Which statement is true about the relationships between the number sets?(1 point)
Not all natural numbers are real numbers.
Whole numbers include all natural numbers and 0.
Some irrational numbers are also integers.
All rational numbers are also integers.
Whole numbers include all natural numbers and 0.
Which of the following is true about −9?(1 point)
It is both an integer and a whole number.
It is a whole number but not an integer.
It is an integer but not a rational number.
It is an integer but not a whole number.
It is both an integer and a whole number.
A sign in a gas station advertises gas at the rate of $3.39 9/10 per gallon. How many dollars does a gallon cost, writing the price strictly as a decimal?(1 point)
To convert the price to a decimal, we can round up the fraction 9/10 to the nearest whole number. Then, we add it to the whole number part of the price.
$3.39 + 1 = $4.39
Thus, a gallon costs $4.39.
Which set of numbers is always rational?(1 point)
The set of integers is always rational.
The height of an equilateral triangle equals 3√/2 times the side length. Using 7/4 for the square root of 3, what is the approximate height in meters of an equilateral triangle with a side length of 8 meters?(1 point)
Given that the height of an equilateral triangle is equal to 3√/2 times the side length, and using 7/4 as the approximate value for the square root of 3, we can calculate the height as follows:
height = (3√/2) * side length
height = (3 * (7/4))/2 * 8
To calculate the height, we can simplify the expression:
height = (21/4)*(8/2)
height = (21/4)*4
height = 21
Therefore, the approximate height in meters of an equilateral triangle with a side length of 8 meters is 21.
The diagonal of a square equals 2–√ times the side length. Using 1.4 for the square root of 2, what is the approximate length in feet from one corner to an opposite corner of a square park that is 700 feet on a side?(1 point)
Given that the diagonal of a square is equal to 2–√ times the side length, and using 1.4 as the approximate value for the square root of 2, we can calculate the length from one corner to an opposite corner as follows:
diagonal = (2 - √2) * side length
diagonal = (2 - 1.4) * 700
To calculate the length, we can simplify the expression:
diagonal = 0.6 * 700
diagonal = 420
Therefore, the approximate length in feet from one corner to an opposite corner of a square park that is 700 feet on a side is 420.
Using a number line, what whole number and rational number to the nearest half is 40−−√ between?(1 point)
40−−√ is between
and
.
To determine the whole number and rational number that are closest to √40, we need to find the whole number and rational number that are closest to the square root of 40.
The square root of 40 is approximately 6.32.
To the nearest half, the whole number and rational number that √40 is between are:
6 and 6.5
Therefore, 40−−√ is between 6 and 6.5.
Using a number line, what whole number and rational number to the nearest half is 58−−√ between?(1 point)
Responses
To determine the whole number and rational number that are closest to √58, we need to find the whole number and rational number that are closest to the square root of 58.
Using a calculator, we find that the square root of 58 is approximately 7.62.
To the nearest half, the whole number and rational number that √58 is between are:
7 and 7.5
Therefore, 58−−√ is between 7 and 7.5.
Using rational approximations, what is an inequality statement comparing 81−−√ and 8–√8? Enter <, >, or =.(1 point)
The inequality is: 81−−√
8–√8.
To compare 81−−√ and 8–√8 using rational approximations, we can simplify the expressions:
81−−√ is approximately 9
8–√8 is approximately 6.83
Therefore, the inequality statement comparing 81−−√ and 8–√8 would be:
9 > 6.83
Thus, the inequality is: 81−−√ > 8–√8.
Express 811 as a repeating decimal. Mark the cycle of repeating digits with a bar.(1 point)
To express 811 as a repeating decimal, we will divide 811 by the appropriate power of 10. Since there is no remainder in this division, the decimal will terminate.
811 ÷ 1000 = 0.811
Therefore, 811 can be expressed as the decimal 0.811.
