6. the diagram shows the number of instruments in the string section of an orchestra. there are 7 cellos, 4 double basses, and 11 each of first violins, second violins, and violas.
a. use ratio notation and decimal notation to describe the relationship between the number of double basses and the total number of instruments in the string section.
b. is the relationship in part a rational or irrational? give two different justifications for your answer.
7. the ratio of 8th - graders in a chess club to the total number of members is 0.45. write the decimal as a fraction. show your work.
8. critique reasoning a student claimed that the number - 4 is not rational because it is not a ratio of two integers. do you agree or disagree? explain.
9. use repeated reasoning pi (π) is an irrational number. what can you say about the numbers π/2, π/3, π/4, and so on? are these rational or irrational?

Question

Accepted Answer

### 6.
#### A.
# Explanation:
## Step1: Calculate total number of string - section instruments
Add the number of each type of instrument. Total = 7 (cellos)+4 (double - basses)+11 (first violins)+11 (second violins)+11 (violas)=44.
## Step2: Find the ratio in ratio notation
The ratio of double - basses to the total number of instruments in the string section is 4:44, which simplifies to 1:11.
## Step3: Find the ratio in decimal notation
Divide the number of double - basses by the total number of instruments. 4÷44 = 0.0909... = 0.\overline{09}
# Answer:
Ratio notation: 1:11; Decimal notation: 0.\overline{09}

#### B.
# Explanation:
## Justification 1: Definition of rational numbers
A rational number is a number that can be written as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b
eq0$. The ratio of double - basses to the total number of instruments is $\frac{4}{44}=\frac{1}{11}$, where $a = 1$ and $b = 11$ are integers and $b
eq0$.
## Justification 2: Decimal representation
The decimal 0.\overline{09} is a repeating decimal. All repeating decimals are rational numbers. A repeating decimal can be converted into a fraction. Let $x = 0.\overline{09}$, then $100x=9.\overline{09}$, and $100x - x=9.\overline{09}-0.\overline{09}$, $99x = 9$, $x=\frac{9}{99}=\frac{1}{11}$
# Answer:
The relationship in Part A is rational.

### 7.
# Explanation:
## Step1: Write the decimal as a fraction
The decimal 0.45 can be written as $\frac{45}{100}$ since 0.45 means 45 hundredths.
## Step2: Simplify the fraction
Find the greatest common divisor (GCD) of 45 and 100, which is 5. Divide both the numerator and the denominator by 5. $\frac{45\div5}{100\div5}=\frac{9}{20}$
# Answer:
$\frac{9}{20}$

### 8.
# Explanation:
A rational number is defined as a number that can be written in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b
eq0$. The number - 4 can be written as $\frac{-4}{1}$, where $a=-4$ and $b = 1$ are integers and $b
eq0$.
# Answer:
Disagree. - 4 is a rational number because it can be written as $\frac{-4}{1}$, which is a ratio of two integers.

### 9.
# Explanation:
Since $\pi$ is an irrational number (a non - repeating, non - terminating decimal), when we consider numbers like $\frac{\pi}{2},\frac{\pi}{3},\frac{\pi}{4}$, etc., they are also irrational. If we assume $\frac{\pi}{n}$ (where $n$ is a non - zero integer) is rational, then $\pi=n\times\frac{\pi}{n}$ would be rational (because the product of two rational numbers is rational), which contradicts the fact that $\pi$ is irrational.
# Answer:
The numbers $\frac{\pi}{2},\frac{\pi}{3},\frac{\pi}{4}$, and so on are irrational.

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QUESTION IMAGE

Question

Explanation:

6.

A.

Step1: Calculate total number of string - section instruments

Step2: Find the ratio in ratio notation

Step3: Find the ratio in decimal notation

Justification 1: Definition of rational numbers

Justification 2: Decimal representation

Step1: Write the decimal as a fraction

Step2: Simplify the fraction

Answer:

B.