**Natural numbers (N):** The counting numbers {1, 2, 3, …}, are called natural numbers. Some authors include 0, so that the natural numbers are {0, 1, 2, 3, …}.

**Whole numbers(W):** The numbers {0, 1, 2, 3, …}.

**Integers (Z):** Positive and negative counting numbers, as well as zero:{…, -2, -1, 0, 1, 2,…}.

**Rational numbers (Q):** Numbers that can be expressed as a ratio of an integer to a non-zero integer. All integers are rational, but the converse is not true.

**Real numbers (R):** Numbers that have decimal representations that have a finite or infinite sequence of digits to the right of the decimal point. All rational numbers are real, but the converse is not true.

For all natural numbers m, n and p the following hold.

m + n = n + m (**Commutative property of addition)**

m + (n + p) = (m + n) + p (**Associative property of addition)**

m x n = n x m **(Commutative property of multiplication)**

m x (n x p) = (m x n) x p **(Associative property of multiplication)**

m x (n + p) = (m x n) + (m x p) (**distributive property)**

** **

**Every non-empty subset of natural numbers of N (or W) has the smallest element.**

This is called the **well ordering property **of natural numbers.

We know, the set of all Positive and negative counting numbers, as well as zero:{…, -2, -1, 0, 1, 2,…}. is called whole numbers. Denoted by Z. If m and n are two whole numbers, with the extension of addition and multiplication, we have to following properties:

**Closure property:**for all integers a, b, both a + b and b + a also integers;**Commutative property:**for all integers a, b

a + b = b + a

a x b = b x a

**Associative property:**for all integers a, b, c

a + (b + c) = (a + b) + c

a x (b x c) = (a x b) x c

**Distributive property:**for all integers a, b, c

a x (b + c) = (a x b) + (a x c)

**Cancellation law:**if a, b, c are integers such that, c ≠ 0 and ac = bc

Then a = b.

**EXERCISE 1.3.1**

- Identify the property in the following statements:

(i) 2+(3+4)=(2+3)+4

**Solution:**

Associative property of addition

(ii) 2.8=8.2

**Solution:**

Commutative property of multiplication.

(iii) 8.(6+5)=(8.6)+(8.5)

**Solution:**

Distributive property

- Find the additive inverses of the following integers:

(i) 6

**Solution:**

(-6) is the additive inverse of 6.

(ii) 9

**Solution: **

(-9) is the additive inverse of 9.

(iii) 123

**Solution:** (-123) is the additive inverse of 123.

(iv) -76

**Solution: **

76 is the additive inverse of -76.

(v) -85

**Solution: **

-85 is the additive inverse of 85.

(vi) 1000

**Solution: **

(-1000) is the additive inverse of 1000.

- Find the integer m in the following:

(i) m + 6 = 8

**Solution:**

m = 8 – 6

Thus, m = 2

(ii) m + 25 = 15

**Solution:**

m = 15 – 25

Thus, m = -10

(iii) m – 40 = 26

**Solution: **

m = 26 – 40

Thus, m = + 14

(iv) m + 28 = -49

**Solution:**

m = -49 -28

Thus, m = -77

- Write in the following in increasing order:

21,-8, 26, 85, 38, -333, -210, 0, 2011

**Solution:**

-333, -210, -26, -8, 0, 21, 33, 85, 2011

- Write the following in decreasing order: 85, 210, -58, 2011, -1024, 528, 364, -10000, 12

**Solution:**

2011, 528, 364, 210, 85, 12, -58, -1024, -10000

1.3.2 Rational numbers

Rational numbers (Q): Numbers that can be expressed as a ratio of an integer to a non-zero integer. All integers are rational, but the converse is not true.

Clearly, the numbers of the form p/q, where p and q are natural numbers.

