Rational numbers are numbers that can be expressed in the form of p/q, where p and q are integers and q is not equal to zero. Essentially, they are fractions.
Key Concepts:
- Representation: Rational numbers can be positive, negative, or zero. They can be represented on a number line.
- Equivalent Rational Numbers: Different fractions can represent the same rational number. For example, 1/2, 2/4, and 3/6 are equivalent.
- Standard Form: A rational number is in standard form when its numerator and denominator have no common factors other than 1 and the denominator is positive.
- Comparison: Rational numbers can be compared using their decimal representations or by finding a common denominator.
- Operations: You can add, subtract, multiply, and divide rational numbers using specific rules.
- Properties: Rational numbers follow properties like closure, commutativity, associativity, and distributivity under addition and multiplication.
Important Points to Remember:
- Not all numbers are rational. Numbers like √2 and π are irrational.
- Division by zero is undefined.
- The concept of rational numbers builds upon previous knowledge of fractions, integers, and number lines.
By understanding these concepts, you can work confidently with rational numbers and solve various mathematical problems.
Exercise 8.1
1. List five rational numbers between:
(i) -1 and 0
(ii) -2 and -1
(iii) −4/5 and −2/3
(iv) 1/2 and 2/3
Ans :
(i) -1 and 0
Five rational numbers between -1 and 0 are:
- -1/2
- -1/3
- -2/5
- -3/7
- -4/9
(ii) -2 and -1
Five rational numbers between -2 and -1 are:
- -11/6
- -5/3
- -7/4
- -9/5
- -17/10
(iii) -4/5 and -2/3
To find rational numbers between these two, let’s convert them to fractions with a common denominator. The LCM of 5 and 3 is 15.
- -4/5 = -12/15
- -2/3 = -10/15 Five rational numbers between -12/15 and -10/15 are:
- -11/15
- -23/30
- -7/10
- -19/30
- -4/9
(iv) 1/2 and 2/3
To find rational numbers between these two, let’s convert them to fractions with a common denominator. The LCM of 2 and 3 is 6.
- 1/2 = 3/6
- 2/3 = 4/6 Five rational numbers between 3/6 and 4/6 are:
- 7/12
- 2/3
- 5/8
- 11/18
- 17/24
2. Write four more rational numbers in each of the following patterns:
Ans :
(i) -3/5, -6/10, -9/15, -12/20, …
We can observe that the numerator is increasing by 3 in each term, and the denominator is increasing by 5.
So, the next four rational numbers are:
- -15/25
- -18/30
- -21/35
- -24/40
(ii) -1/4, -2/8, -3/12, …
Here, the numerator is increasing by 1, and the denominator is increasing by 4 in each term.
So, the next four rational numbers are:
- -4/16
- -5/20
- -6/24
- -7/28
(iii) -1/6, 2/-12, 3/-18, 4/-24, …
The numerator is increasing by 1 in each term, and the denominator is decreasing by -6.
So, the next four rational numbers are:
- 5/-30
- 6/-36
- 7/-42
- 8/-48
(iv) -2/3, 2/-3, 4/-6, 6/-9, …
The numerator is increasing by 2 in each term, and the denominator is decreasing by -3.
So, the next four rational numbers are:
- 8/-12
- 10/-15
- 12/-18
- 14/-21
3. Give four rational numbers equivalent to:
Ans :
(i) -2/7
Four equivalent rational numbers are:
- -4/14 (multiply both numerator and denominator by 2)
- -6/21 (multiply both numerator and denominator by 3)
- -8/28 (multiply both numerator and denominator by 4)
- -10/35 (multiply both numerator and denominator by 5)
(ii) 5/-3
Four equivalent rational numbers are:
- -10/6 (multiply both numerator and denominator by 2)
- -15/9 (multiply both numerator and denominator by 3)
- -20/12 (multiply both numerator and denominator by 4)
- -25/15 (multiply both numerator and denominator by 5)
(iii) 4/9
Four equivalent rational numbers are:
- 8/18 (multiply both numerator and denominator by 2)
- 12/27 (multiply both numerator and denominator by 3)
- 16/36 (multiply both numerator and denominator by 4)
- 20/45 (multiply both numerator and denominator by 5)
4. Draw a number line and represent the following rational numbers on it:
Ans :
5. The points P, Q, R, S, T, U, A and B on the number line are such that, TR = RS = SU and AP = PQ = QB. Name the rational numbers represented by P, Q, R and S.
