GSEB Class 8 Maths Solutions Chapter 1 Rational Numbers Exercise 1.1

Question 1. Using appropriate properties find
(i) \( -\frac{2}{3} \times \frac{3}{5}+\frac{5}{2}-\frac{3}{5} \times \frac{1}{6} \)
(ii) \( \frac{2}{5} \times\left(-\frac{3}{7}\right)-\frac{1}{6} \times \frac{3}{2}+\frac{1}{14} \times \frac{2}{5} \)
Answer:
(i) We are asked to simplify the expression using appropriate properties:
\( -\frac{2}{3} \times \frac{3}{5}+\frac{5}{2}-\frac{3}{5} \times \frac{1}{6} \)
Rearranging the terms to group common factors (Using commutative property):
\( = \left[-\frac{2}{3} \times \frac{3}{5}+\left(\frac{-3}{5}\right) \times \frac{1}{6}\right]+\frac{5}{2} \)
Taking \( \left(\frac{-3}{5}\right) \) as common from the first two terms (Using distributivity property):
\( = \left(\frac{-3}{5}\right)\left[\frac{2}{3}+\frac{1}{6}\right]+\frac{5}{2} \)
Adding the fractions inside the bracket:
\( = \left(\frac{-3}{5}\right)\left[\frac{4+1}{6}\right]+\frac{5}{2} \)
\( = \left(\frac{-3}{5}\right)\left[\frac{5}{6}\right]+\frac{5}{2} \)
Multiplying the fractions:
\( = \frac{-3}{5} \times \frac{5}{6}+\frac{5}{2} \)
\( = \frac{-1}{2}+\frac{5}{2} \)
Adding the fractions:
\( = \frac{-1+5}{2} \)
\( = \frac{4}{2} \)
\( = 2 \)
(ii) We are asked to simplify the expression using appropriate properties:
\( \frac{2}{5} \times\left(-\frac{3}{7}\right)-\frac{1}{6} \times \frac{3}{2}+\frac{1}{14} \times \frac{2}{5} \)
First, simplify the middle term \( -\frac{1}{6} \times \frac{3}{2} \):
\( = \frac{2}{5} \times\left(-\frac{3}{7}\right)-\frac{1}{4}+\frac{1}{14} \times \frac{2}{5} \)
Rearranging the terms to group common factors (Using commutative property):
\( = \frac{2}{5} \times\left(\frac{-3}{7}\right)+\frac{1}{14} \times \frac{2}{5}-\frac{1}{4} \)
Taking \( \frac{2}{5} \) as common from the first two terms (Using distributivity property):
\( = \frac{2}{5}\left[\frac{-3}{7}+\frac{1}{14}\right]-\frac{1}{4} \)
Adding the fractions inside the bracket:
\( = \frac{2}{5}\left[\frac{-6+1}{14}\right]-\frac{1}{4} \)
\( = \frac{2}{5}\left[\frac{-5}{14}\right]-\frac{1}{4} \)
Multiplying the fractions:
\( = \frac{-1}{7}-\frac{1}{4} \)
Finding a common denominator and subtracting:
\( = \frac{-4-7}{28} \)
\( = \frac{-11}{28} \)
In simple words: For the first part, we grouped similar parts and used rules like moving numbers around (commutative) and sharing numbers (distributive) to make the sum easier to solve. For the second part, we followed similar steps of grouping and simplifying to get the final answer.

Exam Tip: Remember to clearly state which property you are using at each step, especially when the question asks to use "appropriate properties". This helps to show your understanding and gets you full marks.

Question 2. Write the additive inverse of each of the following:
1. \( \frac { 2 }{ 8 } \)
2. \( \frac {5}{9} \)
3. \( \frac {-6}{ -5 } \)
4. \( \frac { 2 }{ -9 } \)
5. \( \frac { 19 }{ -6 } \)
Answer:

Exam Tip: To find the additive inverse, simply change the sign of the number. For fractions, if the fraction is overall negative, its additive inverse is positive, and vice versa. Always simplify fractions first if possible.

Question 3. Verify that \( -(-x) = x \) for:
1. \( x = \frac { 11 }{ 15 } \)
2. \( x = - \frac { 13 }{ 17 } \)
Answer:
(i) Given \( x = \frac { 11 }{ 15 } \)
We need to verify that \( -(-x) = x \).
First, find \( -x \):
\( -x = - \frac { 11 }{ 15 } \)
Now, find \( -(-x) \):
\( -(-x) = - \left( - \frac { 11 }{ 15 } \right) = \frac { 11 }{ 15 } \)
Since \( \frac { 11 }{ 15 } = x \), we have verified that \( -(-x) = x \).
(ii) Given \( x = - \frac { 13 }{ 17 } \)
We need to verify that \( -(-x) = x \).
First, find \( -x \):
\( -x = - \left( - \frac { 13 }{ 17 } \right) = \frac { 13 }{ 17 } \)
Now, find \( -(-x) \):
\( -(-x) = - \left( \frac { 13 }{ 17 } \right) = - \frac { 13 }{ 17 } \)
Since \( - \frac { 13 }{ 17 } = x \), we have verified that \( -(-x) = x \).
In simple words: To verify \( -(-x) = x \), we first find the negative of \( x \), and then find the negative of that result. The final answer should be the original \( x \). This property means that two negative signs cancel each other out, bringing you back to the starting value.

