GSEB Class 8 Maths Solutions Chapter 1 સંમેય સંખ્યાઓ Exercise 1.1

Question 1. યોગ્ય ગુણધર્મનો ઉપયોગ કરી કિંમત શોધો.
(i) \( -\frac{2}{3} \times \frac{3}{5}+\frac{5}{2}-\frac{3}{5} \times \frac{1}{6} \)
Answer:
(i) Given expression: \( -\frac{2}{3} \times \frac{3}{5}+\frac{5}{2}-\frac{3}{5} \times \frac{1}{6} \)
We can rearrange the terms to group common factors:
\( = \left(-\frac{2}{3} \times \frac{3}{5}\right) - \left(\frac{3}{5} \times \frac{1}{6}\right) + \frac{5}{2} \)
\( = -\frac{2}{5} - \frac{1}{10} + \frac{5}{2} \)
To add and subtract these fractions, we find a common denominator, which is 10.
\( = -\frac{2 \times 2}{5 \times 2} - \frac{1 \times 1}{10 \times 1} + \frac{5 \times 5}{2 \times 5} \)
\( = -\frac{4}{10} - \frac{1}{10} + \frac{25}{10} \)
Now, we combine the numerators:
\( = \frac{-4 - 1 + 25}{10} \)
\( = \frac{-5 + 25}{10} \)
\( = \frac{20}{10} \)
\( = 2 \)
In simple words: First, we multiply the fractions in each pair. Then, we find a common bottom number (denominator) for all the results. Finally, we add and subtract the top numbers (numerators) to get the final answer.

Exam Tip: Remember to apply the order of operations (BODMAS/PEMDAS) correctly, performing multiplication before addition and subtraction. Look for opportunities to simplify fractions early.

Question 1. યોગ્ય ગુણધર્મનો ઉપયોગ કરી કિંમત શોધો.
(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:
(ii) Given expression: \( \frac{2}{5} \times\left(-\frac{3}{7}\right)-\frac{1}{6} \times \frac{3}{2}+\frac{1}{14} \times \frac{2}{5} \)
We rearrange the terms using the Commutative Property:
\( = \frac{2}{5} \times\left(-\frac{3}{7}\right) + \frac{1}{14} \times \frac{2}{5} - \frac{1}{6} \times \frac{3}{2} \)
Next, we use the Distributive Property for the first two terms:
\( = \frac{2}{5} \times \left(-\frac{3}{7} + \frac{1}{14}\right) - \left(\frac{1}{6} \times \frac{3}{2}\right) \)
Inside the parenthesis, find the LCM of 7 and 14, which is 14:
\( = \frac{2}{5} \times \left(\frac{-3 \times 2}{7 \times 2} + \frac{1}{14}\right) - \frac{3}{12} \)
\( = \frac{2}{5} \times \left(\frac{-6}{14} + \frac{1}{14}\right) - \frac{1}{4} \)
\( = \frac{2}{5} \times \left(\frac{-6 + 1}{14}\right) - \frac{1}{4} \)
\( = \frac{2}{5} \times \left(\frac{-5}{14}\right) - \frac{1}{4} \)
Now, perform the multiplication:
\( = \frac{2 \times (-5)}{5 \times 14} - \frac{1}{4} \)
\( = \frac{-10}{70} - \frac{1}{4} \)
Simplify the first fraction:
\( = -\frac{1}{7} - \frac{1}{4} \)
Find a common denominator for 7 and 4, which is 28:
\( = -\frac{1 \times 4}{7 \times 4} - \frac{1 \times 7}{4 \times 7} \)
\( = -\frac{4}{28} - \frac{7}{28} \)
Combine the numerators:
\( = \frac{-4 - 7}{28} \)
\( = \frac{-11}{28} \)
In simple words: We first put similar parts together using a rearrangement rule. Then, we share out a common number using the distribution rule. We find a common bottom number to combine fractions, and finally multiply and subtract to get the answer.

Exam Tip: For complex expressions, always look for common factors or properties like commutative and distributive rules to simplify the problem before performing calculations. This often makes the process easier and less prone to errors.

