Chapter-1 Rational Numbers exampler

Explanation:

A rational number is defined as a number that can be expressed as p/q where p and q are integers and q is not equal to 0.

Natural numbers are counting numbers starting from 1, such as 1, 2, 3, 4, 5, ...

Whole numbers are natural numbers and 0, such as 0, 1, 2, 3, 4, 5, ...

Integers are whole numbers and their negatives, such as ..., -3, -2, -1, 0, 1, 2, 3, ...

Therefore, a number that can be expressed as p/q, where p and q are integers and q is not equal to 0, is a (d) rational number.

Explanation:

The correct option is (d) p and q are integers and p ≠ 0 also q ≠ 0.

A rational number is defined as any number that can be expressed as a ratio of two integers p and q, where q is not equal to 0. Hence, p and q must be integers, as mentioned in options (a) and (b). However, option (c) is incorrect because it allows q to be zero, which is not allowed since division by zero is undefined.

Therefore, the correct option is (d), which specifies that both p and q must be integers and neither p nor q can be zero.

Explanation:

The correct option is (a) rational numbers are closed under addition.

The expression (3/8) + (-5/7) = (-19/56) shows that the sum of two rational numbers is also a rational number. Here, both (3/8) and (-5/7) are rational numbers because they can be expressed as a ratio of two integers, and their sum (-19/56) is also a rational number because it can also be expressed as a ratio of two integers.

Therefore, this expression illustrates that the set of rational numbers is closed under addition, which means that the sum of any two rational numbers is also a rational number. This corresponds to option (a)

Explanation:

The correct option is (d) rational numbers are closed under division.

Rational numbers are defined as any number that can be expressed as a ratio of two integers p and q, where q is not equal to 0. Therefore, any rational number can be written as p/q.

Option (a) is true because the sum of two rational numbers is always a rational number, which means that the set of rational numbers is closed under addition.

Option (b) is true because the difference between two rational numbers is also a rational number, which means that the set of rational numbers is closed under subtraction.

Option (c) is true because the product of two rational numbers is also a rational number, which means that the set of rational numbers is closed under multiplication.

Option (d) is not true because division by zero is undefined, and therefore the set of rational numbers is not closed under division. In other words, if we divide a non-zero rational number by zero, we get an undefined result.

Therefore, option (d) is the correct answer as it is not true.

Explanation:

The correct option is (a) addition of rational numbers is commutative.

The expression (-3/8) + (1/7) = (1/7) + (-3/8) shows that the order in which two rational numbers are added does not affect the result. Here, both expressions on either side of the equation yield the same result, which means that the addition of rational numbers is commutative.

Option (a) correctly identifies this property of rational numbers, which means that the order of addition can be changed without affecting the result.

Option (b) is not applicable here as it deals with the closure property of rational numbers under addition.

Option (c) deals with the grouping of numbers in addition, which is not applicable in this case.

Option (d) deals with the distributive property of rational numbers under addition and multiplication, which is also not applicable in this case.

Therefore, option (a) is the correct answer, as it correctly identifies the commutativity property of rational numbers under addition.

Explanation:

The correct option is (b) [(2/3) × ((-6/7) × (3/5))] = [(2/3) × ((3/5) × (-6/7))].

Associativity is a property that deals with the grouping of numbers in an operation, such that the order of grouping does not affect the result. For rational numbers, the associativity property states that the order of grouping of rational numbers in multiplication does not affect the result.

Let's evaluate the expressions given in the options:

(a) [(2/3) × ((-6/7) × (3/5))] = [((2/3) × (-6/7)) × (3/5)] LHS = (2/3) × ((-6/7) × (3/5)) = (2/3) × (-18/35) = -12/35 RHS = ((2/3) × (-6/7)) × (3/5) = (-4/7) × (3/5) = -12/35

Therefore, option (a) is true and satisfies the associativity property of rational numbers.

(b) [(2/3) × ((-6/7) × (3/5))] = [(2/3) × ((3/5) × (-6/7))] LHS = (2/3) × ((-6/7) × (3/5)) = (2/3) × (-18/35) = -12/35 RHS = (2/3) × ((3/5) × (-6/7)) = (2/3) × (-18/35) = -12/35

Therefore, option (b) is also true and satisfies the associativity property of rational numbers.

(c) [(2/3) × ((-6/7) × (3/5))] = [((3/5) × (2/3)) × (-6/7)] LHS = (2/3) × ((-6/7) × (3/5)) = (2/3) × (-18/35) = -12/35 RHS = ((3/5) × (2/3)) × (-6/7) = (2/5) × (-6/7) = -12/35

Therefore, option (c) is also true and satisfies the associativity property of rational numbers.

(d) [((2/3) × (-6/7)) × (3/5)] = [((-6/7) × (2/3)) × (3/5)] LHS = ((2/3) × (-6/7)) × (3/5) = (-4/7) × (3/5) = -12/35 RHS = ((-6/7) × (2/3)) × (3/5) = (-4/7) × (3/5) = -12/35

Therefore, option (d) is also true and satisfies the associativity property of rational numbers.

Hence, all the given options demonstrate the associativity property of rational numbers under multiplication, but option (b) is the answer to the question.

Explanation:

(a) the identity for addition of rational numbers.

The identity element of an operation is the value which, when used with that operation on any other value, results in that value. For example, in addition, the identity element is 0 since adding 0 to any number gives the same number.

In the case of rational numbers, the identity element for addition is 0 because adding 0 to any rational number gives the same rational number.

For example:

0 + (3/4) = (3/4) + 0 = 3/4

Therefore, option (a) is the correct answer.

Explanation:

(a) the identity for addition of rational numbers.

As we know, the identity element of an operation is the value which, when used with that operation on any other value, results in that value.

In the case of rational numbers, the identity element for addition is 0, while the identity element for multiplication is 1. Therefore, 1 is the identity for multiplication of rational numbers.

For example:

1 × (3/4) = (3/4) × 1 = 3/4

Therefore, option (c) is incorrect and the correct answer is option (a), which states that 1 is the identity for addition of rational numbers.

Explanation:

(b) 7/19

The additive inverse of a rational number is the value that when added to the given rational number results in 0.

Let x be the additive inverse of -7/19. Then, we have:

x + (-7/19) = 0

x = 7/19

Therefore, the additive inverse of -7/19 is 7/19.