**Example 1: **Add and multiply 1/3 and 8/5

**Solution:**

^{ 1}/_{3 +}^{ 8}/_{5 = }^{(5+8×3)}/_{15 }= ^{(5+24)}/_{15 }= ^{29}/_{15}

_{Their product is,}

^{1}/_{3}^{ x 8}/_{5 = }^{(1×8)}/_{15 }= ^{(8)}/_{15 }= ^{8}/_{15}

_{ }

**EXERCISE 1.3.2**

- Write down ten rational numbers which are equivalent to 𝟓/𝟕 and the denominator not exceeding 80.

**Solution: **

^{5}/_{7 }x^{ 2}/_{2 = }^{10}/_{14 }

^{5}/_{7 }x^{ 3}/_{3 = }^{15}/_{21}

^{5}/_{7 }x^{ 4}/_{4 = }^{20}/_{28}

^{5}/_{7 }x^{ 5}/_{5 = }^{25}/_{35}

^{5}/_{7 }x^{ 6}/_{6 = }^{30}/_{42 }

^{5}/_{7 }x^{ 7}/_{7 = }^{35}/_{49}

^{5}/_{7 }x^{ 8}/_{8 = }^{40}/_{56}

^{5}/_{7 }x^{ 9}/_{9 = }^{45}/_{63 }

^{5}/_{7 }x^{ 10}/_{10 = }^{50}/_{70}

^{5}/_{7 }x^{ 11}/_{11 = }^{55}/_{77 }

- Write down 15 rational numbers which are equivalent to and the numerator not exceeding 180.

**Solution:**

^{11}/_{5 }x^{ 2}/_{2 = }^{22}/_{10 }

^{11}/_{5 }x^{ 3}/_{3 = }^{33}/_{15}

^{11}/_{5 }x^{ 4}/_{4 = }^{44}/_{20}

^{11}/_{5 }x^{ 5}/_{5 = }^{55}/_{25}

^{11}/_{5 }x^{ 6}/_{6 = }^{66}/_{30 }

^{11}/_{5 }x^{ 7}/_{7 = }^{77}/_{35}

^{11}/_{5 }x^{ 8}/_{8 = }^{88}/_{40}

^{11}/_{5 }x^{ 9}/_{9 = }^{99}/_{45 }

^{11}/_{5 }x^{ 10}/_{10 = }^{110}/_{50}

^{11}/_{5 }x^{ 11}/_{11 = }^{121}/_{55 }

^{11}/_{5 }x^{ 12}/_{12 = }^{132}/_{60}

^{11}/_{5 }x^{ 13}/_{13 = }^{143}/_{65 }

^{11}/_{5 }x^{ 14}/_{14 = }^{154}/_{70 }

^{11}/_{5 }x^{ 15}/_{15 = }^{165}/_{75 }

^{11}/_{5 }x^{ 16}/_{16 = }^{176}/_{80 }

- Write down the ten positive numbers such that the sum of numerator and denominator of each is 11. Write them in decreasing order.

**Solution:**

Number:

^{10}/_{1},^{9}/_{2}, ^{8}/_{3},^{ 7}/_{4},^{ 6}/_{5},^{5}/_{6}, ^{4}/_{7},^{ 3}/_{8},^{2}/_{9}, ^{1}/_{10}

Decreasing order:

^{10}/_{1},^{9}/_{2}, ^{8}/_{3},^{ 7}/_{4},^{ 6}/_{5},^{5}/_{6}, ^{4}/_{7},^{ 3}/_{8},^{2}/_{9}, ^{1}/_{10}

- Write down the ten positive numbers such that the numerator and denominator for each them is -2. Write them in increasing order.

**Solution:**

^{Increasing order:}

^{1}/_{3},^{2}/_{4}, ^{3}/_{5 },^{ 4}/_{6},^{ 5}/_{7},^{6}/_{8}, ^{7}/_{9},^{ 8}/_{10},^{9}/_{11}, ^{10}/_{12}

- Is
^{3}/_{(-2) }a rational number? If so, how do you write it in a form conforming to the definition of a rational number (that is, the denominator as a positive integer)?

**Solution:**

^{3}/_{(-2) } is a rational number. It should be written as ^{-3}/_{2 } to be rational number.

