 (since \(7^2 = 49\) and \(8^2 = 64\)).
For example:
\[ \sqrt{50} \]
The number 50 is between 49 and 64. Therefore:
\[ \sqrt{50} \approx 7.07 \]
This is an irrational number between 7 and 8, explained by the fact that it is the square root of a non-perfect square (50).
Problem 4: Find a rational number between \(2 \frac{1}{6}\) and \(2 \frac{1}{2}\). Write your answer as a decimal.
First, convert the mixed numbers to improper fractions:
\[ 2 \frac{1}{6} = \frac{2 \cdot 6 + 1}{6} = \frac{12 + 1}{6} = \frac{13}{6} \] \[ 2 \frac{1}{2} = \frac{2 \cdot 2 + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2} = \frac{15}{6} \]
Now, we need to find a rational number between \( \frac{13}{6} \) and \( \frac{15}{6} \).
The average is:
\[ \text{Average} = \frac{\frac{13}{6} + \frac{15}{6}}{2} = \frac{\frac{28}{6}}{2} = \frac{28}{12} = \frac{14}{6} = \frac{7}{3} \]
Now converting that to decimal:
\[ \frac{7}{3} \approx 2.33 \]
Thus, a rational number between \(2 \frac{1}{6}\) and \(2 \frac{1}{2}\) is approximately \(2.33\).
Problem 5: Find a rational number between \( \pi \) and \( \sqrt{12} \).
First, let's get approximate decimal values for \( \pi \) and \( \sqrt{12} \):
\[ \pi \approx 3.14 \] \[ \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \approx 2 \cdot 1.732 \approx 3.464 \]
Now, we want to find a rational number between \( 3.14 \) and \( 3.464 \).
One simple choice is to take the average:
\[ \text{Average} = \frac{3.14 + 3.464}{2} \approx \frac{6.604}{2} \approx 3.302 \]
A rational number with this value can be represented as a fraction.
\[ \text{Rational number} \approx 3.302 \Rightarrow \frac{3302}{1000} = \frac{1651}{500} \]
Thus, a rational number between \( \pi \) and \( \sqrt{12} \) is approximately \( 3.302 \) or exactly \( \frac{1651}{500} \).
Summary of Answers:
- Rational number between 7 and 8: \( \frac{15}{2} \)
- Irrational number between \( 2 \frac{1}{16} \) and \( 4 \frac{1}{9} \): \( \sqrt{400} \)
- Irrational number between 7 and 8 as a square root: \( \sqrt{50} \)
- Rational number between \( 2 \frac{1}{6} \) and \( 2 \frac{1}{2} \): \( 2.33 \) (or \( \frac{7}{3} \))
- Rational number between \( \pi \) and \( \sqrt{12} \): \( \frac{1651}{500} \) (or approximately \( 3.302 \))