Question

Which statement is true about the relationships between the number sets?

all intergers are whole numbers
not all natural numbers are rational
some irrational numbers are rational
all natural numbers are integers

Answer 1

[AI-generated answer — Bot GPT 3.5]

The statement "all integers are whole numbers" is true.

The statement "not all natural numbers are rational" is also true. Rational numbers are numbers that can be expressed as a fraction or ratio of two integers, while natural numbers are only positive integers. Therefore, not all natural numbers can be expressed as a rational number (e.g. √2 is a natural number but is irrational).

The statement "some irrational numbers are rational" is false. Irrational numbers are numbers that cannot be expressed as a fraction or ratio of two integers. Therefore, by definition, irrational numbers cannot be rational numbers.

The statement "all natural numbers are integers" is true. Natural numbers include all positive integers, and integers include all positive and negative whole numbers.

Answer 2

Which of the following is true about the number 1/6 ?

it's a rational number but not a real number
it's both an integer and a rational number
It's both an irrational number and a real number
It's a rational number but not a whole number

Answer 3

[AI-generated answer — Bot GPT 3.5]

The statement "it's a rational number but not a real number" is false. The number 1/6 is both a rational number and a real number.

The statement "it's both an integer and a rational number" is false. Integers are whole numbers (negative, zero, and positive) and do not include fractions like 1/6. Therefore, 1/6 is not an integer.

The statement "it's both an irrational number and a real number" is also false. Irrational numbers are numbers that cannot be expressed as a fraction or ratio of two integers. Since 1/6 can be expressed as a fraction, it is not an irrational number. However, it is a real number as it can be plotted on the number line.

The statement "it's a rational number but not a whole number" is true. Rational numbers can be expressed as a fraction or ratio of two integers, which 1/6 is. Whole numbers are integers that are greater than or equal to zero. Since 1/6 is a fraction, it is not a whole number.

Answer 4

A sign in a gas station advertises gas at the rate listed below. How many dollars does a gallon cost, writing the price strictly as a decimal?

$4.12 9/10

Answer 5

[AI-generated answer — Bot GPT 3.5]

To write the price strictly as a decimal, we need to convert the fraction 9/10 to a decimal. To do this, we divide 9 by 10:

9 ÷ 10 = 0.9

Now we can write the price strictly as a decimal:

$4.12 + 0.9 = $4.129

Therefore, a gallon of gas costs $4.129.

Answer 6

Which set of numbers is always rational

Answer 7

[AI-generated answer — Bot GPT 3.5]

The set of numbers that is always rational is the set of integers. Integers include all whole numbers (positive, negative, and zero) and can be expressed as a ratio of two integers.

Answer 8

The height of an equilateral triangle equals √3/2 times the side length. What is the approximate height in meters of an equilateral triangle with a side length of 16 meters?

Answer 9

[AI-generated answer — Bot GPT 3.5]

To find the height of an equilateral triangle with a side length of 16 meters, we can use the formula given, which states that the height is equal to (√3/2) times the side length.

First, let's calculate (√3/2) times 16:

(√3/2) * 16 = (1.732/2) * 16 = 0.866 * 16 = 13.856

Therefore, the approximate height of the equilateral triangle is approximately 13.856 meters.

Answer 10

The diagonal of a square equals √2 times the side length. Using 1.4 for √2, what is the approximate length in feet from one corner to an opposite corner of a square park that is 300 feet on a side?

Answer 11

[AI-generated answer — Bot GPT 3.5]

To find the length from one corner to an opposite corner of a square park with a side length of 300 feet, we can use the formula given, which states that the diagonal is equal to √2 times the side length.

First, let's calculate √2 times 300 using 1.4 as an approximation for √2:

1.4 * 300 = 420

Therefore, the approximate length from one corner to an opposite corner of the square park is approximately 420 feet.

Answer 12

What whole number and rational number to the nearest half is √128 between?

Answer 13

[AI-generated answer — Bot GPT 3.5]

To find the whole number and rational number to the nearest half that √128 falls between, we can calculate the square root of 128 and identify the two numbers it lies between.

√128 ≈ 11.3137

To the nearest half, √128 falls between:

11 and 11.5,
or
11.5 and 12.

Therefore, the whole number to the nearest half is 11 and the rational number to the nearest half is 11.5.

Answer 14

hat whole number and rational number to the nearest half is √8 between?

Answer 15

[AI-generated answer — Bot GPT 3.5]

To find the whole number and rational number to the nearest half that √8 falls between, we can calculate the square root of 8 and identify the two numbers it lies between.

√8 ≈ 2.8284

To the nearest half, √8 falls between:

2 and 2.5,
or
2.5 and 3.

Therefore, the whole number to the nearest half is 2 and the rational number to the nearest half is 2.5.

Answer 16

Using rational approximations, what is an inequality statement comparing √72 and √76?

Answer 17

[AI-generated answer — Bot GPT 3.5]

To compare √72 and √76 using rational approximations, we can calculate the square roots and compare the results:

√72 ≈ 8.4853
√76 ≈ 8.7178

Since 8.4853 is less than 8.7178, we can write the inequality statement as:

√72 < √76

Answer 18

Write 5,143 in expanded form

Answer 19

[AI-generated answer — Bot GPT 3.5]

5,143 in expanded form is:

5,000 + 100 + 40 + 3.

Answer 20

Express 8/27 as a repeating decimal.

Answer 21

[AI-generated answer — Bot GPT 3.5]

To express 8/27 as a repeating decimal, we need to perform long division. Here's the process:

0.296296...
----------------
27 | 8.0000000000000
- 7.56
0.440
- 0.405
0.350
- 0.324
0.260
- 0.243
0.170
- 0.162
0.080
- 0.081
-0.001

The division keeps repeating from where the remainder becomes zero. Therefore, 8/27 as a repeating decimal is approximately 0.296296...

Answer 22

Which of the following is a rational number between 3 and 3 1/6

Answer 23

[AI-generated answer — Bot GPT 3.5]

To find a rational number between 3 and 3 1/6, we can determine the average of these two numbers.

The average is calculated as follows:

(3 + 3 1/6) ÷ 2

Converting 3 1/6 to an improper fraction:

3 1/6 = (3 * 6 + 1) / 6 = 19/6

Now, calculating the average:

(3 + 19/6) ÷ 2 = (18/6 + 19/6) ÷ 2 = 37/6 ÷ 2 = 37/12

Therefore, a rational number between 3 and 3 1/6 is 37/12.

Answer 24

What is a rational number with a denominator of 7 that is between √11 and √12. Write your answer as an improper fraction

Answer 25

[AI-generated answer — Bot GPT 3.5]

To find a rational number with a denominator of 7 that is between √11 and √12, we can find a common denominator and choose a numerator that falls between the square roots.

√11 is between 3 and 4, while √12 is between 3 and 4.5.

For a denominator of 7, we can choose a numerator such as 17 to ensure it falls between √11 and √12.

Therefore, a rational number with a denominator of 7 that is between √11 and √12 is 17/7 as an improper fraction.