Explanation:

(a) a positive rational number.

The multiplicative inverse of a rational number is the value that when multiplied by the given rational number results in 1.

Let x be a negative rational number. Then, its multiplicative inverse y will satisfy:

x × y = 1

Multiplying both sides by -1, we get:

(-x) × (-y) = 1

Therefore, the multiplicative inverse of a negative rational number (-x) is a positive rational number (-y), which satisfies:

(-x) × (-y) = 1

Explanation:

(a) identity for addition of rational numbers.

The identity element of an operation is the value which, when used with that operation on any other value, results in that value. In the case of rational numbers, the identity element for addition is 0.

Therefore, if x + 0 = 0 + x = x, then 0 is the identity for addition of rational numbers.

Option (b) is incorrect because the additive inverse of x is -x, not 0.

Option (c) is incorrect because the multiplicative inverse of x is 1/x, not 0.

Option (d) is incorrect because the reciprocal of x is 1/x, not 0.

Explanation:

(c) 21/8

To find the multiplicative inverse of a rational number, we need to find a value that when multiplied by the given rational number results in 1.

Let x be the given rational number, i.e., x = 8/21.

To find the multiplicative inverse of x, we need to find a value y such that:

x × y = 1

Substituting the value of x, we get:

(8/21) × y = 1

Multiplying both sides by the reciprocal of 8/21, we get:

y = (21/8) × 1/ (8/21) = 21/8

Therefore, to get the product 1 when multiplying (8/21), we need to multiply it by 21/8.

Explanation:

(b) x

The expression -(-x) represents the negation of the negation of x. In other words, it means taking the opposite of the opposite of x.

The opposite of x is -x, and the opposite of -x is x. Therefore, we have:

-(-x) = x

Hence, the answer is (b) x.

Explanation:

(a) 8/7

represents the rational number -7/8. The multiplicative inverse of a rational number p/q is q/p, except when p=0.

Therefore, the multiplicative inverse of -7/8 is -8/7. But we need to express the answer as a positive fraction. Since -8/7 is negative, we can multiply both the numerator and denominator by -1 to get:

-8/7 = (-1)(8/7) = -(8/7)

So, the positive multiplicative inverse of -7/8 is 7/(-8) = -7/8. However, this is not one of the options given.

Therefore, the correct answer is (a) 8/7, which is the reciprocal of -7/8.

Explanation:

Since the additive identity of rational numbers is 0, we have:

x + 0 = x

Therefore, the answer is (a) x

Explanation:

The reciprocal of a number is its multiplicative inverse, which means that when you multiply the number by its reciprocal, the result is 1.

The reciprocal of 1 is 1/1 or simply 1, since 1 multiplied by 1 equals 1.

Therefore, the answer is (a) 1.

Explanation:

The reciprocal of -1 is -1

Explanation:

The reciprocal of 0 is not defined. Division by 0 is undefined in mathematics. Therefore, option (d) Not defined is the correct answer.

Explanation:

The reciprocal of any rational number p/q, where p and q are integers and q , is q/p. Therefore, the answer is option (d).

Explanation:

a) x

If y be the reciprocal of rational number x, i.e. y = 1/x

x = 1/y

Then,

Reciprocal of y = x

Explanation:

The reciprocal of (-3/8) x (-7/13) is (1/(-3/8)) x (1/(-7/13))

= (-8/3) x (-13/7)

= (8/3) x (13/7)

= 104/21

Therefore, the answer is (a) 104/21.

Explanation:

(a) – (1/4) × {(2/3) + (-4/7)} = [-(1/4) × (2/3)] + [(-1/4) × (-4/7)] is an example of distributive property of multiplication over addition for rational numbers.

Explain:

Using distributive property of multiplication over addition, we have:

-(1/4) × {(2/3) + (-4/7)} = -(1/4) × (2/3) + -(1/4) × (-4/7)

= [-(1/4) × (2/3)] + [(-1/4) × (-4/7)]

Thus, option (a) is the correct answer.

Explanation:

(d) infinitely many rational numbers can be found between two given rational numbers. This is because the set of rational numbers is dense in the real number system, meaning that there is always a rational number between any two real numbers, including two rational numbers. We can keep finding rational numbers between two given rational numbers by taking the average of the two numbers and simplifying the result. This process can be repeated infinitely many times to find an infinite number of rational numbers between the given two.

Explanation:

(a) Between x and y.

Reason: The expression (x+y)/2 represents the arithmetic mean of x and y. 

 Since x and y are both rational numbers, their arithmetic mean will also be a rational number. Moreover, this rational number will lie between x and y because it is the midpoint of the interval with endpoints x and y. 

Therefore, option (a) is the correct answer.

Explanation:

Option (b) is always true.

If we take the arithmetic mean of two rational numbers x and y, i.e., (x+y)/2, then it is always a rational number between x and y. This is because the numerator, which is the sum of x and y, is always even, and the denominator, which is 2, is also even. Therefore, the fraction (x+y)/2 is always a rational number.

Option (a) is not always true, as (x-y)/2 may or may not be a rational number between x and y. For example, if x=1 and y=2, then (x-y)/2=-0.5 is not a rational number between x and y.

Option (c) is not always true, as (xy)/2 may or may not be a rational number between x and y. For example, if x=1 and y=2, then (xy)/2=1 is not a rational number between x and y.

Option (d) is not always true, as (x/y)/2 may or may not be a rational number between x and y. For example, if x=1 and y=2, then (x/y)/2=0.25 is not a rational number between x and y.

Explanation:

We need to find an equivalent fraction of 5/7 whose numerator is 45.

Let the denominator of the equivalent fraction be x. Then,

5/7 = 45/x

Cross-multiplying, we get:

5x = 7 × 45

5x = 315

x = 315/5

x = 63

Therefore, the equivalent fraction of 5/7, whose numerator is 45 is 45/63.

Explanation:

To find the equivalent rational number of 7/9 whose denominator is 45, we need to multiply both the numerator and denominator of 7/9 by the same factor that will make the denominator equal to 45.

We can find this factor by dividing 45 by the denominator of 7/9, which is 9.

