Samacheer Kalvi Class 8 Maths Solutions Chapter 1 Numbers Exercise 1.3

Tamilnadu Samacheer Kalvi 8th Maths Solutions Chapter 1 Numbers Ex 1.3

Question 1. Verify the closure property for addition and multiplication for the rational numbers \( \frac{-5}{7} \) and \( \frac{8}{9} \).
Answer:
To verify the closure property for addition:
Let \( a = \frac{-5}{7} \) and \( b = \frac{8}{9} \)
Now, we add \( a \) and \( b \):
\( a + b = \frac{-5}{7} + \frac{8}{9} \)
To add these fractions, we find a common denominator, which is \( 7 \times 9 = 63 \).
\( a + b = \frac{(-5 \times 9) + (8 \times 7)}{7 \times 9} \)
\( a + b = \frac{-45 + 56}{63} \)
\( a + b = \frac{11}{63} \)
Since \( \frac{11}{63} \) is a rational number, the closure property holds for addition.

To verify the closure property for multiplication:
Let \( a = \frac{-5}{7} \) and \( b = \frac{8}{9} \)
Now, we multiply \( a \) and \( b \):
\( a \times b = \frac{-5}{7} \times \frac{8}{9} \)
To multiply fractions, we multiply the numerators and the denominators.
\( a \times b = \frac{-5 \times 8}{7 \times 9} \)
\( a \times b = \frac{-40}{63} \)
Since \( \frac{-40}{63} \) is a rational number, the closure property also holds for multiplication.
In simple words: The closure property means if you add or multiply two rational numbers, the answer will always be another rational number. We showed this is true for both addition and multiplication with the given fractions.

🎯 Exam Tip: Remember to always find a common denominator for addition and subtraction of fractions, and multiply numerators with numerators, and denominators with denominators for multiplication.

Question 2. Verify the commutative property for addition and multiplication for the rational numbers \( \frac{-10}{11} \) and \( \frac{-8}{33} \).
Answer:
To verify the commutative property for addition:
Let \( a = \frac{-10}{11} \) and \( b = \frac{-8}{33} \)
First, calculate \( a + b \):
\( a + b = \frac{-10}{11} + \frac{-8}{33} \)
The common denominator is 33.
\( a + b = \frac{(-10 \times 3) + (-8 \times 1)}{33} \)
\( a + b = \frac{-30 - 8}{33} \)
\( a + b = \frac{-38}{33} \) .....(1)
Next, calculate \( b + a \):
\( b + a = \frac{-8}{33} + \frac{-10}{11} \)
The common denominator is 33.
\( b + a = \frac{(-8 \times 1) + (-10 \times 3)}{33} \)
\( b + a = \frac{-8 - 30}{33} \)
\( b + a = \frac{-38}{33} \) .....(2)
From (1) and (2), \( a + b = b + a \). So, addition is commutative for rational numbers.

To verify the commutative property for multiplication:
Let \( a = \frac{-10}{11} \) and \( b = \frac{-8}{33} \)
First, calculate \( a \times b \):
\( a \times b = \frac{-10}{11} \times \frac{-8}{33} \)
\( a \times b = \frac{(-10) \times (-8)}{11 \times 33} \)
\( a \times b = \frac{80}{363} \) .....(3)
Next, calculate \( b \times a \):
\( b \times a = \frac{-8}{33} \times \frac{-10}{11} \)
\( b \times a = \frac{(-8) \times (-10)}{33 \times 11} \)
\( b \times a = \frac{80}{363} \) .....(4)
From (3) and (4), \( a \times b = b \times a \). So, multiplication is commutative for rational numbers.
In simple words: The commutative property means you can swap the order of numbers when you add or multiply them, and the answer stays the same. We proved this works for both adding and multiplying these two rational numbers.

🎯 Exam Tip: Always show both sides of the equation (a+b and b+a, or a×b and b×a) separately and clearly state if they are equal to verify the property.

