Question
. Vocabulary How can you show that a number is a rational
number?
2. Which shows 0.2¯3 as a fraction?
𝖠 ___2
33
𝖡 ___7
33
𝖢 ___23
99
𝖣 ___7
30
3. Approximate √__8to the nearest hundredth. Show your work.
4. Solve the equation m2 = 14.
5. A fish tank is in the shape of a cube. Its volume is 125 ft3. What is the area
of one face of the tank?
6. Write 1.¯12 as a mixed number. Show your work.
number?
2. Which shows 0.2¯3 as a fraction?
𝖠 ___2
33
𝖡 ___7
33
𝖢 ___23
99
𝖣 ___7
30
3. Approximate √__8to the nearest hundredth. Show your work.
4. Solve the equation m2 = 14.
5. A fish tank is in the shape of a cube. Its volume is 125 ft3. What is the area
of one face of the tank?
6. Write 1.¯12 as a mixed number. Show your work.
Answers
GPT-4o mini
Sure! Let's go through each of your questions step-by-step.
### 1. How can you show that a number is a rational number?
A number is considered a rational number if it can be expressed as the quotient of two integers \( \frac{a}{b} \), where \( a \) is an integer, \( b \) is a nonzero integer. For example, numbers like \( \frac{1}{2}, 0.75, \) and \( -3 \) (which can be written as \( \frac{-3}{1} \)) are rational numbers. You can also show that a decimal is rational if it either terminates after a certain number of digits (like 0.5) or repeats in a consistent pattern (like 0.333…).
### 2. Which shows \( 0.2\overline{3} \) as a fraction?
To convert \( 0.2\overline{3} \) to a fraction:
Let \( x = 0.2\overline{3} \).
1. Multiply by 10 to move the decimal:
\[
10x = 2.3\overline{3}
\]
2. Multiply by 100 (to isolate the repeating part):
\[
100x = 23.\overline{3}
\]
3. Now, subtract the first equation from the second:
\[
100x - 10x = 23.\overline{3} - 2.3\overline{3}
\]
\[
90x = 21
\]
\[
x = \frac{21}{90} = \frac{7}{30}
\]
Thus the answer is \( \text{𝖣 } \frac{7}{30} \).
### 3. Approximate \( \sqrt{8} \) to the nearest hundredth. Show your work.
To approximate \( \sqrt{8} \):
1. Calculate \( \sqrt{8} \):
\[
\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \sqrt{2} = 2\sqrt{2}
\]
2. Know that \( \sqrt{2} \) is approximately \( 1.414 \), so:
\[
2\sqrt{2} \approx 2 \times 1.414 = 2.828
\]
3. Rounding 2.828 to the nearest hundredth gives us \( 2.83 \).
### 4. Solve the equation \( m^2 = 14 \).
To solve for \( m \):
1. Take the square root of both sides:
\[
m = \pm \sqrt{14}
\]
Thus, the solutions are \( m = \sqrt{14} \) and \( m = -\sqrt{14} \).
### 5. A fish tank is in the shape of a cube. Its volume is \( 125 \, \text{ft}^3 \). What is the area of one face of the tank?
1. The volume \( V \) of a cube is given by \( V = s^3 \), where \( s \) is the length of a side.
\[
s^3 = 125
\]
\[
s = \sqrt[3]{125} = 5 \, \text{ft}
\]
2. The area of one face of the cube (which is a square) is given by \( s^2 \):
\[
\text{Area} = s^2 = 5^2 = 25 \, \text{ft}^2
\]
### 6. Write \( 1.\overline{12} \) as a mixed number. Show your work.
Let \( x = 1.\overline{12} \).
1. Multiply by 100 to eliminate the repeating decimals:
\[
100x = 112.\overline{12}
\]
2. Subtract the original \( x \):
\[
100x - x = 112.\overline{12} - 1.\overline{12}
\]
\[
99x = 111
\]
\[
x = \frac{111}{99} = \frac{37}{33}
\]
3. Converting this to a mixed number:
\[
1 + \frac{4}{33}
\]
Thus, \( 1.\overline{12} = 1 \frac{4}{33} \).
Let me know if you need further explanations or help!
### 1. How can you show that a number is a rational number?
A number is considered a rational number if it can be expressed as the quotient of two integers \( \frac{a}{b} \), where \( a \) is an integer, \( b \) is a nonzero integer. For example, numbers like \( \frac{1}{2}, 0.75, \) and \( -3 \) (which can be written as \( \frac{-3}{1} \)) are rational numbers. You can also show that a decimal is rational if it either terminates after a certain number of digits (like 0.5) or repeats in a consistent pattern (like 0.333…).
### 2. Which shows \( 0.2\overline{3} \) as a fraction?
To convert \( 0.2\overline{3} \) to a fraction:
Let \( x = 0.2\overline{3} \).
1. Multiply by 10 to move the decimal:
\[
10x = 2.3\overline{3}
\]
2. Multiply by 100 (to isolate the repeating part):
\[
100x = 23.\overline{3}
\]
3. Now, subtract the first equation from the second:
\[
100x - 10x = 23.\overline{3} - 2.3\overline{3}
\]
\[
90x = 21
\]
\[
x = \frac{21}{90} = \frac{7}{30}
\]
Thus the answer is \( \text{𝖣 } \frac{7}{30} \).
### 3. Approximate \( \sqrt{8} \) to the nearest hundredth. Show your work.
To approximate \( \sqrt{8} \):
1. Calculate \( \sqrt{8} \):
\[
\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \sqrt{2} = 2\sqrt{2}
\]
2. Know that \( \sqrt{2} \) is approximately \( 1.414 \), so:
\[
2\sqrt{2} \approx 2 \times 1.414 = 2.828
\]
3. Rounding 2.828 to the nearest hundredth gives us \( 2.83 \).
### 4. Solve the equation \( m^2 = 14 \).
To solve for \( m \):
1. Take the square root of both sides:
\[
m = \pm \sqrt{14}
\]
Thus, the solutions are \( m = \sqrt{14} \) and \( m = -\sqrt{14} \).
### 5. A fish tank is in the shape of a cube. Its volume is \( 125 \, \text{ft}^3 \). What is the area of one face of the tank?
1. The volume \( V \) of a cube is given by \( V = s^3 \), where \( s \) is the length of a side.
\[
s^3 = 125
\]
\[
s = \sqrt[3]{125} = 5 \, \text{ft}
\]
2. The area of one face of the cube (which is a square) is given by \( s^2 \):
\[
\text{Area} = s^2 = 5^2 = 25 \, \text{ft}^2
\]
### 6. Write \( 1.\overline{12} \) as a mixed number. Show your work.
Let \( x = 1.\overline{12} \).
1. Multiply by 100 to eliminate the repeating decimals:
\[
100x = 112.\overline{12}
\]
2. Subtract the original \( x \):
\[
100x - x = 112.\overline{12} - 1.\overline{12}
\]
\[
99x = 111
\]
\[
x = \frac{111}{99} = \frac{37}{33}
\]
3. Converting this to a mixed number:
\[
1 + \frac{4}{33}
\]
Thus, \( 1.\overline{12} = 1 \frac{4}{33} \).
Let me know if you need further explanations or help!