45 ÷ 9 = 5

So, we need to multiply both the numerator and denominator of 7/9 by 5 to get the equivalent rational number with denominator 45:

(7/9) × (5/5) = 35/45

Therefore, the equivalent rational number of 7/9, whose denominator is 45, is 35/45.

Explanation:

We need to compare (15/20) and (35/40).

To compare two rational numbers with different denominators, we need to make their denominators equal. Here, we can see that the LCM of 20 and 40 is 40. So, we need to convert both the fractions into equivalent fractions with denominator 40.

(15/20) = (15/20) × (2/2) = (30/40)

(35/40) = (35/40) × (1/1) = (35/40)

Now, we can see that (35/40) > (30/40), so the greater number is (35/40).

Therefore, the answer is (35/40).

Explanation:

The reciprocal of a positive rational number is also a positive rational number.

Explanation:

The reciprocal of a negative rational number is negative.

Explanation:

Zero does not have a reciprocal. If we multiply any non-zero number by its reciprocal, we get 1, but if we try to find the reciprocal of 0, we run into a problem because there is no number we can multiply by 0 to get 1. Therefore, the reciprocal of 0 is undefined.

Explanation:

The numbers 1 and -1 are their own reciprocal.

Reciprocal of 1 = 1/1 = 1

Reciprocal of -1 = 1/-1 = -1

Explanation:

If y be the reciprocal of x, then we have:

y = 1/x

Squaring both sides, we get:

Y2 = 1/x2

Taking the reciprocal of both sides, we get:

1/y2 = x2

Hence, the reciprocal of y2 in terms of x is x2.

Explanation:

Reciprocal of a number a is 1/a.

Reciprocal of (2/5) × (-4/9) is the reciprocal of the product (2/5) × (-4/9).

We know that the reciprocal of a product is the product of the reciprocals of the factors.

Therefore, the reciprocal of (2/5) × (-4/9) is the product of the reciprocals of 2/5 and -4/9:

reciprocal of (2/5) × (-4/9) = (1/(2/5)) × (1/(-4/9))

= (5/2) × (-9/4)

= -45/8

Hence, the reciprocal of (2/5) × (-4/9) is -45/8.

Explanation:

We can simplify the left-hand side of the equation as follows: (213 × 657)-1 = (139941)-1

To find the reciprocal of 139941, we can simply take the reciprocal of the number: (139941)-1 = 1/139941

Now, we can substitute the value of the reciprocal of (213 × 657) in terms of 213: (213 × 657)-1 = 213-1 × (1/139941)

Simplifying the right-hand side of the equation, we get: (213 × 657)-1 = (1/213) × (1/139941)

Therefore, (213 × 657)-1 = (1/213) × (1/139941)

Explanation:

The negative of 1 is -1.

Explanation:

For rational numbers (a/b), (c/d) and (e/f) we have (a/b) × ((c/d) + (e/f)) = ((a/b) × (c/d)) + ((a/b) × (e/f))

Explanation:

More than

Explanation:

Infinite

Explanation:

On the number line, zero is located in the middle and all numbers to the right of zero are positive, while all numbers to the left of zero are negative.

The rational number 1/3 is positive and is located to the right of zero, while the rational number -1/3 is negative and is located to the left of zero. Therefore, both 1/3 and -1/3 are on the sides of zero on the number line.

Explanation:

Positive

Explanation:

Order

Explanation:

- 7/5

Explanation:

The multiplicative inverse of 4/3 is ¾.

Explanation:

 1011/100.

Explanation:

(1/5) × [(2/7) + (3/8)] = [(1/5) × (2/7)] + [(1/5) × (3/8)]

∵ From the rule of distributive law of multiplication [a × (b + c) = (a × b) + (a × c)]

Explanation:

The two rational numbers lying between –2 and –5 with denominator as 1 are -3 and -4.

Explanation:

This statement is not true.

 A rational number is any number that can be expressed as the ratio of two integers, where the denominator is not zero. Therefore, the denominator y of x/y can be any non-zero integer, not necessarily a whole number. For example, 1/2 is a rational number, but its denominator 2 is not a whole number.

Explanation:

False

q cannot be equal to zero as division by zero is undefined. However, p can be equal to zero, in which case the rational number would be 0/q = 0.

Explanation:

This statement is true.

 In the fraction r/s, s represents the denominator, and a denominator of zero is undefined. Therefore, s cannot be equal to zero for r/s to be a valid rational number.

Explanation:

True.

We can see that 2/3 is less than 5/6, which in turn is less than 1. Therefore, 5/6 lies between 2/3 and 1.

Explanation:

False

We can simplify 5/10 by dividing both numerator and denominator by their common factor 5, giving us 1/2. So, 5/10 is equal to 1/2.

So, 5/6 does not lies between ½ and 1.

Explanation:

The statement is true.

We can write -7/2 as -3.5, which is between -3 and -4 on the number line.

Explanation:

True

The multiplicative inverse of a/b is b/a only if a and b are both non-zero. If a is equal to 0, then a/b is undefined and does not have a multiplicative inverse. Therefore, the statement should be revised to:

"If a and b are non-zero, the multiplicative inverse of a/b is b/a."

Explanation:

This is false. The multiplicative inverse of a rational number is the reciprocal of that number. The reciprocal of -3/5 is -5/3, not 5/3.

Explanation:

False

The additive inverse of 1/2 is -1/2, not -2.

Explanation:

This statement is true.

The additive inverse of a rational number a/b is -a/b. So, if x/y is the additive inverse of c/d, we have:

x/y + c/d = 0

Multiplying both sides by yd, we get:

xd + cy = 0

This means that x/y and c/d are two rational numbers whose sum is zero, which is the definition of additive inverse.

Explanation:

The statement is false. For any rational number x, x+1 is not equal to x. We can see this easily by taking any specific rational number, for example, x = 2/3. Then x+1 = 2/3 + 1 = 5/3, which is not equal to x = 2/3.

Explanation:

False.

If x/y is the additive inverse of c/d, then we have x/y + c/d = 0. Therefore,

x/y - c/d = x/y + (-c/d) = x/y + (-1) × (c/d) = x/y + (-1) × c/d

= x/y - c/d

So, it is not necessary that x/y - c/d = 0, it depends on the values of x, y, c, and d.

Explanation:

This statement is false. The reciprocal of a non-zero rational number q/p is p/q, not q/p.

