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)
The price advertised is $3.399 per gallon.
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NO! not all fractions are rational. "A fraction is rational if it can be expressed as a ratio of two integers." But if you put pi/2 that would not be a rational fraction, as you cannot express it as a ratio of two integers!
To find the height, we can multiply the side length by the square root of 3/2.
Approximating the square root of 3/2 as 7/4, we have:
Height = (7/4)(8) = 14 meters.
To find the whole number and rational number to the nearest half that 58−−√ is between, we can look at the number line.
The square root of 58 is approximately 7.62 (rounded to two decimal places).
To the left of 7.62, the whole number and rational numbers to the nearest half are 7 and 7.5.
To the right of 7.62, the whole number and rational numbers to the nearest half are 8 and 8.5.
Therefore, 58−−√ is between 7 and 7.5.
Which set of numbers is always rational?(1 point)
Responses
integers
integers
decimals
decimals
positive numbers
positive numbers
fractions
fractions
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)
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)
To find the length of the diagonal, we can multiply the side length by 2−√.
Approximating the square root of 2 as 1.4, we have:
Diagonal = (2−√)(700) = (2−1.4)(700) = 0.6(700) = 420 feet.
Using a number line, what whole number and rational number to the nearest half is 40−−√ between?(1 point)
40−−√ is between _ and _.
To find the whole number and rational number to the nearest half that 40−−√ is between, we can look at the number line.
The square root of 40 is approximately 6.32 (rounded to two decimal places).
To the left of 6.32, the whole number and rational numbers to the nearest half are 6 and 6.5.
To the right of 6.32, the whole number and rational numbers to the nearest half are 6.5 and 7.
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
between 8.5 and 9
between 8.5 and 9
between 7 and 7.5
between 7 and 7.5
between 7.5 and 8
between 7.5 and 8
between 8 and 8.5
Using rational approximations, what is an inequality statement comparing −−√81 and –√88? Enter <, >, or =.(1 point)
The inequality is: −−√81 _ –√88.
To compare the square roots of 81 and 88 using rational approximations, we can use the fact that the square root of 81 is 9 and the square root of 88 is approximately 9.38 (rounded to two decimal places).
Therefore, the inequality statement is: −−√81 < –√88.
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:
0.72 <-- 8 ÷ 11 = 0.72 (round to two decimal places)
- 0.66 <-- Subtract 6 from 8, bring down the next digit
-------
0.60 <-- Bring down a zero, divide 60 by 11
- 0.55
-------
0.50
- 0.44
-------
0.60
The division keeps repeating with a cycle of 6, as indicated by the repeating decimal pattern. Therefore, 8/11 can be expressed as 0.72 bar.
Convert 0.72¯¯¯¯¯ to a rational number in simplest form.(1 point)
To convert the repeating decimal 0.72¯¯¯¯¯ to a rational number in simplest form, we can set x = 0.72¯¯¯¯¯ and subtract it from 100 times itself to eliminate the repeating part.
100x = 72.72¯¯¯¯¯
- x = 0.72¯¯¯¯¯
---------------
99x = 72
Simplifying further, we can divide both sides by 99:
99x/99 = 72/99
x = 8/11
Therefore, the rational number equivalent of 0.72¯¯¯¯¯ is 8/11 in simplest form.
Which of the following is a rational number between 5 and 5 1/4?(1 point)
Responses
513
5 Start Fraction 1 over 3 end fraction
517
5 Start Fraction 1 over 7 end fraction
525
5 Start Fraction 2 over 5 end fraction
26−−√
To find a rational number between 5 and 5 1/4, we can find a fraction with a denominator that falls between the denominators of 5 (4) and 5 1/4 (5).
The fractions 5 1/3, 5 1/7, and 5 2/5 all have denominators between 4 and 5. However, to find a fraction that lies strictly between the two given numbers, we can choose the fraction 5 1/3.
Therefore, the rational number between 5 and 5 1/4 is 5 1/3.
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 is between 7–√ and 8–√, we can use the fact that the square root of 7 is approximately 2.65, and the square root of 8 is approximately 2.83.
To get a rational number with a denominator of 7, we need to multiply both numbers by 7:
7–√ ≈ 7 * (2.65) = 18.55
8–√ ≈ 7 * (2.83) = 19.81
Since we want a rational number between these two values, we can choose the value from the open interval (18.55, 19.81). One possible choice is the value 19.
Thus, a rational number with a denominator of 7 that is between 7–√ and 8–√ is 19/7, which is an improper fraction.
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 that every whole number is a rational number, but not every rational number is a whole number.
A rational number is any number that can be expressed as a fraction, where the numerator and denominator are integers. Whole numbers include all positive integers (including zero) and do not include fractions or decimals.
All whole numbers can be expressed as a fraction, with the denominator being 1. For example, 3 can be expressed as the fraction 3/1, and 0 can be expressed as the fraction 0/1. Therefore, all whole numbers are rational numbers.
However, not all rational numbers are whole numbers. Examples of rational numbers that are not whole numbers include 1/2, 3/4, or 2/7. These fractions have numerators and denominators that are integers, but they represent values that are not whole numbers.