Question 2. નીચે આપેલ સંખ્યાની વિરોધી સંખ્યા લખો:
(i) \( \frac {2}{8} \)
Answer:
(i) The opposite number of \( \frac {2}{8} \) is \( -\frac {2}{8} \).
In simple words: To find the opposite number, just change its sign from positive to negative, or negative to positive.

Exam Tip: The opposite of a number 'a' is '-a'. This means their sum is always zero.

Question 2. નીચે આપેલ સંખ્યાની વિરોધી સંખ્યા લખો:
(ii) \( \frac {-5}{9} \)
Answer:
(ii) The opposite number of \( \frac {-5}{9} \) is \( \frac {5}{9} \).
In simple words: If a number is negative, its opposite is the same number but positive.

Exam Tip: Remember that the absolute value remains the same, only the direction on the number line changes.

Question 2. નીચે આપેલ સંખ્યાની વિરોધી સંખ્યા લખો:
(iii) \( \frac {-6}{-5} \)
Answer:
(iii) The fraction \( \frac {-6}{-5} \) simplifies to \( \frac {6}{5} \). The opposite number of \( \frac {6}{5} \) is \( -\frac {6}{5} \).
In simple words: First, make sure the fraction is in its simplest form. If both top and bottom numbers are negative, the fraction becomes positive. Then, change the sign of the simplified fraction to find its opposite.

Exam Tip: Always simplify fractions before finding their opposite to prevent confusion. Two negative signs cancel each other out to make a positive.

Question 2. નીચે આપેલ સંખ્યાની વિરોધી સંખ્યા લખો:
(iv) \( \frac {2}{-9} \)
Answer:
(iv) The opposite number of \( \frac {2}{-9} \) is \( \frac {2}{9} \).
In simple words: A negative sign can be placed anywhere in a fraction (top, bottom, or in front). To find the opposite, just flip that negative sign.

Exam Tip: Remember that \( \frac{a}{-b} = -\frac{a}{b} \). So, \( \frac{2}{-9} \) is the same as \( -\frac{2}{9} \), and its additive inverse (opposite) will be \( \frac{2}{9} \).

Question 2. નીચે આપેલ સંખ્યાની વિરોધી સંખ્યા લખો:
(v) \( \frac {19}{-6} \)
Answer:
(v) The opposite number of \( \frac {19}{-6} \) is \( \frac {19}{6} \).
In simple words: When you see a negative sign in the bottom part of a fraction, it means the whole fraction is negative. To find its opposite, simply make the entire fraction positive.

Exam Tip: Treat a fraction like \( \frac{19}{-6} \) as \( -\frac{19}{6} \). Its additive inverse will be \( +\frac{19}{6} \).

Question 3. ચકાસણી કરો : -(-x) = x
(i) \( x = \frac {11}{15} \)
Answer:
(i) Given \( x = \frac {11}{15} \).
First, find the negative of \( x \), which is \( -x \).
\( -x = -\frac{11}{15} \)
Now, find the negative of \( -x \), which is \( -(-x) \).
\( -(-x) = -\left(-\frac{11}{15}\right) \)
A double negative becomes a positive, so:
\( -(-x) = \frac{11}{15} \)
Since \( x = \frac{11}{15} \), we can say that \( -(-x) = x \).
In simple words: We start with a number, then make it negative. After that, we make it negative again. This brings us back to the original number.

Exam Tip: The principle \( -(-a) = a \) holds true for all real numbers. It demonstrates that taking the additive inverse twice returns the original number.

Question 3. ચકાસણી કરો : -(-x) = x
(ii) \( x = \left(\frac{-13}{17}\right) \)
Answer:
(ii) Given \( x = \left(\frac{-13}{17}\right) \).
First, determine the negative of \( x \), which is \( -x \).
\( -x = -\left(\frac{-13}{17}\right) \)
The two negative signs cancel each other, making the term positive:
\( -x = \frac{13}{17} \)
Next, find the negative of \( -x \), which is \( -(-x) \).
\( -(-x) = -\left(\frac{13}{17}\right) \)
So, \( -(-x) = -\frac{13}{17} \).
Since \( x = -\frac{13}{17} \), it shows that \( -(-x) = x \).
In simple words: We begin with a negative number. Then, we find its opposite, which makes it positive. Finally, we find the opposite of that positive number, bringing us back to our initial negative number.

