Rational numbers

The governing concept is the closure of rational numbers under addition. Let the two rational numbers be a/b and c/d, where a, b, c and d are integers and b and d are non-zero. Their sum is a/b + c/d = (ad + bc)/bd. The numerator ad + bc is an integer, and the denominator bd is a non-zero integer, so the result is again of the form of an integer divided by a non-zero integer. Hence it is rational, making option B correct. The sum is not necessarily irrational, natural or zero. For example, 1/2 + 1/3 = 5/6, which is rational but is neither a natural number nor zero. This confirms the general closure property.

If the two rational numbers are \(\frac{a}{b}\) and \(\frac{c}{d}\), where \(b,d\neq 0\), their product is \(\frac{ac}{bd}\). Since \(ac\) and \(bd\) are integers and \(bd\neq 0\), the product is rational. Therefore, option A is correct. The product may be positive, negative, or zero; it is not necessarily only negative or an integer. Exam tip: remember that rational numbers are closed under addition, subtraction, and multiplication.

Direct answer: Option B, 0. An additive identity is a number which leaves another number unchanged when added to it. Let any rational number be \(r\). Then \(r+0=r\), so zero does not alter the value. This is why 0 is called the identity for addition. Option A, 1, is not the additive identity because \(r+1\) is usually different from \(r\); 1 is the multiplicative identity because \(r\times1=r\). Option B is correct because adding zero keeps every rational number unchanged. Option C, -1, changes the number to \(r-1\), so it is not an identity. Option D, 2, changes the number to \(r+2\), so it also fails. The word “additive” is an important clue: look for the number used in addition, not multiplication. For example, \(\frac{5}{7}+0=\frac{5}{7}\) and \(-3+0=-3\). Remember: additive identity is 0; multiplicative identity is 1.

Multiplying any rational number by (1) gives the same number. Hence (1) is the multiplicative identity.

A rational number is in standard form when the numerator and denominator have no common factor other than 1 and the denominator is positive. The negative sign may initially appear in the denominator, but it is conventionally moved to the numerator. To simplify the fraction, divide 8 and 12 by their greatest common divisor, which is 4.

Thus, \\(\frac{8}{-12}=\frac{-8}{12}=\frac{-2}{3}\\). The denominator in \\(\frac{-2}{3}\\) is positive, and 2 and 3 have no common factor, so the fraction is in standard form. Option A is therefore correct. Option B has an equivalent value but does not follow the standard positive-denominator convention, while C is not fully simplified.

Both the numerator and denominator are negative. A fraction with like signs is positive, so \(\frac{-5}{-6}=\frac{5}{6}\). Options A and B represent negative values, while option D is the reciprocal. Exam tip: two negative signs in a fraction make the result positive.

In a fraction, the number above the line is called the numerator, while the number below it is the denominator. In \(\frac{4}{5}\), 4 is above the line, so the numerator is 4; 5 is the denominator. Exam tip: identify the numerator by looking at the top number of the fraction.

In a fraction, the top number is the numerator and the bottom number is the denominator. In \(\frac{13}{17}\), 13 is the numerator and 17 is below it, so 17 is the denominator. Exam tip: To identify the denominator, look at the number written below the fraction bar.

The fraction is \\(\frac{0}{-9}\\). Division asks which number multiplied by the denominator gives the numerator. Since \\((-9)\times 0=0\\), the value is 0. More generally, zero divided by any nonzero number equals zero. The denominator here is -9, which is nonzero, so the fraction is defined. Therefore option B is correct.

The negative sign in the denominator does not make the result negative when the numerator is zero; zero has no positive or negative sign in this calculation. The expression would be undefined only if the denominator were zero, not merely because it is negative. Thus it is incorrect to choose -9 or “undefined.” The answer can also be seen from \\(\frac{0}{9}=0\\) and changing the denominator's sign still leaves the quotient zero.

A rational number is written in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q\neq 0\). Since \(\frac{2}{2}=1\), option A is correct. \(\frac{0}{1}=0\), \(\frac{3}{4}\neq 1\), and \(\frac{1}{0}\) is undefined. Exam tip: 1 can be written as \(\frac{n}{n}\) for any non-zero integer \(n\).

Dividing both the numerator and denominator of \(\frac{6}{14}\) by \(2\) gives \(\frac{3}{7}\). Thus, \(\frac{6}{14}=\frac{3}{7}\), so the two fractions are equivalent. Option D is incorrect because fractions represent rational numbers. Exam tip: To check whether two fractions are equivalent, divide or multiply their numerator and denominator by the same non-zero number.

The HCF of 12 and 15 is 3. Dividing both terms by 3 gives \(\frac{12\div3}{15\div3}=\frac{4}{5}\), which is in simplest form. Neither 2 nor 5 divides both numbers exactly, and 12 is not a divisor of 15. Exam tip: To reduce a fraction, divide its numerator and denominator by their HCF.

Both fractions have the same denominator, 9, so add the numerators and retain the denominator: \(\frac{2}{9}+\frac{1}{9}=\frac{2+1}{9}=\frac{3}{9}\). This can also be simplified to \(\frac{1}{3}\), but among the given options, \(\frac{3}{9}\) is correct. Exam tip: When adding fractions with the same denominator, add only the numerators and keep the denominator unchanged.

