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Rational, Irrational, and Decimal Practice
Fifty original questions covering the complete rational, irrational, and decimal lesson, with explanations and SAT-focused strategy.
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Question 1
Explanation
Natural numbers are the positive counting numbers \(1,2,3,\ldots\), so \(5\) qualifies.
Question 2
Explanation
Whole numbers include \(0\) and the natural numbers. With natural numbers starting at \(1\), only \(0\) fits.
Question 3
Explanation
Integers are whole-number values and their negatives, including zero. Thus \(-7\) is an integer.
Question 4
Explanation
\(0.75=75/100=3/4\), a ratio of integers, so it is rational.
Question 5
Explanation
Because \(11\) is not a perfect square, \(\sqrt{11}\) cannot be expressed as a ratio of integers and is irrational.
Question 6
Explanation
Opposites have equal distance from zero and opposite signs. The opposite of \(-4.6\) is \(4.6\).
Question 7
Explanation
More negative values lie farther left. Since \(-0.8<-0.5<-0.3<0\), \(-0.8\) is farthest left.
Question 8
Explanation
Zero is the origin and is neither positive nor negative. It is rational because \(0=0/1\).
Question 9
Explanation
\(0.125\) has a finite number of digits, so it terminates and is rational.
Question 10
Explanation
The block \(27\) repeats forever in \(0.272727\ldots\), making it a repeating rational decimal.
Question 11
Explanation
\(0.4=4/10=2/5\). Because it is a ratio of integers, it is rational.
Question 12
Explanation
\(25\) is a perfect square, so \(\sqrt{25}=5\), an integer and rational number.
Question 13
Explanation
The tenths digit is \(7\), and the hundredths digit is \(4\), so the tenths digit stays unchanged. The result is \(8.7\).
Question 14
Explanation
The thousandths digit is \(6\), so round the hundredths up. Carrying changes \(3.99\) to \(4.00\).
Question 15
Explanation
The real numbers consist of all rational and irrational numbers on the number line.
Question 16
Explanation
Both \(a\) and \(b\) must be integers, and division by zero is undefined, so \(b\ne0\).
Question 17
Explanation
Approximate \(-\sqrt3\approx-1.732\) and \(-2/3\approx-0.667\), so \(-\sqrt3<-2/3<0<\sqrt2\).
Question 18
Explanation
Because \(5^2=25<30<36=6^2\), taking positive square roots gives \(5<\sqrt{30}<6\).
Question 19
Explanation
Any integer \(n\) can be written as \(n/1\), so every integer is rational. The reverse is not true.
Question 20
Explanation
Divide \(7\) by \(8\): \(7/8=0.875\), a terminating decimal.
Question 21
Explanation
\(\pi\approx3.14159\), which is slightly greater than \(3.14\). The decimal \(3.14\) is only an approximation.
Question 22
Explanation
\(-\sqrt{49}=-7\). Negative integers are not whole or natural, so integer is the smallest listed set.
Question 23
Explanation
The gaps between \(1\)s keep changing, so there is no fixed repeating block and the decimal does not terminate. It is irrational.
Question 24
Explanation
Compare absolute values: \(0.09,0.1,0.125,0.11\). The smallest distance is \(0.09\), so \(-0.09\) is closest.
Question 25
Explanation
Distance is the absolute difference: \(|1.75-(-2.5)|=|4.25|=4.25\).
Question 26
Explanation
\(7/20=0.35\) terminates. In lowest terms its denominator contains only factors \(2\) and \(5\).
Question 27
Explanation
Let \(x=0.666\ldots\). Then \(10x=6.666\ldots\); subtracting gives \(9x=6\), so \(x=2/3\).
Question 28
Explanation
The thousandths digit is \(8\), so increase the hundredths digit from \(4\) to \(5\): \(-6.35\).
Question 29
Explanation
To round to hundredths, inspect the next digit, the thousandths digit.
Question 30
Explanation
\(\sqrt5\approx2.236\) is irrational and lies in the interval. \(9/4=2.25\) is in the interval but rational, so the added condition makes the answer unique.
Question 31
Explanation
\(x+0=x\), so it remains irrational. The others equal \(0,0,1\) (with \(x\ne0\)), all rational.
Question 32
Explanation
A number minus itself is \(0\), so \(\sqrt7-\sqrt7=0\), which is rational.
Question 33
Explanation
\(\sqrt3\cdot\sqrt3=3\), which is rational. Multiplying an irrational by itself can produce a rational number.