Explanation:

False.

If x and y are rational numbers, then y is known as the negative of x

Explanation:

Yes, this statement is true.

The negative of a rational number is the same as multiplying the number by -1. So, for any rational number x, the negative of x is -x.

Then, taking the negative of -x gives:

-(-x) = -1*(-x) = x

Therefore, the negative of the negative of any rational number x is x itself.

Explanation:

True

the negative of 0 exists and is equal to 0.

Explanation:

This statement is false. The negative of 1 is -1, not 1 itself.

Explanation:

False.

For example, if we take x = 2/3 and y = 1/4, then:

x - y = 2/3 - 1/4 = 8/12 - 3/12 = 5/12

y - x = 1/4 - 2/3 = 3/12 - 8/12 = -5/12

Thus, x - y ≠ y - x in general for all rational numbers x and y.

Explanation:

Yes, this is true. The commutative property of multiplication states that the order of factors can be changed without affecting the result. Since multiplication is well-defined for rational numbers, the commutative property holds for rational numbers as well. Therefore, for all rational numbers x and y, we have (x) × (y) = (y) × (x).

Explanation:

The statement is false.

For any rational number x, x × 0 = 0, not x.

Explanation:

The statement is false

. The correct equation is the distributive property of multiplication over addition which states that for any rational numbers x, y, and z:

x × (y + z) = (x × y) + (x × z)

For example, if we take x = 2, y = 3, and z = 4, then:

2 × (3 + 4) = 2 × 7 = 14

and

(2 × 3) + (2 × 4) = 6 + 8 = 14

So, the equation holds true.

Explanation:

False.

Because, for every rational numbers a, b and c, [a × (b + c) = (a × b) + (a × c)]

Explanation:

This statement is not false

 In addition to 1, there is another number that is its own reciprocal, which is -1. This is because (-1) x (-1) = 1.

Explanation:

This statement is false. The reciprocal of -1 is -1 itself, which is a rational number.

Explanation:

False.

The correct form is for any rational number x, (x) × (-1) = – x.

Explanation:

This statement is not true for all rational numbers.

Consider the rational numbers -2 and -1. We have -2 < -1, but their difference, (-2) - (-1) = -2 + 1 = -1, is a negative rational number.

In fact, if x and y are any two rational numbers such that x < y, then y - x is a positive rational number.

Explanation:

Yes, the statement is true.

We know that a rational number is a number that can be expressed in the form p/q, where p and q are integers and q is not equal to zero.

Let x = -a/b and y = -c/d, where a, b, c, and d are positive integers.

Then x + y = (-a/b) + (-c/d) = (-ad/bd) + (-bc/bd) = -(ad+bc)/bd

Since both a, b, c, and d are positive integers, it follows that ad and bc are also positive integers. Therefore, ad + bc is a positive integer.

So, x + y can be expressed in the form -p/q, where p is a positive integer and q is a positive integer.

Thus, x + y is a negative rational number.

Explanation:

False.

Between any two rational numbers there are infinite rational numbers.

Explanation:

False.

Rational numbers are closed under addition, subtraction and multiplication.

Explanation:

False, the subtraction of rational numbers is not commutative.

In general, the order of subtraction matters, and switching the order can lead to different results.

For example, consider the rational numbers 2/3 and 1/4.

If we subtract 1/4 from 2/3, we get:

2/3 - 1/4 = 8/12 - 3/12 = 5/12

But if we subtract 2/3 from 1/4, we get:

1/4 - 2/3 = 3/12 - 8/12 = -5/12

So, we can see that switching the order of subtraction leads to different results, which means that the subtraction of rational numbers is not commutative.

Explanation:

No, the statement is false.

To compare -3/4 and -2, we can convert both of them into fractions with a common denominator.

Multiplying -2 by 4/4, we get:

-2 = -8/4

So, we can compare -3/4 and -8/4:

-3/4 < -8/4

Therefore, -8/4, which is equivalent to -2, is actually smaller than -3/4.

Hence, the statement "-3/4 is smaller than -2" is false.

Explanation:

True.

Because, 0/1 is a rational number.

Explanation:

No, the statement is false.

There are infinite positive rational number on the right side of 0 on the number line.

Explanation:

True, the population of India in 2004-05 is a rational number.

A rational number is any number that can be expressed as a ratio of two integers. The population of India in 2004-05 is a numerical value that can be expressed as a ratio of two integers, namely the total population and 1.

According to the World Bank, the population of India in 2004-05 was approximately 1.096 billion people. We can express this value as the ratio 1,096,000,000/1, which is a ratio of two integers and therefore a rational number.

Therefore, the population of India in 2004-05 is a rational number.

Explanation:

True

Explanation:

No, the statement is false.

The reciprocal of x-1 is 1/(x-1). This is because the reciprocal of a non-zero number a is 1/a.

On the other hand, 1/x is the reciprocal of x, not the reciprocal of x-1.

Therefore, the statement "the reciprocal of x-1 is 1/x" is false.

Explanation:

False.

To determine whether the rational number 57/23 lies to the left or right of zero on the number line, we can compare it to 0.

Since 57/23 is a positive number (the numerator and denominator are both positive), it lies to the right of zero on the number line.

Therefore, the statement "The rational number 57/23 lies to the left of zero on the number line" is false.

Explanation:

False.

To determine whether the rational number 7/-4 (which can be simplified to -7/4) lies to the left or right of zero on the number line, we can compare it to 0.

Since -7/4 is a negative number (the numerator is negative and the denominator is positive), it lies to the left of zero on the number line.

Therefore, the statement "The rational number 7/-4 lies to the right side zero on the number line" is false. Instead, it lies to the left side of zero.

Explanation:

False.

The rational number -8/-3 can be simplified to 8/3. To determine whether it lies to the left or right of zero on the number line, we can compare it to 0.

8/3 is a positive number, since both the numerator and denominator are positive. Therefore, it lies to the right of zero on the number line.

Therefore, the statement "The rational number -8/-3 lies neither to the right nor to the left of zero on the number line" is false. The rational number 8/3 lies to the right of zero on the number line.

Explanation:

True.

To determine whether the rational numbers 1/2 and -1 are on the opposite sides of zero on the number line, we can compare them to 0.