Exam Tip: Remember that a double negative always results in a positive. \( -(-a) = a \). This rule is key for successfully verifying such expressions.

Question 4. Find the multiplicative inverse of the following:
1. -13
2. \( \frac { -13 }{ 19 } \)
3. \( \frac { 1 }{ 5 } \)
4. \( \frac { -5 }{8} \times \frac {-3}{7} \)
5. \( -1 \times \frac { -2 }{ 5 } \)
6. -1
Answer:

Exam Tip: Remember that for a product of fractions, you must first calculate the product to get a single rational number before finding its multiplicative inverse. Also, the multiplicative inverse of 1 is 1, and the multiplicative inverse of -1 is -1.

Question 5. Name the property under multiplication used in each of the following:
1. \( \frac { -4 }{ 5 } \times 1 = 1 \times \frac { -4 }{ 5 } = \frac { -4 }{5} \)
2. \( - \frac {13}{17} \times \frac { -2 }{ 7 } = \frac { -2 }{ 7 } \times \frac { -13 }{ 17 } \)
3. \( \frac{-19}{29} \times \frac{29}{-19} = 1 \)
Answer:

Exam Tip: Be sure to distinguish between additive and multiplicative properties. For multiplication, the identity element is 1, and the inverse results in 1. For addition, the identity element is 0, and the inverse results in 0.

Question 6. Multiply \( \frac {6}{13} \) by the reciprocal of \( \frac { -7 }{ 16 } \)
Answer:
We need to multiply \( \frac {6}{13} \) by the reciprocal of \( \frac { -7 }{ 16 } \).
First, find the reciprocal of \( \frac { -7 }{ 16 } \):
Reciprocal of \( \frac { -7 }{ 16 } \) is \( \frac { 16 }{ -7 } \), which can also be written as \( - \frac { 16 }{ 7 } \).
Now, multiply \( \frac {6}{13} \) by this reciprocal:
\( \frac { 6 }{ 13 } \times \left( - \frac { 16 }{ 7 } \right) \)
\( = \frac { 6 \times (-16) }{ 13 \times 7 } \)
\( = \frac { -96 }{ 91 } \)
In simple words: To solve this, first flip the second fraction to find its reciprocal. Then, multiply the first fraction by this new, flipped fraction. This means multiplying the top numbers together and the bottom numbers together. Make sure to keep track of any negative signs in your answer.

Exam Tip: Always remember that the reciprocal of a negative fraction is also a negative fraction. Carefully multiply numerators and denominators and simplify if possible.

Question 7. Tell what property allows you to compute \( \frac { 1 }{ 3 } \times \left(6 \times \frac{4}{3}\right) \) as \( \left(\frac{1}{3} \times 6\right) \times \frac{4}{3} \)
Answer:
In computing \( \frac { 1 }{ 3 } \times \left(6 \times \frac{4}{3}\right) \) as \( \left(\frac{1}{3} \times 6\right) \times \frac{4}{3} \), we use the associativity property of multiplication.
In simple words: This is allowed by the "associativity" rule. This rule means that when you multiply three or more numbers, you can change how you group them with parentheses, and the final answer will still be the same. The order of numbers stays the same, only the grouping changes.

Exam Tip: The associative property concerns the grouping of numbers, not their order. If the order of the numbers changes, that indicates the commutative property. If only the grouping changes, it's associative.

Question 8. Is \( \frac {8}{9} \) the multiplicative inverse of \( -1\frac {1}{8} \)? Why or Why not?
Answer:
To determine if \( \frac {8}{9} \) is the multiplicative inverse of \( -1\frac {1}{8} \), we first convert the mixed fraction to an improper fraction.
\( -1\frac { 1 }{ 8 } = - \left( \frac{1 \times 8 + 1}{8} \right) = - \frac { 9 }{ 8 } \)
Now, we multiply \( \frac {8}{9} \) by \( - \frac { 9 }{ 8 } \) to see if their product is 1.
\( \frac { 8 }{ 9 } \times \left( - \frac { 9 }{ 8 } \right) = \frac { 8 \times (-9) }{ 9 \times 8 } = \frac { -72 }{ 72 } = -1 \)
Since the product is -1, and not 1, \( \frac {8}{9} \) is not the multiplicative inverse of \( -1\frac {1}{8} \).
The reason is that the product of a number and its multiplicative inverse must always be 1.
In simple words: No, \( \frac {8}{9} \) is not the multiplicative inverse of \( -1\frac {1}{8} \). We changed \( -1\frac {1}{8} \) to \( - \frac {9}{8} \). When we multiplied \( \frac {8}{9} \) by \( - \frac {9}{8} \), we got -1, not 1. For numbers to be multiplicative inverses, their product must be exactly 1.