Ans :
Given:
- TR = RS = SU
- AP = PQ = QB
Observation: The number line is divided into equal segments between the integers.
Solution:
- Identifying the distance between integers: The distance between two consecutive integers on the number line is 1 unit.
- Determining the value of each segment: Since TR = RS = SU and there are 3 equal segments between -2 and 0, each segment represents 1/3 of a unit.
- Finding the coordinates of points:
- Point R: Lies 1 segment to the left of 0. So, R represents -1/3.
- Point S: Lies 2 segments to the left of 0. So, S represents -2/3.
- Point P: Lies 1 segment to the right of 2. So, P represents 7/3.
- Point Q: Lies 2 segments to the right of 2. So, Q represents 8/3.
Therefore, the rational numbers represented by P, Q, R, and S are:
- P = 7/3
- Q = 8/3
- R = -1/3
- S = -2/3
6. Which of the following pairs represent the same rational number?
Ans :
7. Rewrite the following rational numbers in the simplest form:
Ans :
1. (-8)/6
The GCD of 8 and 6 is 2.
Dividing both numerator and denominator by 2, we get:
(-8)/6 = (-4)/3
2. 25/45
The GCD of 25 and 45 is 5.
Dividing both numerator and denominator by 5, we get:
25/45 = 5/9
3. (-44)/72
The GCD of 44 and 72 is 4.
Dividing both numerator and denominator by 4, we get:
(-44)/72 = (-11)/18
Therefore, the rational numbers in their simplest forms are:
- (-8)/6 = (-4)/3
- 25/45 = 5/9
- (-44)/72 = (-11)/18
8. Fill in the boxes with the correct symbol out of >, < and =.
Ans :
i) -5/7 < 2/3
- Convert to like fractions: -15/21 and 14/21
- -15/21 is less than 14/21, so -5/7 is less than 2/3
ii) -4/5 < -5/7
- Convert to like fractions: -28/35 and -25/35
- -28/35 is less than -25/35, so -4/5 is less than -5/7
iii) -7/8 = 14/-16
- -7/8 is already in its simplest form.
- 14/-16 can be simplified to -7/8 by dividing both numerator and denominator by 2.
- -7/8 is equal to -7/8
iv) -8/5 < -7/4
- Convert to like fractions: -32/20 and -35/20
- -32/20 is greater than -35/20, so -8/5 is greater than -7/4
v) 1/-3 > -1/4
- Convert to like fractions: -4/12 and -3/12
- -4/12 is greater than -3/12, so 1/-3 is greater than -1/4
vi) 5/-11 = -5/11
- 5/-11 is equivalent to -5/11 (dividing both numerator and denominator by -1)
vii) 0 < -7/6
- 0 is less than any negative number
Summary
Here are the completed comparisons:
- -5/7 < 2/3
- -4/5 < -5/7
- -7/8 = 14/-16
- -8/5 < -7/4
- 1/-3 > -1/4
- 5/-11 = -5/11
- 0 < -7/6
9. Which is greater in each of the following:
Ans :
i) 2/3, 5/2
- Convert both fractions to have a common denominator:
- 2/3 = 4/6
- 5/2 = 15/6
- Comparing the numerators, 15/6 is greater than 4/6.
- Therefore, 5/2 is greater than 2/3.
ii) -5/6, -4/3
- Convert both fractions to have a common denominator:
- -5/6 = -5/6
- -4/3 = -8/6
- Comparing the numerators, -5/6 is greater than -8/6.
- Therefore, -5/6 is greater than -4/3.
iii) -3/4, 2/-3
- Rewrite 2/-3 as -2/3.
- Convert both fractions to have a common denominator:
- -3/4 = -9/12
- -2/3 = -8/12
- Comparing the numerators, -8/12 is greater than -9/12.