- Earlier you have studied decimals 0.9,0.8. Can you write these as rational numbers?

**Solution: **

Yes, we can write decimals like 0.9, 0.8 as rational numbers. i.e.,

0.9 = ^{9}/_{10 } and 0.8 = ^{8}/_{10}

**1.3.3 Properties of rational numbers**

** **

**Closure property:**

We have learnt earlier, that, for all integers a, b, both a + b and b + a also integers;

Example1: Let us find the sum of ^{5}/_{6 }and ^{11}/_{13 }

**Solution:**

^{5}/_{6} + ^{11}/_{13} = ^{[(5×13)+(11×6)]}/_{(6×13)}

= ^{(65+66)}/_{78}

= ^{131}/_{78}

Example 2: The product of ^{2}/_{11 }and ^{8}/_{7 }:

**Solution:**

^{2}/_{11} + ^{8}/_{7 }= ^{(2×8)}/_{(11×7)}

= ^{16}/_{77}

**The set of all rational numbers is closed under addition and multiplication.**

** **

**Associative property:**

We have learnt earlier, that, the associative property is, for all integers a, b, c

a + (b + c) = (a + b) + c

a x (b x c) = (a x b) x c

Example 3: Consider three rational numbers ^{1}/_{2 }, ^{4}/_{5 }, ^{-6}/_{7 }

**Solution:**

^{1}/_{2 }+( ^{4}/_{5 }+ ^{-6}/_{7}) = ^{1}/_{2 }+( ^{(1×5+4×2)}/_{(5×7)})

= ^{1}/_{2 }+ (^{-2}/_{35 })_{ }

= ^{(35×1+(-2)x2)}/_{(35×2) }

= ^{31}/_{70}

_{On the other hand,}

(^{1}/_{2 }+ ^{4}/_{5}) + ^{(-6)}/_{7} = ^{(1×5+4×2)}/_{(2×5) }+ ^{(-6)}/_{7}

= ^{(13)}/_{(10) }+ ^{(-6)}/_{7}

= ^{[13×7+(-6)x10]}/_{(10×7) }

= ^{31}/_{70}

Similarly, we can find for multiplication.

Thus, we can come to the conclusion that,

**Addition and multiplication are associative on the set of all rational numbers.**

**Commutative property**

We have learnt earlier, that, the commutative property is, for all integers a, b

a + b = b + a

a x b = b x a

Example 4: Let us take two rational numbers, say ^{8}/_{11 }and ^{(-16)}/_{9 }

**Solution:**

^{8}/_{11} + ^{(-16)}/_{9} = ^{(8×9+(-16)x11)}/_{99 } _{ } _{ }

= ^{72-176}/_{99 }

= ^{-104}/_{99}

On the other hand,

^{(-16)}/_{9} + ^{8}/_{11} = ^{((-16)x11+8×9)}/_{99 } _{ } _{ }

= ^{-176+72}/_{99 }

= ^{-104}/_{99}

Example 5: Similarly, we can verify that,

^{8}/_{11} x ^{(-16)}/_{9} = ^{(-16)}/_{9} x ^{8}/_{11}

**Addition and multiplication are commutative on the set of all rational numbers.**

** **

**Distributive property**

We have studied earlier, that, distributive property is, for all integers a, b, c

a x (b + c) = (a x b) + (a x c)

Consider the rational numbers, ^{3}/_{2}, ^{1}/_{2} and ^{1}/_{9}. Observe that,

^{3}/_{2 }x ( ^{1}/_{2 }+ ^{1}/_{9}) = ^{3}/_{2 }x ( ^{11}/_{18 }) = ^{22}/_{54} = ^{11}/_{27}

(^{3}/_{2 }x ^{1}/_{2 }) + ( ^{3}/_{2 }x ^{1}/_{9}) = (^{2}/_{6 }) + ( ^{2}/_{27 }) = ^{11}/_{27 } ** **