1/2 is a positive number, since the numerator is positive and the denominator is positive. Therefore, it lies to the right of zero on the number line.

-1 is a negative number, since the numerator is negative and the denominator is 1. Therefore, it lies to the left of zero on the number line.

Since 1/2 and -1 lie on opposite sides of zero (one to the right and the other to the left), the statement "The rational numbers 1/2 and -1 are on the opposite sides of zero on the number line" is true.

Explanation:

True.

A rational number is a number that can be expressed as the ratio of two integers, where the denominator is not zero. Every fraction can be expressed in this form, with the numerator and denominator being integers (for example, 2/3 is a rational number because 2 and 3 are both integers). Therefore, every fraction is a rational number.

Explanation:

True.

An integer is a whole number that can be expressed without a fractional or decimal component. A rational number, on the other hand, is a number that can be expressed as the ratio of two integers, where the denominator is not zero.

Every integer can be expressed in the form of a fraction with denominator 1. For example, the integer 5 can be expressed as the fraction 5/1. Therefore, every integer can be expressed as the ratio of two integers, and hence every integer is a rational number.

Explanation:

True

Explanation:

The statement "The negative of a negative rational number is a positive rational number" is true.

To see why, suppose we have a negative rational number, say -a/b, where a and b are positive integers.

The negative of -a/b is simply a/b, which is a positive rational number. We can see this by noting that the opposite of a negative number is its positive counterpart.

Therefore, the negative of a negative rational number is indeed a positive rational number.

Explanation:

True.

Let x = 4, y = 2

Then,

= x – y

= 4 – 2

= 2

Explanation:

This statement is false.

While 0 is a rational number, it is not the smallest. There are infinitely many rational numbers that are smaller than 0, such as -1/2, -3/4, -1/100, and so on.

Therefore, the statement "0 is the smallest rational number" is false.

Explanation:

True.

Every whole number is an integer, as every whole number can be expressed as an integer without a fractional or decimal component. For example, 2, 5, 9, and 100 are all whole numbers and integers.

Explanation:

True.

For example, the whole number 3 can be expressed as the fraction 3/1. Since this fraction is expressed as a ratio of two integers, it is a rational number. Therefore, every whole number is also a rational number.

Explanation:

False.

0 is whole number and also a rational number.

Explanation:

True.

The number 0 is the midpoint of the number line and any number to the right of 0 is positive and any number to the left of 0 is negative.

So, 1/2 is to the right of 0 and hence positive, while -5/2 is to the left of 0 and hence negative. Therefore, they are on the opposite sides of 0 on the number line.

Explanation:

True.

Multiplication of rational numbers is commutative, which means that changing the order of the numbers being multiplied does not affect the result. Therefore, (-4/5) × (-6/5) and (-6/5) × (-4/5) are equivalent and give the same result.

Explanation:

To select the rational numbers from the list which are also integers, we need to identify those numbers where the denominator is equal to 1. These are the numbers that have no fractional part and are therefore integers.

Therefore, the rational numbers from the list which are also integers are:

• 3/1 = -3

• 1/1 = 1

• 0/1 = 0

• -1/1 = -1

• -2/1 = -2

Hence, these are the rational numbers from the list which are also integers: -3, -2, -1, 0, 1.

Explanation:

To write a rational number with denominator 4 in its lowest form, we need to find the greatest common divisor (GCD) of the numerator and denominator and divide both by it. If the GCD is not 1 or 4, then the number cannot be written with denominator 4 in its lowest form.

Therefore, the rational numbers in the list that can be written with denominator 4 in their lowest form are:

• 16/4 = 4

• -12/4 = -3

• -20/4 = -5

• 20/4 = 5

Hence, these are the numbers that can be written as rational numbers with denominator 4 in their lowest form: 4, -3, -5, 5.

Explanation:

(a) To find the sum of the given rational numbers, we can rearrange them by grouping the like terms:

Explanation:

(i) To verify – (-x) = x for x = 3/5, we substitute the value of x in the given equation:

Explanation:

Example to show that rational numbers are closed under addition:

Let a = 2/3 and b = 5/7 be two rational numbers. Then, a + b = (2/3) + (5/7) = (14/21) + (15/21) = 29/21. Since 29/21 is also a rational number, we can say that rational numbers are closed under addition.

Example to show that rational numbers are closed under subtraction:

Let a = 7/8 and b = 1/5 be two rational numbers. Then, a - b = (7/8) - (1/5) = (35/40) - (8/40) = 27/40. Since 27/40 is also a rational number, we can say that rational numbers are closed under subtraction.

Example to show that rational numbers are closed under multiplication:

Let a = 3/4 and b = 4/5 be two rational numbers. Then, a * b = (3/4) * (4/5) = 12/20 = 3/5. Since 3/5 is also a rational number, we can say that rational numbers are closed under multiplication.

Rational numbers are not closed under division. Here are two examples to support this:

Example 1: Let a = 3/4 and b = 0. Then, a/b is undefined, since division by 0 is not defined. Therefore, rational numbers are not closed under division.

Example 2: Let a = 2/3 and b = 6/7. Then, a/b = (2/3) / (6/7) = (2/3) * (7/6) = 7/9. Even though the quotient is a rational number, the denominator is not equal to 1, which means that the rational numbers are not closed under division.

Explanation:

(a) To verify the property x + y = y + x for rational numbers, we need to show that x + y = y + x for any two rational numbers x and y.

Let x = 1/2 and y = 1/2 be two rational numbers. Then,

x + y = (1/2) + (1/2) = 1

y + x = (1/2) + (1/2) = 1

Since x + y = y + x for these rational numbers, we can say that the property x + y = y + x holds for rational numbers.

(b) To verify the property x + y = y + x for rational numbers, we need to show that x + y = y + x for any two rational numbers x and y.

Explanation:

(a) We have:

Explanation:

The arrangement of the given rational number is as per the rule of Associative property for Multiplication.

Explanation:

(a) We are given that x × y = y × z

x = 7 and y = ½, then we have:

x × y = y × z 

7 × ½ = ½ × z

 7/2 = z

Now, let y = ½ and z = 7/2, then we have:

x × y = y × z 

x × ½ = ½ × 7/2

 x/2 = 7/4

 x = 7/2

Therefore, we have:

x = 7/2, y = ½, and z = 7/2

And, we can see that x × y = y × z.