Exam Tip: Remember that the product of a number and its multiplicative inverse must be positive 1. A product of -1 indicates that they are additive inverses, or that one number is the reciprocal of the other's negative.

Question 9. Is 0.3 the multiplicative inverse of \( 3\frac { 1 }{ 3 } \)? Why or why not?
Answer:
To determine if 0.3 is the multiplicative inverse of \( 3\frac { 1 }{ 3 } \), we first convert both numbers into fractions.
Convert 0.3 to a fraction:
\( 0.3 = \frac {3}{10} \)
Convert the mixed fraction \( 3\frac { 1 }{ 3 } \) to an improper fraction:
\( 3\frac { 1 }{ 3 } = \frac { (3 \times 3) + 1 }{ 3 } = \frac { 9 + 1 }{ 3 } = \frac { 10 }{ 3 } \)
Now, multiply these two fractions together:
\( \frac { 3 }{ 10 } \times \frac { 10 }{ 3 } = \frac { 3 \times 10 }{ 10 \times 3 } = \frac { 30 }{ 30 } = 1 \)
Since the product of 0.3 and \( 3\frac { 1 }{ 3 } \) is 1, yes, 0.3 is the multiplicative inverse of \( 3\frac { 1 }{ 3 } \).
In simple words: Yes, 0.3 is the multiplicative inverse of \( 3\frac { 1 }{ 3 } \). We changed 0.3 into \( \frac {3}{10} \) and \( 3\frac {1}{3} \) into \( \frac {10}{3} \). When we multiply these two fractions, the answer is 1. When two numbers multiply to 1, they are called multiplicative inverses of each other.

Exam Tip: When dealing with decimals and mixed fractions for multiplicative inverse problems, always convert them to improper fractions first. This makes the multiplication straightforward and reduces the chance of errors.

Question 10. Write:
1. The rational number that does not have a reciprocal.
2. The rational numbers that are equal to their reciprocals.
3. The rational number that is equal to its negative.
Answer:
1. The rational number zero (0) does not have a reciprocal. This is because division by zero is not defined, and a reciprocal involves dividing 1 by the number.
2. The rational numbers 1 and (-1) are equal to their reciprocals, respectively. The reciprocal of 1 is \( \frac{1}{1} = 1 \), and the reciprocal of -1 is \( \frac{1}{-1} = -1 \).
3. The rational number that is equal to its negative is 0. This is because \( -0 = 0 \). No other rational number is equal to its own negative.
In simple words: First, the number that can't be flipped is zero, because you can't divide by zero. Second, the numbers that stay the same when you flip them are 1 and -1. Third, the number that is the same as its opposite (negative version) is zero.

Exam Tip: These are fundamental properties of rational numbers. Understand why 0 has no reciprocal (division by zero is undefined) and why 1 and -1 are unique in being equal to their reciprocals. Also, be clear on why only 0 is equal to its negative.

Question 11. Fill in the blanks:
1. Zero has ____ reciprocal.
2. The numbers ____ and ____ are their own reciprocals.
3. The reciprocal of -5 is ____.
4. Reciprocal of \( \frac {1}{x} \), where x \( \ne \) 0 is ____.
5. The product of two rational numbers is always a ____.
Answer:
1. Zero has **no** reciprocal.
2. The numbers **1** and **-1** are their own reciprocals.
3. The reciprocal of -5 is \( \frac { -1 }{ 5 } \).
4. Reciprocal of \( \frac { 1 }{ x } \), where x \( \ne \) 0 is **x**.
5. The product of two rational numbers is always a **rational number**.
In simple words: We filled in the blanks based on what we know about numbers. Zero doesn't have a flip-side number. Only 1 and -1 stay the same when you flip them. To flip -5, you get one-fifth. If you flip one-over-x, you just get x. When you multiply any two simple fractions or numbers, the answer is always another simple fraction or number.

Exam Tip: These fill-in-the-blanks questions check your understanding of basic definitions and properties of rational numbers. Make sure you are familiar with key terms like 'reciprocal' and 'rational number'.