Question 3. Verify the associative property for addition and multiplication for the rational numbers \( \frac{-7}{9} \), \( \frac{5}{6} \) and \( \frac{-4}{3} \).
Answer:
To verify the associative property for addition:
Let \( a = \frac{-7}{9} \), \( b = \frac{5}{6} \) and \( c = \frac{-4}{3} \)
First, calculate \( (a + b) + c \):
\( (a + b) + c = \left( \frac{-7}{9} + \frac{5}{6} \right) + \frac{-4}{3} \)
For \( \frac{-7}{9} + \frac{5}{6} \), the common denominator is 18.
\( = \left( \frac{(-7 \times 2) + (5 \times 3)}{18} \right) + \frac{-4}{3} \)
\( = \left( \frac{-14 + 15}{18} \right) + \frac{-4}{3} \)
\( = \frac{1}{18} + \frac{-4}{3} \)
For \( \frac{1}{18} + \frac{-4}{3} \), the common denominator is 18.
\( = \frac{1 + (-4 \times 6)}{18} \)
\( = \frac{1 - 24}{18} \)
\( = \frac{-23}{18} \) .....(1)
Next, calculate \( a + (b + c) \):
\( a + (b + c) = \frac{-7}{9} + \left( \frac{5}{6} + \frac{-4}{3} \right) \)
For \( \frac{5}{6} + \frac{-4}{3} \), the common denominator is 6.
\( = \frac{-7}{9} + \left( \frac{5 + (-4 \times 2)}{6} \right) \)
\( = \frac{-7}{9} + \left( \frac{5 - 8}{6} \right) \)
\( = \frac{-7}{9} + \frac{-3}{6} \)
Simplify \( \frac{-3}{6} \) to \( \frac{-1}{2} \).
\( = \frac{-7}{9} + \frac{-1}{2} \)
For \( \frac{-7}{9} + \frac{-1}{2} \), the common denominator is 18.
\( = \frac{(-7 \times 2) + (-1 \times 9)}{18} \)
\( = \frac{-14 - 9}{18} \)
\( = \frac{-23}{18} \) .....(2)
From (1) and (2), \( (a + b) + c = a + (b + c) \). So, addition is associative for rational numbers.

To verify the associative property for multiplication:
Let \( a = \frac{-7}{9} \), \( b = \frac{5}{6} \) and \( c = \frac{-4}{3} \)
First, calculate \( (a \times b) \times c \):
\( (a \times b) \times c = \left( \frac{-7}{9} \times \frac{5}{6} \right) \times \frac{-4}{3} \)
\( = \left( \frac{-7 \times 5}{9 \times 6} \right) \times \frac{-4}{3} \)
\( = \frac{-35}{54} \times \frac{-4}{3} \)
\( = \frac{-35 \times -4}{54 \times 3} \)
\( = \frac{140}{162} \)
Simplify by dividing by 2:
\( = \frac{70}{81} \) .....(3)
Next, calculate \( a \times (b \times c) \):
\( a \times (b \times c) = \frac{-7}{9} \times \left( \frac{5}{6} \times \frac{-4}{3} \right) \)
\( = \frac{-7}{9} \times \left( \frac{5 \times -4}{6 \times 3} \right) \)
\( = \frac{-7}{9} \times \left( \frac{-20}{18} \right) \)
Simplify \( \frac{-20}{18} \) to \( \frac{-10}{9} \).
\( = \frac{-7}{9} \times \frac{-10}{9} \)
\( = \frac{-7 \times -10}{9 \times 9} \)
\( = \frac{70}{81} \) .....(4)
From (3) and (4), \( (a \times b) \times c = a \times (b \times c) \). So, multiplication is associative for rational numbers.
In simple words: The associative property means that how you group numbers in parentheses when adding or multiplying does not change the final answer. We showed this is true for these three rational numbers for both addition and multiplication.

🎯 Exam Tip: When simplifying fractions in intermediate steps, ensure the simplification is correct and applies to both numerator and denominator, as it can make further calculations easier.