**In the set of all rational numbers, multiplication is distributive over addition.**

** **

**Additive identity**

Consider the rational number ^{0}/_{1}, observe that,

^{7}/_{8 }+ ^{0}/_{1} = ^{(7×1+0x8)}/_{8} = ^{7}

^{0}/_{1} + ^{7}/_{8} = ^{(0+7)}/_{8} = ^{7}/_{8}

_{Thus, the rational number }^{0}/_{1 acts as additive }identity. We denote this by 0.** **

**The set of all rational numbers has 0 as additive identity; that is r + 0 = 0 + r = r, for all rational numbers.**

** **

** **

**Multiplicative identity**

Again consider, the rational number, ^{1}/_{1}. We have, for example,

^{11}/_{12} x ^{1}/_{1} = ^{11}/_{12}

^{1}/_{1} x ^{11}/_{12} = ^{11}/_{12}

Thus, the rational number ^{1}/_{1 } is identity with respect to multiplication.

** The set of all rational numbers has 1 as multiplicative identity, that is r x 1 = 1 x r = r, for all rational numbers.**

** **

**Additive inverse**

Take ^{8}/_{13 and }^{-8}/_{13}. If we add these two, we get

^{8}/_{13} + ^{(-8)}/_{13} = ^{(8-8)}/_{13} = ^{0}/_{13} = 0.

This is true for all rational numbers.

**For each rational number r, there exists a rational number, denoted by – r, such that r + (-r) = 0 = (-r) + r.**

** **

**Multiplicative inverse**

We have studied earlier about multiplication inverse. Consider a rational number, ^{7}/_{5} , we see that,

^{7}/_{5} + ^{5}/_{7} = ^{35}/_{35} = 1

This is true for all rational numbers.

**For each rational number r ≠ 0, there exists a rational number, denoted by r ^{-1} (or **

**r x r**

^{-1 }= r^{-1}x r = 1

** **

**The only fundamental operations are addition and multiplication. The subtraction and division are defined in terms of addition and multiplication.**

** **

**EXERCISE 1.3.3**

- Name the property indicated in the following:

(i) 315+115 = 430

**Solution:**

Closure property of addition

(ii) ^{3}/_{4 }+ ^{9}/_{5} = ^{27}/_{20}

**Solution: **

Closure property of multiplication

(iii) 5 + 0 = 0 + 5 = 5

**Solution:**

0 is the additive identity

(iv) ^{8}/_{9} x 1 = ^{8}/_{9}

**Solution:**

1 is the multiplicative identity

(v) ^{8}/_{17} + ^{-8}/_{17 }= 0

**Solution:**

Additive inverse

(vi)^{ 22}/_{23 }+ ^{22}/_{23} = 1_{ }

**Solution:**

Multiplication inverse

- Check the commutative property of addition for the following pairs:

(i)^{ 102}/_{201 },^{ 3}/_{4}

**Solution:**

We know, commutative property, a + b = b + a

Therefore, ^{102}/_{201 }+^{ 3}/_{4} =^{ 3}/_{4} + ^{102}/_{201}

LHS,

^{102}/_{201 }+^{ 3}/_{4 }= ^{(102×4+3×201)}/_{(201×4) }= ^{(408+603)}/_{804 }= ^{1011}/_{804}

RHS,

^{3}/_{4} + ^{102}/_{201} = ^{(102×4+201×3)}/_{(201×4)} = ^{603+408}/_{804} = ^{1011}/_{804}

Therefore, RHS = LHS, Commutative property proved.

(ii) ^{-8}/_{13} , ^{23}/_{27}

**Solution:**

We know, commutative property, a + b = b + a

Therefore, ^{-8}/_{13} + ^{23}/_{27} = ^{-8}/_{13} + ^{23}/_{27} _{LHS,}

^{-8}/_{13} + ^{23}/_{27} = ^{-8}/_{13} + ^{[(-8)x27+23×13]}/_{(13×27)} = ^{-216+299}/_{351} = ^{83}/_{351}

RHS,

^{-8}/_{13} + ^{23}/_{27} = ^{[23×13+(-8)x27]}/_{(13×27)} = ^{299-216}/_{351} = ^{83}/_{351}

Therefore, RHS = LHS, Commutative property proved.

