Question
5 Problems - SHOW ALL WORK!
Isaac knows that ¼ is written as 0.25 in decimal form. How can he find a decimal for 7/4?
Explain why ⅓ is called a rational number. How do you know?
Predict the decimal for 5/9 using mental math.
Divide to check your prediction, and describe the results.
How would you identify this as a rational or irrational number?
Jackson is practicing basketball. He makes a basket from the free-throw line 9 out of 27 shots.
Write the decimal from the fraction of Jackson’s shot that resulted in the basket.
How can you tell if this number is rational or irrational?
What is the difference between terminating and repeating decimals? How does the end behavior of a decimal relate to rational number or irrational number?
What do I need help with?
Isaac knows that ¼ is written as 0.25 in decimal form. How can he find a decimal for 7/4?
Explain why ⅓ is called a rational number. How do you know?
Predict the decimal for 5/9 using mental math.
Divide to check your prediction, and describe the results.
How would you identify this as a rational or irrational number?
Jackson is practicing basketball. He makes a basket from the free-throw line 9 out of 27 shots.
Write the decimal from the fraction of Jackson’s shot that resulted in the basket.
How can you tell if this number is rational or irrational?
What is the difference between terminating and repeating decimals? How does the end behavior of a decimal relate to rational number or irrational number?
What do I need help with?
Answers
GPT-4o mini
Let's solve all the problems step by step, showing all necessary work.
**1. Finding the Decimal for \( \frac{7}{4} \)**
To find the decimal equivalent of \( \frac{7}{4} \), we can perform the division of 7 by 4.
\[
7 \div 4 = 1.75
\]
So, \( \frac{7}{4} = 1.75 \).
**2. Rational Number Explanation**
A rational number is defined as any number that can be expressed as the quotient or fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
For the fraction \( \frac{1}{3} \):
- \( p = 1 \) (an integer)
- \( q = 3 \) (an integer, not zero)
This satisfies the definition of a rational number. Therefore, \( \frac{1}{3} \) is a rational number because it can be expressed in the form of \( \frac{p}{q} \).
**Predicting the Decimal for \( \frac{5}{9} \)**
To predict the decimal for \( \frac{5}{9} \) using mental math, we can recognize that any fraction with a denominator of 9 will likely produce a repeating decimal.
We can predict:
\[
\frac{5}{9} \approx 0.555...\text{ (which we suspect will repeat)}
\]
**Dividing to Check the Prediction**
Now, let’s perform the actual division to check our prediction:
\[
5 \div 9 = 0.555\ldots
\]
So our prediction was confirmed, and we've identified that \( \frac{5}{9} = 0.\overline{5} \) (a repeating decimal).
**Rational or Irrational Number Identification**
A number is rational if it can be expressed as a fraction of two integers. Since \( \frac{5}{9} \) is already in fraction form, it is a rational number. The decimal \( 0.\overline{5} \) is also rational as it can continue indefinitely but repeats.
**3. Jackson's Shot Percentage**
Jackson made 9 out of 27 shots. We can write this as a fraction:
\[
\text{Fraction} = \frac{9}{27}
\]
To find the decimal, we divide:
\[
9 \div 27 = 0.333...
\]
So, the decimal representation of the fraction of Jackson’s shots that resulted in a basket is \( 0.\overline{3} \).
**Rational or Irrational Identification**
We determine that \( 0.333...\) (or \( \frac{1}{3} \)) is indeed a rational number because it can be expressed as a fraction \( \frac{p}{q} \).
**4. Difference Between Terminating and Repeating Decimals**
- **Terminating decimals** are decimals that come to an end—like \( 0.5 \) or \( 1.75 \).
- **Repeating decimals** continue indefinitely with one or more digits repeating—like \( 0.\overline{3} \) or \( 0.666...\).
The key difference is that terminating decimals have a definitive end point while repeating decimals continue infinitely with a set pattern.
**Rational vs. Irrational Numbers**
Rational numbers can be expressed as a fraction of integers (including terminating and repeating decimals). In contrast, **irrational numbers**, such as \( \sqrt{2} \) or \( \pi \), cannot be expressed as a fraction of two integers, and their decimal forms neither terminate nor repeat.
**What you might need help with:**
- Understanding specific examples of rational vs irrational numbers.
- Performing long division for fraction to decimal conversions.
- Recognizing patterns in decimal expansion.
Feel free to ask any specific questions or request further clarification on any topic!
**1. Finding the Decimal for \( \frac{7}{4} \)**
To find the decimal equivalent of \( \frac{7}{4} \), we can perform the division of 7 by 4.
\[
7 \div 4 = 1.75
\]
So, \( \frac{7}{4} = 1.75 \).
**2. Rational Number Explanation**
A rational number is defined as any number that can be expressed as the quotient or fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
For the fraction \( \frac{1}{3} \):
- \( p = 1 \) (an integer)
- \( q = 3 \) (an integer, not zero)
This satisfies the definition of a rational number. Therefore, \( \frac{1}{3} \) is a rational number because it can be expressed in the form of \( \frac{p}{q} \).
**Predicting the Decimal for \( \frac{5}{9} \)**
To predict the decimal for \( \frac{5}{9} \) using mental math, we can recognize that any fraction with a denominator of 9 will likely produce a repeating decimal.
We can predict:
\[
\frac{5}{9} \approx 0.555...\text{ (which we suspect will repeat)}
\]
**Dividing to Check the Prediction**
Now, let’s perform the actual division to check our prediction:
\[
5 \div 9 = 0.555\ldots
\]
So our prediction was confirmed, and we've identified that \( \frac{5}{9} = 0.\overline{5} \) (a repeating decimal).
**Rational or Irrational Number Identification**
A number is rational if it can be expressed as a fraction of two integers. Since \( \frac{5}{9} \) is already in fraction form, it is a rational number. The decimal \( 0.\overline{5} \) is also rational as it can continue indefinitely but repeats.
**3. Jackson's Shot Percentage**
Jackson made 9 out of 27 shots. We can write this as a fraction:
\[
\text{Fraction} = \frac{9}{27}
\]
To find the decimal, we divide:
\[
9 \div 27 = 0.333...
\]
So, the decimal representation of the fraction of Jackson’s shots that resulted in a basket is \( 0.\overline{3} \).
**Rational or Irrational Identification**
We determine that \( 0.333...\) (or \( \frac{1}{3} \)) is indeed a rational number because it can be expressed as a fraction \( \frac{p}{q} \).
**4. Difference Between Terminating and Repeating Decimals**
- **Terminating decimals** are decimals that come to an end—like \( 0.5 \) or \( 1.75 \).
- **Repeating decimals** continue indefinitely with one or more digits repeating—like \( 0.\overline{3} \) or \( 0.666...\).
The key difference is that terminating decimals have a definitive end point while repeating decimals continue infinitely with a set pattern.
**Rational vs. Irrational Numbers**
Rational numbers can be expressed as a fraction of integers (including terminating and repeating decimals). In contrast, **irrational numbers**, such as \( \sqrt{2} \) or \( \pi \), cannot be expressed as a fraction of two integers, and their decimal forms neither terminate nor repeat.
**What you might need help with:**
- Understanding specific examples of rational vs irrational numbers.
- Performing long division for fraction to decimal conversions.
- Recognizing patterns in decimal expansion.
Feel free to ask any specific questions or request further clarification on any topic!