Hence, the property x × y = y × z is verified for x = 7 and y = ½.

(b) To verify the property x × y = y × z using x = 2/3 and y = 9/4, we need to show that:

Explanation:

(a) To verify the property x × (y × z) = (x × y) × z for rational numbers, we can substitute the given values of x, y, and z and check if both sides of the equation evaluate to the same value.

Explanation:

(a) We have to verify that x × (y + z) = x × y + x × z for x, y, z as rational numbers.

Let's substitute the given values and simplify:

x × (y + z) = (-1/2) × [(3/4) + (1/4)] = (-1/2) × (4/4) = -1/2

x × y + x × z = (-1/2) × (3/4) + (-1/2) × (1/4) = (-3/8) + (-1/8) = -4/8 = -1/2

We can see that x × (y + z) = x × y + x × z for the given values. Therefore, the property is verified.

Property used: Distributive property of multiplication over addition.

Hence, verified.

(b) We have to verify the property x × (y + z) = x × y + x × z for x = -½, y = 2/3, z = ¾

Explanation:

(a) Using the distributivity of multiplication of rational numbers over addition, we have:

Explanation:

(a) We need to first simplify the product of rational numbers before adding to another rational number.

Explanation:

The rational number that does not belong with the other three is (-7/3).

All other three rational numbers (-5/11), (-1/2), (-4/9) are proper fractions i.e., their absolute values are less than 1. However, (-7/3) is an improper fraction, i.e., its absolute value is greater than 1. Therefore, (-7/3) is the rational number that does not belong with the other three.

Explanation:

Let the cost of one metre of the wire be x.

Cost of 19/4 metres of wire = ₹ 171/2

So, we can write:

19/4 × x = 171/2

Multiplying both sides by 4, we get:

19x = 342

Dividing both sides by 19, we get:

x = 18

Therefore, the cost of one metre of the wire is ₹18.

Explanation:

Distance travelled by the train = 1445/2 km

Time taken by the train = 17/2 hours

Speed of the train = Distance travelled / Time taken

Speed of the train = (1445/2) / (17/2) = (1445/2) × (2/17) = 85 km/h

Therefore, the speed of the train is 85 km/h.

Explanation:

Given that 16 shirts of equal size can be made out of 24m of cloth.

Let x be the amount of cloth needed for making one shirt.

We can set up a proportion:

16 shirts can be made from 24m of cloth.

Therefore, 1 shirt can be made from (24/16)m = (3/2)m of cloth.

Hence, the amount of cloth needed for making one shirt is 3/2 meters.

Explanation:

Let the total amount of money in Hamid's bank account be x.

According to the given information, 7/11 of x = 77,000

Multiplying both sides by 11/7, we get:

x = (77,000 × 11) / 7 x = 121,000

Therefore, Hamid has ₹ 121,000 in his bank account.

Explanation:

We are given that a 352/3m long rope is cut into equal pieces measuring 22/3 m each.

To find the number of such small pieces, we need to divide the total length of the rope by the length of each small piece.

Number of pieces = (Total length of rope) ÷ (Length of each small piece)

Number of pieces = 352/3 ÷ 22/3 (substituting the given values)

Number of pieces = (352/3) × (3/22)

Number of pieces = 16

Therefore, there are 16 small pieces of length 22/3 m each in the rope.

Explanation:

Let the number of students below average be x. 

Then, according to the question, 1/6 of 48 

= Number of students above average = 8 1/4 of 48

 = Number of students average = 12 

Therefore, number of students below average = Total number of students - (Number of students above average + Number of students average) 

=> x = 48 - (8 + 12) = 28 

Hence, there are 28 students below average in the class.

Explanation:

Let us assume total number of students in the school be x.

From the question it is given that,

The number of students come by car = (2/5) × x

The number of students come by bus = (¼) × x

Remaining students walk to school = x – ((2x/5) + (¼x))

= x – ((8x – 5x)/20)

= x – (13x/20)

= (20x – 13x)/20

= 7x/20

Then, number of students walk to school on their own = (1/3) of (7x/20)

= 7x/60

Since, 224 students come to school on their own.

As per the data given in the question,

= (7x/60) = 224

x = (224 × 60)/7

x = 32 × 60

x = 1920

∴The total number of students in that school is 1920.

Explanation:

Let Huma get ₹ x

So, Seema gets ½ of what Huma gets.

Therefore, Seema gets ₹ x/2

And Hubna gets 1 23 of what Seema gets

So Hubna gets= 5/3 × x/2

                       = 5x/6

In total they receive ₹2016, so sum of each’s share will be 2016

=> x + x/2 + 5x/6 = 2016

=> 6x + 3x + 5x/ 6 = 2016

=> 14x = 2016 × 6

=> x = 12096/14

=> x = 864

So, Huma gets ₹864

Seema gets = ₹864/2

= ₹ 432

Hubna gets 5 × 864 / 6

= ₹ 720

Explanation:

The property that allows us to compare

(2/3) × [¾ × (5/7)] and [(2/3) × (5/7)] × ¾

is the Associative Property of Multiplication.

According to the Associative Property of Multiplication, we can change the grouping of the factors in a multiplication expression without changing the result.

So, we can rearrange the factors in the first expression as follows:

(2/3) × [¾ × (5/7)] = (2/3) × (3/4) × (5/7)

And in the second expression, we can rearrange the factors as follows:

[(2/3) × (5/7)] × ¾ = (2/3) × (5/7) × 3/4

Now we can simplify both expressions using the commutative and associative properties of multiplication:

(2/3) × (3/4) × (5/7) = (2 × 3 × 5) / (3 × 4 × 7) = 30/84

(2/3) × (5/7) × (3/4) = (2 × 5 × 3) / (3 × 7 × 4) = 30/84

So we can see that both expressions simplify to the same value, which is 30/84. Therefore, we can say that

(2/3) × [¾ × (5/7)] = [(2/3) × (5/7)] × ¾

Explanation:

(i) The property used in the following equation is the commutative property of multiplication.

Explanation:

(i) The given number  can be written as = -9/8

Explanation:

To arrange these fractions in descending order, we need to compare their values.

One way to do this is to convert them to decimals and then compare them.