(iii) ^{-7}/_{9} , ^{-18}/_{19}

**Solution:**

We know, commutative property, a + b = b + a

Therefore, ^{-7}/_{9} + ^{-18}/_{19} = ^{(-18)}/_{19} + ^{(-7)}/_{9}

LHS,

^{-7}/_{9} + ^{-18}/_{19} = ^{[(-7)x19+(-18)x9]}/_{(9×19)} = ^{-133-162}/_{171} = ^{-295}/_{171}

RHS,

^{(-18)}/_{19} + ^{(-7)}/_{9} = ^{[(-18)x9+(-7)x19]}/_{(9×19)} = ^{-162-133}/_{171} = ^{-295}/_{171}

Therefore, RHS = LHS, Commutative property proved.

- Check the commutative property of multiplication for the following pairs:

(i)^{ 22}/_{45} , ^{3}/_{4}

**Solution:**

We know commutative property multiplication, a×b = b×a

Therefore, ^{22}/_{45} x ^{3}/_{4} = ^{3}/_{4 }x ^{22}/_{45}

LHS,

^{22}/_{45} x ^{3}/_{4 }= ^{(22×3)}/_{(45×4)} = ^{66}/_{180}

_{RHS,}

^{3}/_{4 }x ^{22}/_{45 }= ^{(22×3)}/_{(45×4)} = ^{66}/_{180}

Therefore, RHS = LHS, Commutative property proved.

(ii) ^{-7}/_{13} , ^{25}/_{27}

**Solution:**

We know commutative property multiplication, a×b = b×a

Therefore, ^{-7}/_{13} x ^{25}/_{27} = ^{25}/_{27} x ^{-7}/_{13}

LHS,

^{-7}/_{13} x ^{25}/_{27} = ^{[25x(-7)]}/_{27×13} = ^{-175}/_{351}

RHS,

^{25}/_{27} x ^{-7}/_{13} = ^{[25x(-7)]}/_{27×13} = ^{-175}/_{351}

Therefore, RHS = LHS, Commutative property proved.

(iii)^{ -8}/_{9}, ^{17}/_{19}

**Solution:**

We know commutative property multiplication, a×b = b×a

Therefore, ^{-8}/_{9 }x ^{17}/_{19} = ^{17}/_{19 }x ^{(-8)}/_{9}

LHS,

^{-8}/_{9 }x ^{17}/_{19} = ^{(-8)x17}/_{9×19} = ^{136}/_{171}

RHS,

^{17}/_{19 }x ^{(-8)}/_{9 }= ^{(-8)x17}/_{9×19} = ^{136}/_{171}

Therefore, RHS = LHS, Commutative property proved

- Check the distributive property for the following triples of rational numbers:

(i) ^{1}/_{8} ,^{ 1}/_{9} , ^{1}/_{10}

**Solution:**

We know distributive property, a (b + c) = ab + ac

Therefore, ^{1}/_{8} x (^{1}/_{9} + ^{1}/_{10}) = (^{1}/_{8} x^{ 1}/_{9}) + (^{1}/_{8} x ^{1}/_{10})

LHS,

^{1}/_{8} x (^{1}/_{9} + ^{1}/_{10}) = ^{1}/_{8} x (^{(10+9)}/_{(9×10)})

= ^{1}/_{8} x (^{19}/_{90})

= (^{19}/_{720})

RHS,

(^{1}/_{8} x^{ 1}/_{9}) + (^{1}/_{8} x ^{1}/_{10}) = (^{1}/_{9×8}) + (^{1}/_{10×8})

= (^{1}/_{72}) + (^{1}/_{40} )

=(^{1+18}/_{720}) = ^{19}/_{720}

Therefore, RHS = LHS, distributive property proved.