- Therefore, 2/-3 is greater than -3/4.
iv) -1/4, 1/4
- Comparing the numerators directly, 1/4 is greater than -1/4.
- Therefore, 1/4 is greater than -1/4.
v) -3 2/7, -3 4/5
- Convert both mixed numbers to improper fractions:
- -3 2/7 = -23/7
- -3 4/5 = -19/5
- Convert both fractions to have a common denominator:
- -23/7 = -115/35
- -19/5 = -133/35
- Comparing the numerators, -115/35 is greater than -133/35.
- Therefore, -3 2/7 is greater than -3 4/5.
Summary
- (i) 5/2
- (ii) -5/6
- (iii) 2/-3
- (iv) 1/4
- (v) -3 2/7
10. Write the following rational numbers in ascending order:
Ans :
(i) -3/5, -2/5, -1/5
(ii) -2/9, -4/3, 1/3
(iii) -3/7, -3/2, -3/4
Exercise 8.2
1. Find the sum:
Ans :
i) 5/4 + (-11/4)
we can add the numerators:
(5 + (-11))/4 = -6/4
Simplifying the fraction:
-6/4 = -3/2
Therefore, 5/4 + (-11/4) = -3/2
ii) 5/3 + 3/5
we need to find a common denominator.
Converting the fractions to have a common denominator:
(5/3)(5/5) + (3/5)(3/3) = 25/15 + 9/15
Now, we can add the numerators:
(25 + 9)/15 = 34/15
Therefore, 5/3 + 3/5 = 34/15
iii) -9/10 + 22/15
Find the LCM of 10 and 15, which is 30.
Convert the fractions to have a common denominator:
(-9/10)(3/3) + (22/15)(2/2) = -27/30 + 44/30
Add the numerators:
(-27 + 44)/30 = 17/30
Therefore, -9/10 + 22/15 = 17/30
iv) -3/-11 + 5/9
Simplify the first fraction:
(-3/-11) = 3/11
Find the LCM of 11 and 9, which is 99.
Convert the fractions to have a common denominator:
(3/11)(9/9) + (5/9)(11/11) = 27/99 + 55/99
Add the numerators:
(27 + 55)/99 = 82/99
Therefore, -3/-11 + 5/9 = 82/99
v) -8/19 + (-2)/57
Find the LCM of 19 and 57, which is 57.
Convert the fractions to have a common denominator:
(-8/19)(3/3) + (-2/57)(1/1) = -24/57 + (-2/57)
Add the numerators:
(-24 + (-2))/57 = -26/57
Therefore, -8/19 + (-2)/57 = -26/57
vi) -2/3 + 0
Adding 0 to any number doesn’t change
Therefore, -2/3 + 0 = -2/3
vii) -2 1/3 + 4 3/5
convert the mixed numbers to improper fractions:
-2 1/3 = -7/3
4 3/5 = 23/5
Convert the fractions to have a common denominator:
(-7/3)(5/5) + (23/5)(3/3) = -35/15 + 69/15
Add the numerators:
(-35 + 69)/15 = 34/15
Therefore, -2 1/3 + 4 3/5 = 34/15
2. Find:
Ans :
i) 7/24 – 17/36
The LCM of 24 and 36 is 72.
7/24 = (7 * 3)/(24 * 3) = 21/72
17/36 = (17 * 2)/(36 * 2) = 34/72
Subtract the numerators:
21/72 – 34/72 = (21 – 34)/72 = -13/72
Therefore, 7/24 – 17/36 = -13/72
ii) 5/63 – (-6/21)
Convert -6/21 to its simplest form
-6/21 = -2/7
Find the common denominator for 5/63 and -2/7:
Convert fractions to equivalent fractions with a denominator of 63:
5/63 remains the same.