Question 34
Explanation
\(-12/3=-4\), an integer; every integer is rational. The other rational choices are not integers.
Question 35
Explanation
The opposite is \(1.8\). Distance is \(|1.8-(-1.8)|=3.6\).
Question 36
Explanation
Since \(4<\sqrt{20}<5\), negating reverses the placement: \(-5<-\sqrt{20}<-4\).
Question 37
Explanation
The third decimal is \(9\), and the next digit is \(6\), so rounding carries to \(0.050\). Three decimal places must be shown.
Question 38
Explanation
The fifth decimal digit is \(8\), so the fourth decimal digit \(2\) rounds up to \(3\): \(2.7183\).
Question 39
Explanation
\(1.4<\sqrt2\approx1.414<1.5=3/2\).
Question 40
Explanation
A nonzero denominator yields quotient zero only when the numerator is zero, so \(a=0\).
Question 41
Explanation
No real number squared equals \(-1\), so \(\sqrt{-1}\) is not real. The other values are real.
Question 42
Explanation
A rational number's decimal expansion either terminates or eventually repeats. Irrational decimals neither terminate nor repeat.
Question 43
Explanation
Let \(x=0.999\ldots\). Then \(10x-x=9\), so \(9x=9\) and \(x=1\).
Question 44
Explanation
Values from \(6.15\) up to but not including \(6.25\) round to \(6.2\). Thus \(6.24\) works.
Question 45
Explanation
The midpoint is the average: \(((-3/2)+(5/2))/2=(2/2)/2=1/2\).
Question 46
Explanation
The integers are \(-2,-1,0,1,2,3\), for a total of \(6\). Neither noninteger endpoint is included.
Question 47
Explanation
\(\sqrt{12}/2=(2\sqrt3)/2=\sqrt3\), irrational. The others simplify to \(3,5,1/2\).
Question 48
Explanation
If \(r+s\) were rational, subtracting rational \(r\) would make \(s\) rational, a contradiction. The other expressions can be rational.
Question 49
Explanation
Since \(3.16^2\approx9.986\), \(\sqrt{10}\approx3.162\). Choice B gives the useful location.
Question 50
Explanation
The tenths digit is \(9\), so the ones digit rounds up: \(149.95\) rounds to \(150\).
Keyboard: use Tab to move, arrow keys to change answer choices, and Enter to check an answer.
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Questions to review
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- Question 1Classify natural numbersEasy
- Question 2Classify whole numbersEasy
- Question 3Classify integersEasy
- Question 4Classify rational numbersEasy
- Question 5Classify irrational numbersEasy
- Question 6OppositesEasy
- Question 7Order real numbersEasy
- Question 8Zero and originEasy
- Question 9Terminating decimalsEasy
- Question 10Repeating decimalsEasy
- Question 11Decimal to fractionEasy
- Question 12Square-root classificationEasy
- Question 13RoundingEasy
- Question 14Rounding with carryingEasy
- Question 15Real number systemEasy
- Question 16Rational definitionMedium
- Question 17Order mixed real numbersMedium
- Question 18Estimate square rootsMedium
- Question 19Number-set relationshipsMedium
- Question 20Fraction to decimalMedium
- Question 21Compare irrational valuesMedium
- Question 22Simplify then classifyMedium
- Question 23Nonrepeating decimalsMedium
- Question 24Number-line distanceMedium
- Question 25Number-line distanceMedium
- Question 26Terminating fraction criterionMedium
- Question 27Repeating decimal to fractionMedium
- Question 28Rounding negativesMedium
- Question 29Rounding place valueMedium
- Question 30Mixed classificationMedium
- Question 31Irrational operationsMedium
- Question 32Operations on irrational numbersMedium
- Question 33Operations on irrational numbersMedium
- Question 34Multiple classificationsMedium
- Question 35Opposites and distanceMedium
- Question 36Negative square-root estimationMedium
- Question 37Rounding precisionMedium
- Question 38RoundingMedium
- Question 39Order mixed formsMedium
- Question 40Rational representationMedium
- Question 41Real number boundaryHard
- Question 42Rational decimal formsHard
- Question 43Repeating decimal equivalenceHard
- Question 44Reverse roundingHard
- Question 45Number-line midpointHard
- Question 46Count integers in intervalHard
- Question 47Simplify then classifyHard
- Question 48Number-set reasoningHard
- Question 49Approximate rootsHard
- Question 50RoundingHard