Converting the fractions to decimals, we get:

• 1/4 = 0.25

• 13/16 = 0.8125

• 5/8 = 0.625

Now we can see that 13/16 is the largest decimal, followed by 5/8 and then 1/4.

Therefore, the descending order of the fractions is:

• 13/16

• 5/8

• 1/4

  
  

Explanation:

Let the other rational number be denoted by x. Then we have:

(7/9) × x = -14/27

Solving for x, we can write:

x = (-14/27) ÷ (7/9) x = (-14/27) × (9/7) x = -2/3

Therefore, the other rational number is -2/3.

Explanation:

Let the numbers to be multiplied be x

- 15/20 × x = - 5/7

x = 5/7 × 20 / 15

x = 20/21

If 20/21 is multipied with – 15/20 we get – 5/7

Explanation:

Let the number to be multiplied with -8/13 be denoted by x. Then we have:

(-8/13) × x = 24

Solving for x, we can write:

x = 24 ÷ (-8/13) x = 24 × (-13/8) x = -39

Therefore, we need to multiply -8/13 by -39 so that the product is 24.

Explanation:

Let total number of student studying in school be x

Students who come by car= 2/5 x

Students who come by bus= ¼ x

Remaining students= x – (2/5 x + ¼ x)

                                  = x – ( 13/20 x)

                                  = 20x – 13x /20

Out of which students who walk by their own= 224 (given)

1/3 (20x – 13x /20) = 224

20x – 13x /20 = 672

7x/20 = 13440

X = 13440/7

X= 1920

Hence, total number of students in school is 1920.

Explanation:

No, it is not possible to find a rational number whose multiplicative inverse is -1.

The multiplicative inverse of a rational number is the number that, when multiplied by the original number, gives a result of 1.

However, there is no rational number that can be multiplied by -1 to give 1. This is because any rational number multiplied by -1 will always give a negative result, whereas the multiplicative inverse of a number must be positive.

Therefore, there is no solution to this problem.

Explanation:

Here are five rational numbers between 0 and 1:

1. 1/4: This is one quarter or 0.25 in decimal form.

2. 3/8: This is three eighths or 0.375 in decimal form.

3. 1/2: This is one half or 0.5 in decimal form.

4. 5/8: This is five eighths or 0.625 in decimal form.

5. 3/4: This is three quarters or 0.75 in decimal form.

Note that a rational number is any number that can be expressed as a ratio of two integers. In the above examples, each rational number is written as a fraction with a numerator and a denominator.

Explanation:

Let's consider the first rational number as +1/5. The absolute value of this number is |+1/5| = +1/5.

Now, we need to find another rational number whose absolute value is also 1/5. Let's consider the second rational number as -1/5. The absolute value of this number is |-1/5| = +1/5.

Therefore, the two rational numbers whose absolute value is 1/5 are +1/5 and -1/5.

Explanation:

We can find the number of pieces by dividing the total length of the rope by the length of each piece.

Number of pieces = Total length of the rope / Length of each piece

Given that the length of one piece is 10/3 meters and the total length of the rope is 40 meters, we can substitute these values into the formula above:

Number of pieces = 40 meters / (10/3 meters)

To divide by a fraction, we can multiply by its reciprocal:

Number of pieces = 40 meters * (3/10)

Number of pieces = 12

Therefore, the number of pieces of length 10/3 meters that can be cut from a 40-meter rope is 12.

Explanation:

To find the length of each piece, we need to divide the total length of the rope by the number of pieces.

Total length of rope = 5½ meters

Number of pieces = 12

Length of each piece = Total length of rope / Number of pieces

We need to convert the mixed number 5½ to an improper fraction before we can divide.

5½ = 11/2 (To convert mixed number to an improper fraction, we multiply the whole number by the denominator of the fraction, and then add the numerator. So, 5 x 2 + 1 = 11 and the denominator remains the same.)

Now, we can substitute the values into the formula:

Length of each piece = 11/2 meters / 12

To divide by a fraction, we can multiply by its reciprocal:

Length of each piece = 11/2 meters * 1/12

Simplifying the fraction:

Length of each piece = 11/24 meters

Therefore, each piece of the 5½ meters long rope is 11/24 meters long when it is cut into 12 equal pieces.

The LCM of the denominators 7, 8, 2 and 5 is 280

∴ 8/7 = [(8×40)/ (7×40)] = (320/280)

(-9/8) = [(-9×35)/ (8×35)] = (-315/280)

(-3/2) = [(-3×140)/ (2×140)] = (-420/280)

(2/5) = [(2×56)/ (56×56)] = (112/280)

Now, 320 > 112 > 0 > -315 > -420

Hence, ⇒ 8/7 > 2/5 > 0 > -9/8 > -3/2

Descending order 8/7, 2/5, 0, -9/8, -3/2

Explanation:

0 ÷ (2/3) = 0 × (3/2)

= 0/2

= 0

Explanation:

We can convert Celsius (°C) to Fahrenheit (°F) using the formula:

(C/5) = (F - 32)/9

where C is the temperature in Celsius and F is the temperature in Fahrenheit.

Given that the temperature in Himachal Pradesh is -16°C, we can substitute this value into the formula:

(-16/5) = (F - 32)/9

Multiplying both sides by 9:

-144/5 = F - 32

Adding 32 to both sides:

F = (-144/5) + 32

Simplifying the fraction:

F = (-144 + 160)/5

F = 16/5

Therefore, the temperature in Fahrenheit is 16/5 or 3.2°F (rounded to one decimal place).

Explanation:

The additive inverse of a number a is the number that, when added to a, gives zero. For a given number a, its additive inverse is -a.

Therefore, the additive inverse of 7 is -7.

The multiplicative inverse of a non-zero number a is the number that, when multiplied by a, gives 1. For a given non-zero number a, its multiplicative inverse is 1/a.

Therefore, the multiplicative inverse of 7 is 1/7.

Now, we need to find the sum of the additive inverse and multiplicative inverse of 7:

(-7) + (1/7)

To add these two fractions, we need to find a common denominator, which is 7:

(-49/7) + (1/7) = (-49 + 1)/7

Simplifying the fraction:

= -48/7

Therefore, the sum of the additive inverse and multiplicative inverse of 7 is -48/7.

Explanation:

Top of Form

The additive inverse of 1/3 is -1/3.