(ii) ^{-4}/_{9} ,^{6}/_{5} ,^{11}/_{10}

**Solution:**

We know distributive property, a (b + c) = ab + ac

Therefore, ^{-4}/_{9} x (^{6}/_{5} + ^{11}/_{10}) = (^{-4}/_{9} x ^{6}/_{5} ) + (^{-4}/_{9} x ^{11}/_{10})

LHS,

^{-4}/_{9} x (^{6}/_{5} + ^{11}/_{10}) = ^{-4}/_{9} x (^{11+12}/_{10×5}) = ^{-4}/_{9} x (^{23}/_{10}) = (^{-92}/_{90})

RHS,

(^{-4}/_{9} x ^{6}/_{5} ) + (^{-4}/_{9} x ^{11}/_{10}) = (^{-24}/_{45} ) + (^{-44}/_{90})= ^{(-48-44)}/_{90} = ^{-92}/_{90}

Therefore, RHS = LHS, distributive property proved.

(iii) ^{3}/_{8} ,0 , ^{13}/_{7}

**Solution: **

We know distributive property, a (b + c) = ab + ac

Therefore, ^{3}/_{8} x (0 + ^{13}/_{7}) = (^{3}/_{8} x 0) + (^{3}/_{8} x^{13}/_{7})

LHS,

^{3}/_{8} x (0 + ^{13}/_{7}) = ^{3}/_{8} x (^{13}/_{7}) = ^{13×3}/_{7×8} = ^{39}/_{56}

RHS,

(^{3}/_{8} x 0) + (^{3}/_{8} x^{13}/_{7}) = ^{3}/_{8} x (^{13}/_{7}) = ^{13×3}/_{7×8} = ^{39}/_{56}

Therefore, RHS = LHS, distributive property proved.

- Find the additive inverse of each of the following numbers:

^{8}/_{5} , ^{6}/_{10}, ^{-3}/_{8} , ^{-16}/_{3}, ^{-4}/_{1}

**Solution:**

Additive inverse of ^{8}/_{5} , ^{6}/_{10}, ^{-3}/_{8} , ^{-16}/_{3}, ^{-4}/_{1} are ^{-8}/_{5} , ^{-6}/_{10}, ^{3}/_{8} , ^{16}/_{3}, ^{4}/_{1 }respectively.

- Find the multiplicative inverse of each of the following numbers:

2 , ^{6}/_{11}, ^{-8}/_{15} , ^{19}/_{18}, ^{1}/_{1000}

**Solution: **

Multiplicative inverse of 2 , ^{6}/_{11}, ^{-8}/_{15} , ^{19}/_{18}, ^{1}/_{1000} are ^{1}/_{2}, ^{11}/_{6} , ^{-15}/_{8}, ^{18}/_{19} , 1000 respectively.

** **

**1.3.4 Representation of rational numbers on the Number.**

Earlier, we have seen how to represent integers on a line. We choose an infinite line and fix some point on the line. This is denoted by 0. Fix a unit of length and on both sides of 0; go on marking points at equal unit distance.

We can also use the same number line to represent rational numbers.

For example,

**Between any two distinct rational numbers, there is another rational number.**

** **

** **

**Exercise 1.3.4**

- Represent the following rational numbers on the number line:

^{-8}/_{5 }, ^{3}/_{8} , ^{2}/_{7}, ^{12}/_{5 }, ^{45}/_{13}

**Solution:**

^{-8}/_{5}

^{3}/_{8}

^{ }

^{2}/_{7}

^{12}/_{5}

^{45}/_{13}

- Write the following rational numbers in ascending order:

^{3}/_{4} , ^{7}/_{12} ,^{ 15}/_{11} , ^{22}/_{19} ,^{ 101}/_{100} , ^{-4}/_{5} ,^{ -102}/_{81} , ^{-13}/_{7}

**Solution: **

Ascending order

^{-13}/_{7} ,^{ -102}/_{81} , ^{-4}/_{5} , ^{7}/_{12} ,^{ 3}/_{4}, ,^{ 101}/_{100}, ^{22}/_{19 },^{ 15}/_{11}

Method:

We know,

^{3}/_{4} , ^{7}/_{12} ,^{ 15}/_{11} , ^{22}/_{19} ,^{ 101}/_{100} , ^{-4}/_{5} ,^{ -102}/_{81} , ^{-13}/_{7} is equal to 0.75, 0.5, 1.3, 1.1, 1.01, -0.3, -1.8, -1.2 respectively.

