Number line

Correct answer: B. \(-1.11\)

Explanation: \(-1.11\) can be written as \(-1.110\) up to three decimal places. Comparing \(-1.110\) and \(-1.101\), \(-1.110\) is more negative, so it lies further left on the number line and is smaller. \(-1.101\) is closer to zero and is therefore greater. Exam tip: for negative decimals, the number with the greater magnitude is the smaller number.

Correct answer: B. 12

Explanation: The distance between two numbers on a number line is their absolute difference: \(\left|-13-(-1)\right|=\left|-12\right|=12\). Therefore, the correct answer is 12. Choosing 11 misses one unit. Exam tip: a distance is always non-negative.

Correct answer: C. Both are equal

Explanation: The correct answer is that both are equal. A zero added at the end of a decimal does not change its value, so \(-0.70=-0.7\). Therefore, neither number is greater; they represent the same point on the number line. Exam tip: Adding or removing trailing zeros in a decimal does not change its value.

Correct answer: A. 3.5

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-3+10}{2}=\frac{7}{2}=3.5\). Therefore, 3.5 is correct. Although 4 is close, it is not at an equal distance from both numbers. Exam tip: add the two numbers and divide by 2 to find their midpoint.

Correct answer: C. 8

Explanation: On a number line, the distance between two numbers is the positive value of their difference. Thus, \(|-16-(-8)|=|-8|=8\). Therefore, 8 is the correct answer. Choosing 7 would result from an incorrect subtraction. Exam tip: a distance is always non-negative.

Correct answer: A. ( -2.01 )

Explanation: Among negative numbers, the number with the greater magnitude is smaller. Here, \(2.01=2.010\), and \(2.010>2.001\). Therefore, \(-2.010<-2.001\), so \((-2.01)\) is smaller. \((-2.001)\) is closer to zero and is therefore greater. Exam tip: Add trailing zeros when needed to compare decimal places.

Correct answer: A. 5

Explanation: The midpoint of two numbers on a number line is their average:
\(\frac{-5+15}{2}=\frac{10}{2}=5\). Therefore, 5 is correct. The value 10 is the distance between the numbers, not their midpoint. Exam tip: add the two numbers first, then divide by 2.

Correct answer: A. 15

Explanation: The distance between two numbers on a number line is the absolute value of their difference. Thus, \(|6-(-9)|=|15|=15\). Therefore, 15 is correct. A result such as 14 may come from handling the negative sign incorrectly. Exam tip: always take the absolute value of the difference, because distance cannot be negative.

Correct answer: A. Infinitely many irrational numbers lie between any two distinct rational numbers.

Explanation: If \(a<b\) are rational, then \(a+(b-a)\sqrt2/n\), for \(n\ge2\), is an irrational number between them. Different values of \(n\) give infinitely many such numbers. Exam tip: both rational and irrational numbers are dense on the number line.

Correct answer: A. \(OB^2=OA^2+AB^2=2^2+1^2=5\), so \(OB=\sqrt{5}\)

Explanation: Triangle \(OAB\) is right-angled. By Pythagoras, \(OB^2=2^2+1^2=5\), hence \(OB=\sqrt{5}\). Transferring this radius from \(O\) to the number line marks the required point. Exam tip: verify the sum of squares, not the sum of sides.

Correct answer: B. \(-7\)

Explanation: For negative numbers, the number closer to zero is greater. \(-7\) is closer to zero than \(-7.7\), so \(-7 > -7.7\). Choosing \(-7.7\) is incorrect because it lies to the left of \(-7\) on the number line. Exam tip: on a number line, the number farther to the right is always greater.

Correct answer: A. \(-4\)

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-12+4}{2}=\frac{-8}{2}=-4\). Hence, \(-4\) is correct. \(-8\) is the sum of the two numbers, not their midpoint. Exam tip: Add the two numbers first and then divide by 2 to find the midpoint.

Correct answer: C. 16

Explanation: On a number line, the distance between two numbers is the positive value of their difference. Thus, \(\lvert -18-(-2)\rvert=\lvert -16\rvert=16\). Therefore, 16 is correct. Choosing 18 results from not subtracting the negative number correctly. Exam tip: always take the absolute value of the difference for distance.