-2/7 = (-2 * 9)/(7 * 9) = -18/63
Add the numerators:
5/63 – (-18/63) = 5/63 + 18/63 = 23/63
Therefore, 5/63 – (-6/21) = 23/63
iii) -6/13 – (7/15)
Find the common denominator for -6/13 and 7/15:
Convert fractions to equivalent fractions with a denominator of 195:
-6/13 = (-6 * 15)/(13 * 15) = -90/195
7/15 = (7 * 13)/(15 * 13) = 91/195
Subtract the numerators:
-90/195 – 91/195 = (-90 – 91)/195 = -181/195
Therefore, -6/13 – (7/15) = -181/195
iv) -3/8 – 7/11
Find the common denominator for -3/8 and 7/11:
Convert fractions to equivalent fractions with a denominator of 88:
-3/8 = (-3 * 11)/(8 * 11) = -33/88
7/11 = (7 * 8)/(11 * 8) = 56/88
Subtract the numerators:
-33/88 – 56/88 = (-33 – 56)/88 = -89/88
Therefore, -3/8 – 7/11 = -89/88
v) -2 1/9 – 6
-2 1/9 = -19/9
Convert 6 to a fraction with a denominator of 9:
6 = 54/9
Subtract the fractions:
-19/9 – 54/9 = (-19 – 54)/9 = -73/9
Therefore, -2 1/9 – 6 = -73/9
3. Find the product:
Ans :
i) 9/2 × (-7/4)
Multiply the numerators and denominators:
(9 * -7) / (2 * 4) = -63/8
Therefore, 9/2 × (-7/4) = -63/8
ii) 3/10 × (-9)
Rewrite -9 as a fraction: -9 = -9/1
Multiply the numerators and denominators:
(3 * -9) / (10 * 1) = -27/10
Therefore, 3/10 × (-9) = -27/10
iii) -6/5 × 9/11
Multiply the numerators and denominators:
(-6 * 9) / (5 * 11) = -54/55
Therefore, -6/5 × 9/11 = -54/55
iv) 3/7 × (-2/5)
Multiply the numerators and denominators:
(3 * -2) / (7 * 5) = -6/35
Therefore, 3/7 × (-2/5) = -6/35
v) 3/11 × 2/5
Multiply the numerators and denominators:
(3 * 2) / (11 * 5) = 6/55
Therefore, 3/11 × 2/5 = 6/55
vi) 3/-5 × -5/3
Multiply the numerators and denominators:
(3 * -5) / (-5 * 3) = -15/-15 = 1
Therefore, 3/-5 × -5/3 = 1
4. Find the value of:
Ans :
i) (-4) + 2/3
-4 can be written as -12/3.
Now, we can add the fractions:
-12/3 + 2/3 = -10/3
Therefore, (-4) + 2/3 = -10/3
ii) -3/5 + 2
2 can be written as 10/5.
Now, we can add the fractions:
-3/5 + 10/5 = 7/5
Therefore, -3/5 + 2 = 7/5
iii) -4/5 + (-3)
-3 can be written as -15/5.
Now, we can add the fractions:
-4/5 + (-15/5) = -19/5
Therefore, -4/5 + (-3) = -19/5
iv) -1/8 + 3/4
The least common multiple (LCM) of 8 and 4 is 8.
-1/8 remains the same.
3/4 can be converted to 6/8.
Now, we can add the fractions:
-1/8 + 6/8 = 5/8
Therefore, -1/8 + 3/4 = 5/8
v) -2/13 + 1/7
The least common multiple (LCM) of 13 and 7 is 91.
-2/13 can be converted to -14/91.
1/7 can be converted to 13/91.
Now, we can add the fractions:
-14/91 + 13/91 = -1/91
Therefore, -2/13 + 1/7 = -1/91
vi) -7/12 + (-2/13)
The least common multiple (LCM) of 12 and 13 is 156.
-7/12 can be converted to -91/156.
-2/13 can be converted to -24/156.
Now, we can add the fractions:
-91/156 + (-24/156) = -115/156
Therefore, -7/12 + (-2/13) = -115/156
vii) 3/13 + (-4/65)
The least common multiple (LCM) of 13 and 65 is 65.
3/13 can be converted to 15/65.
Now, we can add the fractions:
15/65 + (-4/65) = 11/65
Therefore, 3/13 + (-4/65) = 11/65