- Write 5 rational number between
^{2}/_{5}and^{3}/_{5}having the same denominators.

**Solution:**

We know,

^{2}/_{5} = 0.4 and ^{3}/_{5} = 0.6

Now, we have to find 5 rational numbers between 0.4 and 0.6. We get,

^{2}/_{5} = 0.4 < ^{2.1}/_{5} = 0.42 < ^{2.2}/_{5} = 0.44 < ^{2.4}/_{5} = 0.48 < ^{2.6}/_{5} = 0.52 <^{2.8}/_{5} = 0.56 < ^{3}/_{5} = 0.6

Therefore, the rational numbers between ^{2}/_{5} and ^{3}/_{5} are ^{2.1}/_{5} , ^{2.2}/_{5} , ^{2.4}/_{5} ,^{ 2.6}/_{5} and ^{2.8}/_{5}

- How many positive rational numbers less than 1 are there such that the sum of the numerator and denominator does not exceed 10?

**Solution:**

^{1}/_{2 },^{1}/_{3 },^{ 1}/_{4 },^{1}/_{5 },^{ 1}/_{6 },^{1}/_{7 },^{ 1}/_{8 },^{1}/_{9 },^{2}/_{3 },^{2}/_{5 },^{2}/_{7 },^{3}/_{4 },^{ 3}/_{7 },^{3}/_{5 },^{ 1}/_{2 },^{4}/_{5 },

Therefore, only 15 positive rational number are possible such that they are less than 1 and the sum of the numerator and denominator does not exceed 10.

- Suppose m/n and p/q are two positive rational numbers. Where does
^{(m+n)}/_{(n+q) }lie, with respect to m/n and p/q?

**Solution:**

It is given that, m/n and p/ q are 2 numbers.

If ^{m}/_{n }= ^{p}/_{q }i.e., if ^{m}/_{n }= ^{p}/_{q }=^{1}/_{2 }

Then,

^{(m+n)}/_{(n+q)} = ^{(1+1)}/_{(2+2)} = ^{2}/_{4} = ^{1}/_{2}

Therefore, if ^{m}/_{n }= ^{p}/_{q then }^{(m+n)}/_{(n+q) }=^{ m}/_{n }= ^{p}/_{q}

If ^{m}/_{n }and ^{p}/_{q }are 2 distinct numbers. i.e., ^{m}/_{n }= ^{1}/_{5 }and ^{p}/_{q }= ^{1}/_{3 }then,

^{(m+n)}/_{(n+q)} = ^{(1+1)}/_{(5+3)} = ^{(2)}/_{(8)} = ^{1}/_{4}

Therefore, if ^{m}/_{n }and ^{p}/_{q }are 2 distinct numbers then ^{(m+n)}/_{(n+q) }lies between ^{m}/_{n }and ^{p}/_{q }

i.e,

- How many rational numbers are there strictly between 0 and 1 such that the denominator of the rational number is 80?

**Solution:**

^{1}/_{80} , ^{2}/_{80} , ^{3}/_{80} , ^{4}/_{80} … ^{78}/_{80} , ^{79}/_{80}

Therefore, there are 79 positive rational numbers.

- How many rational numbers are there strictly between 0 and 1 with the property that the sum of the numerator and denominator is 70?

**Solution: **

^{1}/_{69} , ^{2}/_{68} , ^{3}/_{67} , ^{4}/_{66} … ^{68}/_{2} , ^{69}/_{1}

Therefore, there are 69 rational numbers.

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