Question 4. Verify the distributive property \( a \times (b + c) = (a \times b) + (a \times c) \) for the rational numbers \( a = \frac{-1}{2} \), \( b = \frac{2}{3} \) and \( c = \frac{-5}{6} \).
Answer:
Given the rational numbers \( a = \frac{-1}{2} \), \( b = \frac{2}{3} \) and \( c = \frac{-5}{6} \).
First, calculate the Left Hand Side (LHS): \( a \times (b + c) \)
\( a \times (b + c) = \frac{-1}{2} \times \left( \frac{2}{3} + \frac{-5}{6} \right) \)
For \( \frac{2}{3} + \frac{-5}{6} \), the common denominator is 6.
\( = \frac{-1}{2} \times \left( \frac{(2 \times 2) + (-5 \times 1)}{6} \right) \)
\( = \frac{-1}{2} \times \left( \frac{4 - 5}{6} \right) \)
\( = \frac{-1}{2} \times \frac{-1}{6} \)
\( = \frac{(-1) \times (-1)}{2 \times 6} \)
\( = \frac{1}{12} \) .....(1)
Next, calculate the Right Hand Side (RHS): \( (a \times b) + (a \times c) \)
\( (a \times b) + (a \times c) = \left( \frac{-1}{2} \times \frac{2}{3} \right) + \left( \frac{-1}{2} \times \frac{-5}{6} \right) \)
For the first part: \( \frac{-1}{2} \times \frac{2}{3} = \frac{-1 \times 2}{2 \times 3} = \frac{-2}{6} = \frac{-1}{3} \)
For the second part: \( \frac{-1}{2} \times \frac{-5}{6} = \frac{-1 \times -5}{2 \times 6} = \frac{5}{12} \)
Now add these two results:
\( = \frac{-1}{3} + \frac{5}{12} \)
The common denominator is 12.
\( = \frac{(-1 \times 4) + 5}{12} \)
\( = \frac{-4 + 5}{12} \)
\( = \frac{1}{12} \) .....(2)
From (1) and (2), we have \( a \times (b + c) = (a \times b) + (a \times c) \) is true. Hence, multiplication is distributive over addition for rational numbers.
In simple words: The distributive property shows how multiplication works with addition. It says you can multiply a number by a sum, or you can multiply the number by each part of the sum separately and then add the results. Both ways give the same answer. We checked and found this rule holds true for the given rational numbers.

🎯 Exam Tip: Always calculate the Left Hand Side (LHS) and Right Hand Side (RHS) of the distributive property separately and show that they are equal. Simplify fractions at each step to avoid errors.

Question 5. Verify the identity property for addition and multiplication for the rational numbers \( \frac{15}{19} \) and \( \frac{-18}{25} \).
Answer:
To verify the identity property for addition:
For any rational number \( a \), \( a + 0 = 0 + a = a \). Here, 0 is the additive identity.
For \( \frac{15}{19} \):
\( \frac{15}{19} + 0 = \frac{15+0}{19} = \frac{15}{19} \)
For \( \frac{-18}{25} \):
\( \frac{-18}{25} + 0 = \frac{-18+0}{25} = \frac{-18}{25} \)
The identity property for addition is verified.

To verify the identity property for multiplication:
For any rational number \( a \), \( a \times 1 = 1 \times a = a \). Here, 1 is the multiplicative identity.
For \( \frac{15}{19} \):
\( \frac{15}{19} \times 1 = \frac{15 \times 1}{19} = \frac{15}{19} \)
For \( \frac{-18}{25} \):
\( \frac{-18}{25} \times 1 = \frac{-18 \times 1}{25} = \frac{-18}{25} \)
The identity property for multiplication is verified.
In simple words: The identity property means that there's a special number you can add to any number (zero for addition) or multiply by any number (one for multiplication) that doesn't change the original number. We showed that adding zero or multiplying by one keeps the rational numbers the same.

🎯 Exam Tip: Remember that 0 is the additive identity and 1 is the multiplicative identity. Clearly show the steps for both numbers and both operations.

Question 6. Verify the additive and multiplicative inverse property for the rational numbers \( \frac{-7}{17} \) and \( \frac{17}{27} \).
Answer:
To verify the additive inverse property:
For any rational number \( a \), its additive inverse is \( -a \), such that \( a + (-a) = 0 \).
For \( \frac{-7}{17} \): The additive inverse is \( -(\frac{-7}{17}) = \frac{7}{17} \).
\( \frac{-7}{17} + \frac{7}{17} = \frac{-7 + 7}{17} = \frac{0}{17} = 0 \)
For \( \frac{17}{27} \): The additive inverse is \( -\frac{17}{27} \).
\( \frac{17}{27} + \left( -\frac{17}{27} \right) = \frac{17 - 17}{27} = \frac{0}{27} = 0 \)
The additive inverse property is verified for both rational numbers.