Exam Tip: Be careful with multiple negative signs. An even number of negative signs results in a positive value, while an odd number results in a negative value.

Question 4. નીચે આપેલ સંખ્યાનો વ્યસ્ત જણાવો:
(i) -13
Answer:
(i) The reciprocal of -13 is \( \frac {1}{-13} \), which can also be written as \( -\frac {1}{13} \).
In simple words: To find the reciprocal of a whole number, you put 1 over that number. If the number is negative, its reciprocal also stays negative.

Exam Tip: The reciprocal of a non-zero number 'a' is '1/a'. The product of a number and its reciprocal is always 1.

Question 4. નીચે આપેલ સંખ્યાનો વ્યસ્ત જણાવો:
(ii) \( \frac {-13}{19} \)
Answer:
(ii) The reciprocal of \( \frac {-13}{19} \) is \( \frac {19}{-13} \), which is also \( -\frac {19}{13} \).
In simple words: To get the reciprocal of a fraction, simply flip it upside down. The negative sign stays with the fraction.

Exam Tip: When finding the reciprocal of a negative fraction, the negative sign usually remains with the numerator or is placed in front of the entire fraction.

Question 4. નીચે આપેલ સંખ્યાનો વ્યસ્ત જણાવો:
(iii) \( \frac {1}{5} \)
Answer:
(iii) The reciprocal of \( \frac {1}{5} \) is \( \frac {5}{1} \), which simplifies to 5.
In simple words: To find the reciprocal of a fraction, you just turn it over. So, the bottom number becomes the top number.

Exam Tip: The reciprocal of a unit fraction (a fraction with 1 as the numerator) is simply its denominator.

Question 4. નીચે આપેલ સંખ્યાનો વ્યસ્ત જણાવો:
(iv) \( \frac{-5}{8} \times \frac{-3}{7} \)
Answer:
(iv) First, multiply the given fractions:
\( \frac{-5}{8} \times \frac{-3}{7} = \frac{(-5) \times (-3)}{8 \times 7} = \frac{15}{56} \)
Now, find the reciprocal of \( \frac {15}{56} \).
The reciprocal of \( \frac {15}{56} \) is \( \frac {56}{15} \).
In simple words: We first multiply the two fractions together. Then, we flip the resulting fraction upside down to get its reciprocal.

Exam Tip: Always perform any operations (like multiplication) first to simplify the expression into a single number before finding its reciprocal.

Question 4. નીચે આપેલ સંખ્યાનો વ્યસ્ત જણાવો:
(v) -1 x \( \frac {-2}{5} \)
Answer:
(v) First, multiply the given numbers:
\( -1 \times \frac{-2}{5} = \frac{(-1) \times (-2)}{5} = \frac{2}{5} \)
Now, find the reciprocal of \( \frac {2}{5} \).
The reciprocal of \( \frac {2}{5} \) is \( \frac {5}{2} \).
In simple words: We start by multiplying the number by negative one. This changes its sign. Then, we flip the fraction to find its reciprocal.

Exam Tip: Multiplying by -1 simply changes the sign of the number. Once simplified, finding the reciprocal involves inverting the fraction.

Question 4. નીચે આપેલ સંખ્યાનો વ્યસ્ત જણાવો:
(vi) -1
Answer:
(vi) The reciprocal of -1 is \( \frac {1}{-1} \), which is also -1.
In simple words: The only number whose reciprocal is itself is -1 (and also 1).

Exam Tip: Remember that 1 and -1 are unique as they are their own reciprocals.

Question 5. નીચે આપેલ ગુણાકારની ક્રિયામાં કયા ગુણધર્મનો ઉપયોગ થયેલ છે તે જણાવો.
(i) \( -\frac {4}{5} \times 1 = 1 \times -\frac {4}{5} = -\frac {4}{5} \)
Answer:
(i) This equation shows that when any number is multiplied by 1, the result is the number itself. This property is known as the Multiplicative Identity or the identity element for multiplication.
In simple words: When you multiply anything by one, it stays the same. So, one is like a mirror for multiplication.

Exam Tip: The number 1 is the multiplicative identity because multiplying any number by 1 does not change the number's value.