The multiplicative inverse of a non-zero number a is the number that, when multiplied by a, gives 1. For a given non-zero number a, its multiplicative inverse is 1/a.

Therefore, the multiplicative inverse of 1/3 is 3.

Now, we need to find the product of the additive inverse and multiplicative inverse of 1/3:

(-1/3) x (3)

Multiplying these two fractions:

= -1

Therefore, the product of the additive inverse and multiplicative inverse of 1/3 is -1.

Explanation:

(a) We have to find out the difference of wingspan of an Albatross and wingspan of a Sea gull.

Explanation:

Given,

Diameter of circle=1 14 cm or 5/4cm = breadth of rectangle

Length of rectangle=8 34 cm or 35/4 cm

Length of aluminium strip = 8¾ cm = 35/4 cm

∴The number of full circles cut from the aluminum strip = (35/4) ÷ (5/4)

= (35/4) × (4/5)

= (7/1) × (1/1)

= 7 circles

Radius of circle = (5/ (4 × 2)) = 5/8 cm

Area to be cut by one circle = πr2

= (22/7) × (5/8)2

= (22/7) × (25/64) cm2

Now, area to be cut by 7 full circles = 7 × (22/7) × (25/64)

= (22 × 25)/64

= 550/64 cm2

Area of the aluminum strip = length × breadth

= (35/4) × (5/4) cm2

= (175/16) cm2

∴The wastage of aluminum strip = (175/16) – (550/64)

= (700 – 550)/64

= 150/64

= 75/32 cm2

Explanation:

The first recipe requires 1/2 cup of sugar.

The second recipe requires 2 tablespoons of sugar, and 1 tablespoon is equivalent to 1/16 cup. So, the second recipe requires:

2 × 1/16 = 1/8 cup of sugar

To find out how much more sugar the first recipe requires, we need to subtract the amount of sugar in the second recipe from the amount of sugar in the first recipe.

1/2 cup − 1/8 cup = 4/8 cup − 1/8 cup = 3/8 cup

Therefore, the first recipe requires 3/8 cup more sugar than the second recipe.

Explanation:

(a) To find how farther Soni hopped than Nancy, we need to subtract the distance covered by Nancy from the distance covered by Soni.

Distance covered by Soni = 1/20 km Distance covered by Nancy = 1/32 km

Difference = 1/20 km - 1/32 km = (8/160 - 5/160) km = 3/160 km

Therefore, Soni hopped 3/160 km farther than Nancy.

(b) To find the total distance covered by Seema and Megha, we need to add the distance covered by both of them.

Distance covered by Seema = 1/25 km Distance covered by Megha = 1/40 km

Total distance covered = 1/25 km + 1/40 km = (8/200 + 5/200) km = 13/200 km

Therefore, the total distance covered by Seema and Megha is 13/200 km.

(c) To find who walked farther, Nancy or Megha, we need to compare their distances.

Distance covered by Nancy = 1/32 km Distance covered by Megha = 1/40 km

Since 1/32 km is greater than 1/40 km, Nancy walked farther than Megha.

Explanation:

From the table the distance between Himgaon and Rawalpur = 98¾ km = 395/4 km

The distance between Sonapur and Ramgarh =
= 122/3 km

Then,

Difference of the distance between Himgaon and Rawalpur to Sonapur and Ramgarh,

= ((395/4) – (122/3))

= (1185 – 488)/ 12

= 697/12

=
km

(b) From the table,

Explanation:

(a) The rational number expressing the amount of paper recycled is 5/11.

Explanation:

To express the given numbers as rational numbers, we need to find their equivalent fractions in the form p/q:

97.28 cm = 9728/100 = 1216/125 98.44 cm = 9844/100 = 2461/250 98.04 cm = 9804/100 = 2451/250 97.94 cm = 9794/100 = 487/50

Now we can arrange these rational numbers in ascending order:

1216/125 < 2451/250 < 2451/250 < 487/50

So the overall widths in ascending order are:

97.28 cm < 98.04 cm < 98.44 cm < 97.94 cm

Explanation:

Given, height of existing roller coaster = 2/3m

 Let the height of new roller coaster be h.

 It is given that the new roller coaster is 3/5 times the height of the existing coaster. 

Therefore, h = (3/5) x (2/3) = 6/15 = 2/5 m

 Hence, the height of the new roller coaster will be 2/5 m.

Explanation:

To write the given decimals in the form of rational numbers, we need to express them as fractions with a common denominator and simplify them. We can take the common denominator as 10000, then:

• 2.6924 = 26924/10000 = 13462/5000

• 0.6096 = 6096/10000 = 3054/5000

• -6.9088 = -69088/10000 = -17272/2500

• -8.636 = -8636/1000 = -2189/250

Therefore, the rational numbers in the required form are:

• May: 13462/5000

• June: 3054/5000

• July: -17272/2500

• August: -2189/250

  
  

Explanation:

Kerala; Punjab; HP; Maharashtra; Haryana; Tamil Nadu; West Bengal; Karnataka; Gujarat; Andhra Pradesh; Bihar; Rajasthan; UP; Orissa; Assam; MP

Explanation:

Given, the length of the skirt = 287/8 cm and the length of the hem = 25/8 cm

When the hem is let down, the new length of the skirt will be:

New length = length of skirt + length of hem

New length = 287/8 + 25/8

New length = (287 + 25)/8

New length = 312/8

New length = 39 cm

Therefore, the length of the skirt will be 39 cm if the hem is let down.

Explanation:

Let x be the allowance received by each. 

Manavi saves 1/2 of her allowance and spends 1/4 of it at the mall. 

So, the amount Manavi saves is: 1/2 x ₹1260 = ₹630 

The amount Manavi spends at the mall is: 1/4 x ₹1260 = ₹315 

Therefore, the amount left with Manavi is: ₹1260 - ₹630 - ₹315 = ₹315

Kuber saves 1/3 of his allowance and spends 3/5 of it at the mall.

 So, the amount Kuber saves is: 1/3 x ₹1260 = ₹420

 The amount Kuber spends at the mall is: 3/5 x ₹1260 = ₹756

 Therefore, the amount left with Kuber is: ₹1260 - ₹420 - ₹756 = ₹84

Hence, Manavi has ₹315 left and Kuber has ₹84 left.