The governing concept is the additive inverse of a rational number. The opposite-sign form of a number x is -x, and x plus -x equals zero. Here x = -7/3, so its opposite is -(-7/3) = 7/3. The check is (-7/3) + (7/3) = 0, confirming that option A is correct. Option B is the reciprocal of -7/3, not its additive inverse. Option C is the reciprocal of the positive value 7/3. Option D has two negative signs, so it simplifies to 7/3 and represents the same value, but option A is the standard and clearest form requested. The key distinction is between changing a sign and taking a reciprocal.

Both numbers are positive fractions with the same numerator, 3. When the same positive number is divided into fewer equal parts, each part is larger. Therefore, a denominator of 4 gives larger parts than a denominator of 5. This is the key idea for comparing positive fractions with equal numerators.

To verify it numerically, use a common denominator of 20. We get \(3/4=15/20\) and \(3/5=12/20\). Since 15 is greater than 12, \(3/4\) is greater than \(3/5\). Thus option B is correct. Option A reverses the comparison, while the fractions are neither equal nor negative.

The direct answer is C, \\(\frac{-1}{6}\\). On a number line, among negative numbers the number closer to zero is greater. Compare the magnitudes: \\(\frac{1}{3}=\frac{2}{6}\\), so \\(\frac{-1}{3}=\frac{-2}{6}\\). Now compare \\(\frac{-2}{6}\\) and \\(\frac{-1}{6}\\): the latter is closer to zero, so it is greater. Equivalently, their decimal values are approximately \\(-0.333\\) and \\(-0.167\\); \\(-0.167\\) is greater. Option A is wrong because \\(\frac{-1}{3}\\) is farther left on the number line. Option B is wrong because the fractions have different values. Option C is correct because \\(\frac{-1}{6}>\frac{-1}{3}\\). Option D is wrong because the values are given exactly, so the comparison can be determined. Common mistake: thinking that the larger denominator automatically gives a smaller negative number without considering the sign. For negative fractions with the same numerator magnitude, the one closer to zero is larger.

To change the denominator from \(8\) to \(16\), multiply it by \(2\). To keep the fraction equivalent, the numerator must also be multiplied by \(2\): \(5\times2=10\). Therefore, the correct answer is \(10\). Exam tip: In equivalent fractions, multiply or divide both the numerator and denominator by the same non-zero number.

The middle value is ( \frac{\frac{2}{3}+\frac{4}{5}}{2}=\frac{11}{15} ). You can take the average to find a number between two numbers.

The greatest common divisor of 10 and 25 is 5. Dividing both the numerator and denominator by 5 gives \(\frac{10\div5}{25\div5}=\frac{2}{5}\). Option A, \(\frac{1}{5}\), is incorrect because the numerator and denominator must both be divided by the same common factor. Exam tip: To reduce a fraction to its simplest form, divide its numerator and denominator by their greatest common divisor.

Since \(16\) is a perfect square, \(\sqrt{16}=4\). The number \(4\) can be written as \(\frac{4}{1}\), so it is rational. Therefore, option B is correct. It is not irrational because its value is an exact integer, and it is certainly a real, defined number. Exam tip: the square root of a positive perfect square is an integer and hence rational.

A repeating decimal is always rational because it can be expressed as the ratio of two integers. Here, \(2.\overline{3}=2.333\ldots=\frac{7}{3}\), so it is a rational real number. Irrational decimals are non-terminating and non-repeating. Exam tip: identify a repeating decimal as rational and a non-terminating, non-repeating decimal as irrational.

Option B is correct because −5 is an integer, and every integer can be written as \(\frac{p}{q}\) with \(q\neq 0\), so it is rational. Also, 0.4 is a terminating decimal: \(0.4=\frac{4}{10}=\frac{2}{5}\), so it is rational. In contrast, \(\sqrt{3}\), \(\sqrt{7}\), and \(\sqrt{11}\) are irrational, and \(\pi\) is also irrational. Exam tip: Every integer and every terminating or recurring decimal is a rational number.

\(22/7\) is the ratio of two integers with a non-zero denominator, so it is a rational number. Its value is very close to π, but it is not exactly equal to π. Therefore, option A is correct; option B is wrong because \(22/7\) is rational, and option D is wrong because \(22/7 = 3\frac{1}{7}\), which is not an integer. Exam tip: Before calling a fraction an integer, check whether its numerator is exactly divisible by its denominator.

The governing concept is the definition of coprime, or relatively prime, numbers. Two integers are called coprime when their only positive common factor is 1. Therefore their highest common factor, also called greatest common divisor, is 1. Option C is correct. The numbers themselves need not be prime: for example, 8 and 15 are both composite in the broad sense of ordinary integers, yet their common factors include only 1, so HCF(8,15) = 1. A value of 2 or 3 would mean that both numbers share that factor, which would contradict coprimality. Zero is not the HCF of ordinary nonzero coprime numbers. This property is important in rational-number proofs because a fraction in lowest form has a numerator and denominator whose HCF is 1.