Question 5. નીચે આપેલ ગુણાકારની ક્રિયામાં કયા ગુણધર્મનો ઉપયોગ થયેલ છે તે જણાવો.
(ii) \( -\frac{13}{17} \times \frac{-2}{7}=\frac{-2}{7} \times \frac{-13}{17} \)
Answer:
(ii) This equation demonstrates that changing the order of the numbers in multiplication does not change the result. This is known as the Commutative Property of Multiplication.
In simple words: You can multiply numbers in any order, and the answer will be exactly the same.

Exam Tip: The commutative property states that for any two numbers 'a' and 'b', \( a \times b = b \times a \). This applies to both addition and multiplication.

Question 5. નીચે આપેલ ગુણાકારની ક્રિયામાં કયા ગુણધર્મનો ઉપયોગ થયેલ છે તે જણાવો.
(iii) \( \frac{-19}{29} \times \frac{29}{-19}=1 \)
Answer:
(iii) This equation shows that when a number is multiplied by its reciprocal, the result is always 1. This property is known as the Multiplicative Inverse Property or the existence of reciprocals.
In simple words: When you multiply a number by its upside-down version, you always get one. This means they are inverses of each other.

Exam Tip: For any non-zero rational number \( \frac{a}{b} \), its multiplicative inverse (reciprocal) is \( \frac{b}{a} \), such that \( \frac{a}{b} \times \frac{b}{a} = 1 \).

Question 6. સંખ્યા \( \frac {6}{13} \) ને \( \frac {-7}{16} \) ના વ્યસ્ત વડે ગુણો.
Answer:
First, we need to find the reciprocal of \( \frac {-7}{16} \).
The reciprocal of \( \frac {-7}{16} \) is \( \frac {16}{-7} \), which is also \( -\frac {16}{7} \).
Now, we 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: We first flip the second fraction to find its reciprocal. Then, we multiply the first fraction by this flipped fraction, multiplying the top numbers together and the bottom numbers together.

Exam Tip: Be careful with negative signs when finding reciprocals and performing multiplication. A negative number multiplied by a positive number results in a negative product.

Question 7. \( \frac{1}{3} \times\left(6 \times \frac{4}{3}\right) \) ની \( \left(\frac{1}{3} \times 6\right) \times \frac{4}{3} \) રીતે ગણતરી કયા ગુણધર્મના ઉપયોગથી કરી શકાય તે જણાવો.
Answer:
To change the grouping of numbers in a multiplication problem from \( \frac{1}{3} \times\left(6 \times \frac{4}{3}\right) \) to \( \left(\frac{1}{3} \times 6\right) \times \frac{4}{3} \), we use the Associative Property of Multiplication.
In simple words: This rule means that when you multiply three or more numbers, you can group them differently without changing the final answer.

Exam Tip: The associative property states that for any numbers 'a', 'b', and 'c', \( a \times (b \times c) = (a \times b) \times c \). This property is fundamental for simplifying expressions.

Question 8. શું \( \frac {8}{9} \) એ સંખ્યા \( -1\frac {1}{8} \) નો વ્યસ્ત છે? કેમ અથવા કેમ નહીં?
Answer:
No, \( \frac {8}{9} \) is not the reciprocal of \( -1\frac {1}{8} \).
First, convert the mixed fraction \( -1\frac {1}{8} \) to an improper fraction:
\( -1\frac {1}{8} = -\frac{(1 \times 8) + 1}{8} = -\frac{9}{8} \)
For two numbers to be reciprocals of each other, their product must be 1.
Let's multiply \( \frac {8}{9} \) by \( -\frac{9}{8} \):
\( \frac {8}{9} \times \left(-\frac{9}{8}\right) = \frac{8 \times (-9)}{9 \times 8} = \frac{-72}{72} = -1 \)
Since the product is -1 and not 1, \( \frac {8}{9} \) is not the reciprocal of \( -1\frac {1}{8} \).
In simple words: To check if two numbers are reciprocals, you multiply them. If the answer is 1, they are. In this case, when we multiply them, we get negative one, not one, so they are not reciprocals.

Exam Tip: Always remember that the product of a number and its multiplicative inverse (reciprocal) must be exactly 1, not -1 or any other value.