Correct answer: A. There are infinitely many rational and infinitely many irrational numbers

Explanation: For \(a<b\), both rational and irrational numbers are dense between them, so each type occurs infinitely often. \((a+b)/2\) gives only one midpoint, not the full set. Exam tip: density means that every interval contains such numbers.

Correct answer: B. ( -4 )

Explanation: On a number line, the number farther to the right is greater.

definitely
-4 is closer to zero than -4.4 and lies to its right on the number line. Therefore, ( -4 ) is greater. “Both are equal” is incorrect because -4.0 and -4.4 are different numbers. Exam tip: Among negative numbers, the number with the smaller absolute value is greater.

Correct answer: B. 12

Explanation: On a number line, the distance between two numbers is the absolute value of their difference: \(|2-(-10)|=|12|=12\). Therefore, the correct answer is 12.
-12 may arise as a signed difference, but distance can never be negative. Exam tip: always use the absolute value of the difference when finding distance.

Correct answer: A. \(-\sqrt{3}\)

Explanation: Since \(1<\sqrt{3}<2\), multiplying by \(-1\) reverses the inequalities: \(-2<-\sqrt{3}<-1\). In contrast, \(-\sqrt{5}\) is less than \(-2\). Exam tip: use nearby perfect squares to locate square roots.

Correct answer: A. 4

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-6+14}{2}=\frac{8}{2}=4\). Therefore, 4 is the correct answer. For example, 5 is not equally distant from -6 and 14. Exam tip: add the two numbers and divide the result by 2 to find the midpoint.

Correct answer: B. 6

Explanation: The distance between two numbers on a number line is their absolute difference: \(|-11-(-5)|=|-6|=6\). Hence, the correct answer is 6. Getting 5 would mean the signs of the negative numbers have not been handled correctly. Exam tip: always take the absolute value of the difference, since distance cannot be negative.

Correct answer: B. -0.11

Explanation: Write the decimals with the same number of places: \(-0.11=-0.110\). Comparing \(-0.110\) and \(-0.101\), \(-0.110\) is more negative, so it lies farther left on the number line and is smaller. Therefore, \(-0.11\) is correct. \(-0.101\) is closer to zero, so it is greater. Exam tip: When comparing negative decimals, add trailing zeros if needed and then compare place values.

Correct answer: A. 0

Explanation: The midpoint of two numbers is found using \(\frac{a+b}{2}\). Here, \(\frac{-8+8}{2}=\frac{0}{2}=0\). Therefore, 0 is at an equal distance from -8 and 8. Option 4 is incorrect because it is not the average of -8 and 8. Exam tip: The midpoint of opposite numbers \(a\) and \(-a\) is always 0.

Correct answer: B. 21

Explanation: The distance between two numbers on a number line is the absolute value of their difference: \(|7-(-14)|=|21|=21\). Therefore, the correct answer is 21. Choosing 20 usually results from missing the sign change while subtracting a negative number. Exam tip: always use \(|a-b|\) to find distance.

Correct answer: C. Both are equal

Explanation: Writing -0.2 as -0.20 only adds a zero to the right of the decimal part; it does not change the value of the number. Hence, -0.2 = -0.20, so both are equal. Options A and B are not correct because neither number is greater than the other. Exam tip: Zeros added at the end of a decimal do not change its value.

Correct answer: B. \(-0.00005\)

Explanation: For negative numbers, the number closer to zero is greater. \(-0.00005\) is closer to zero than \(-0.0005\); therefore, \(-0.00005 > -0.0005\). They are not equal because their decimal values are different. Exam tip: On a number line, the number farther to the right is greater.

Correct answer: A. -2

Explanation: The midpoint of two numbers is their average: \(\frac{-13+9}{2}=\frac{-4}{2}=-2\). Hence, \(-2\) is correct. \(-1\) can result from an error while adding the signed numbers. Exam tip: add the numbers with their signs first, then divide by 2.