Express 8/11 as a repeating decimal. Mark the cycle of repeating digits with a bar.(1 point)
To express 8/11 as a repeating decimal, we can use long division:
0.7
-----------
11 | 8.00
- 7 7
-------
30
- 2 2
-------
80
- 77
-------
30
- 22
-------
80
The division of 8 by 11 results in a repeating pattern of 7 as the quotient:
8/11 = 0.7(27)
Therefore, 8/11 can be expressed as the repeating decimal 0.7272... with the repeating digits 72 marked with a bar.
Convert 0.72¯¯¯¯¯ to a rational number in simplest form.(1 point)
To convert the repeating decimal 0.72¯¯¯¯¯ to a rational number, we can use the method of repeating decimals.
Let x = 0.72¯¯¯¯¯
To eliminate the repeating part, we can multiply both sides of the equation by a power of 10. Since the repeating part consists of 2 digits, we can multiply by 100:
100x = 72.7272¯¯¯¯¯
Next, we subtract the original equation from the multiplied equation to eliminate the repeating part:
100x - x = 72.7272¯¯¯¯¯ - 0.72¯¯¯¯¯
99x = 72
Dividing both sides by 99, we have:
x = 72/99
To simplify this fraction, we can factor out the greatest common divisor (GCD) of 72 and 99, which is 9:
x = (8 * 9)/(11 * 9)
x = 8/11
Therefore, the decimal 0.72¯¯¯¯¯ can be expressed as the rational number 8/11 in simplest form.
Which of the following is a rational number between 5 and 514?(1 point)
Responses
To find a rational number between 5 and 514, we can choose any fraction between them. Let's choose the simplest fraction possible, which is 6/1.
6/1 = 6
Therefore, 6 is a rational number between 5 and 514.
Which of the following is a rational number between 5 and 5/14?(1 point)
Responses
To find a rational number between 5 and 5/14, we can choose any fraction between them. Let's choose the fraction 1/2.
1/2 is a rational number between 5 and 5/14.
Therefore, 1/2 is a rational number between 5 and 5/14.
Which of the following is a rational number between 5 and 514?(1 point)
26−−√
5 2/5
5 1/7
5 1/3
To determine which of the given options is a rational number between 5 and 514, we need to find a rational number that lies between these values.
The square root of 26 is approximately 5.1. Therefore, 26−−√ is not a rational number between 5 and 514.
To compare the remaining options, we can convert them to decimals:
5 2/5 = 5.4
5 1/7 = 5.142857...
5 1/3 = 5.333...
Among these options, 5 1/7 = 5.142857... is the only rational number that lies between 5 and 514.
Therefore, 5 1/7 is the rational number between 5 and 514.
What is a rational number with a denominator of 7 that is between 7–√ and 8–√. Write your answer as an improper fraction.(1 point)
To find a rational number with a denominator of 7 that lies between 7–√ and 8–√, we need to find the whole number part of each radical expression.
7–√ is approximately 2.64, which means the whole number part is 2.
8–√ is approximately 2.83, which means the whole number part is also 2.
To find a rational number with a denominator of 7 between these values, we can use the whole number part and the common denominator.
The whole number part is 2, so we will use 2 as the numerator with a denominator of 7, resulting in the improper fraction:
2/7
Therefore, 2/7 is a rational number with a denominator of 7 that lies between 7–√ and 8–√.
Describe the relationship between rational numbers and whole numbers. Are all rational numbers also whole numbers? Are all whole numbers also rational numbers? Give examples to support your answer.(4 points)
The relationship between rational numbers and whole numbers is as follows:
1. Are all rational numbers also whole numbers?
No, not all rational numbers are whole numbers. Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. Whole numbers, on the other hand, include all natural numbers (positive integers) and zero. Whole numbers do not have a fractional or decimal component.
Example:
3/2 is a rational number but not a whole number because it has a fractional component.
2. Are all whole numbers also rational numbers?
Yes, all whole numbers are also rational numbers. This is because whole numbers can be expressed as fractions with a denominator of 1. Any number with a denominator of 1 is considered a rational number.
Example:
5 is a whole number, and it can also be expressed as 5/1, which is a rational number.
In conclusion, while all whole numbers are rational numbers, not all rational numbers are whole numbers.