To verify the multiplicative inverse property:
For any non-zero rational number \( a \), its multiplicative inverse (or reciprocal) is \( \frac{1}{a} \), such that \( a \times \frac{1}{a} = 1 \).
For \( \frac{-7}{17} \): The multiplicative inverse is \( \frac{17}{-7} \).
\( \frac{-7}{17} \times \frac{17}{-7} = \frac{(-7) \times 17}{17 \times (-7)} = \frac{-119}{-119} = 1 \)
For \( \frac{17}{27} \): The multiplicative inverse is \( \frac{27}{17} \).
\( \frac{17}{27} \times \frac{27}{17} = \frac{17 \times 27}{27 \times 17} = \frac{459}{459} = 1 \)
The multiplicative inverse property is verified for both rational numbers.
In simple words: The inverse property means you can find a number that, when added to the original number, gives zero (additive inverse), or when multiplied by the original number, gives one (multiplicative inverse). We found these inverse numbers for both given rational numbers.

🎯 Exam Tip: Always remember that the additive inverse is the negative of the number, and the multiplicative inverse is the reciprocal (flipping the fraction), and ensure your final results for addition are 0 and for multiplication are 1.

Objective Type Questions

Question 7. Closure property is not true for division of rational numbers because of the number
(a) 1
(b) 1
(c) 0
(d) \( \frac { 1 }{ 2 } \)
Answer: (c) 0
In simple words: If you try to divide any number by zero, it doesn't make sense; it's undefined. This means that if you divide two rational numbers, and one of them is zero, the answer is not always a rational number.

🎯 Exam Tip: Always remember that division by zero is undefined, which is why the closure property fails for division of rational numbers.

Question 8. \( \frac{1}{2}-\left(\frac{3}{4}-\frac{5}{6}\right) \neq\left(\frac{1}{2}-\frac{3}{4}\right)-\frac{5}{6} \) illustrates that subtraction does not satisfy the property for rational numbers.
(a) commutative
(b) closure
(c) distributive
(d) associative
Answer: (d) associative
In simple words: The associative property lets you group numbers differently when you add or multiply without changing the answer. This problem shows that when you subtract, changing how you group the numbers will give you a different answer, so subtraction is not associative.

🎯 Exam Tip: Understanding the order of operations in parentheses is key. When the grouping of numbers changes the result, the operation is not associative.

Question 9. Which of the following illustrates the inverse property for addition?
(a) \( \frac{1}{8}-\frac{1}{8}=0 \)
(b) \( \frac{1}{0}+\frac{1}{8}=\frac{1}{4} \)
(c) \( \frac{1}{8}+0=\frac{1}{8} \)
(d) \( \frac{1}{8}-0=\frac{1}{8} \)
Answer: (a) \( \frac{1}{8}-\frac{1}{8}=0 \)
In simple words: The inverse property for addition states that if you add a number to its opposite, the result is always zero. Option (a) shows this perfectly: adding \( \frac{1}{8} \) to its negative, \( -\frac{1}{8} \), gives zero.

🎯 Exam Tip: The additive inverse of a number is simply its opposite (the number with the sign changed). Their sum must always be zero for the property to hold.

Question 10. \( \frac{3}{4} \times\left(\frac{1}{2}-\frac{1}{4}\right)=\frac{3}{4} \times \frac{1}{2}- \frac{3}{4} \times \frac{1}{4} \) illustrates that multiplication is distributive over
(a) addition
(b) subtraction
(c) multiplication
(d) division
Answer: (b) subtraction
In simple words: This equation shows that when you multiply a number by a difference (one number minus another), you can get the same answer by multiplying the first number by each part of the difference separately, and then subtracting those results. This is called the distributive property over subtraction.

🎯 Exam Tip: The key indicator for the distributive property is how an operation outside parentheses affects operations inside. Here, multiplication distributes over the subtraction inside the parentheses.