Question 9. શું 0.3 એ \( 3\frac {1}{3} \) નો વ્યસ્ત છે? કેમ અથવા કેમ નહીં?
Answer:
Yes, 0.3 is the reciprocal of \( 3\frac {1}{3} \).
First, convert 0.3 to a fraction:
\( 0.3 = \frac{3}{10} \)
Next, convert the mixed fraction \( 3\frac {1}{3} \) to an improper fraction:
\( 3\frac {1}{3} = \frac{(3 \times 3) + 1}{3} = \frac{10}{3} \)
Now, multiply these two fractions:
\( \frac{3}{10} \times \frac{10}{3} = \frac{3 \times 10}{10 \times 3} = \frac{30}{30} = 1 \)
Since the product of 0.3 and \( 3\frac {1}{3} \) is 1, they are reciprocals of each other.
In simple words: We change the decimal and the mixed number into fractions. If multiplying these two fractions gives us one, then they are reciprocals. In this problem, they do multiply to one, so they are indeed reciprocals.

Exam Tip: When dealing with decimals or mixed numbers, convert them into improper fractions first to easily find their reciprocal or check relationships.

Question 10. લખોઃ
(i) એવી સંમેય સંખ્યા કે જેનો વ્યસ્ત ન હોય.
Answer:
(i) The rational number that does not have a reciprocal is 0 (zero).
In simple words: Zero is the only number that you cannot flip over to make a reciprocal because you cannot divide by zero.

Exam Tip: Remember that division by zero is undefined, so zero cannot have a multiplicative inverse (reciprocal).

Question 10. લખોઃ
(ii) એવી સંમેય સંખ્યાઓ કે જે તેના વ્યસ્તને સમાન હોય.
Answer:
(ii) The rational numbers that are equal to their own reciprocals are 1 and -1.
For example, the reciprocal of 1 is \( \frac{1}{1} = 1 \), and the reciprocal of -1 is \( \frac{1}{-1} = -1 \).
In simple words: Only the numbers one and negative one are the same as their flipped versions.

Exam Tip: This property is unique to 1 and -1. All other numbers have a reciprocal that is different from themselves.

Question 10. લખોઃ
(iii) એવી સંમેય સંખ્યા કે જે તેની વિરોધી સંખ્યાને સમાન હોય.
Answer:
(iii) The rational number that is equal to its own opposite is 0 (zero).
For example, the opposite of 0 is -0, which is also 0.
In simple words: The only number that is the same as its negative version is zero.

Exam Tip: Zero is the only number that is neither positive nor negative, and it serves as the additive identity, meaning \( a + 0 = a \).

Question 11. નીચેની ખાલી જગ્યા પૂરોઃ
(i) શૂન્યનો વ્યસ્ત ..........
Answer:
(i) શૂન્યનો વ્યસ્ત નથી.
In simple words: Zero has no reciprocal.

Exam Tip: This is a common point of confusion. Remember that division by zero is undefined.

Question 11. નીચેની ખાલી જગ્યા પૂરોઃ
(ii) સંખ્યાઓ .......... અને .......... પોતાના જ વ્યસ્ત છે.
Answer:
(ii) સંખ્યાઓ 1 અને -1 પોતાના જ વ્યસ્ત છે.
In simple words: The numbers 1 and -1 are special because they are their own reciprocals.

Exam Tip: These are the only two numbers whose reciprocal equals themselves. All other numbers have a different reciprocal.

Question 11. નીચેની ખાલી જગ્યા પૂરોઃ
(ii) -5ની વ્યસ્ત સંખ્યા ........ છે.
Answer:
(ii) -5ની વ્યસ્ત સંખ્યા \( \underline{\frac {-1}{5}} \) છે.
In simple words: To find the reciprocal of -5, simply write it as a fraction with 1 on top and -5 on the bottom.

Exam Tip: The reciprocal of any non-zero integer 'n' is always \( \frac{1}{n} \). If 'n' is negative, its reciprocal is also negative.

Question 11. નીચેની ખાલી જગ્યા પૂરોઃ
(iv) \( \frac{1}{x} \) ની વ્યસ્ત સંખ્યા .........., કે જ્યાં \( x \neq 0 \).
Answer:
(iv) \( \frac{1}{x} \) ની વ્યસ્ત સંખ્યા \( x \) છે, કે જ્યાં \( x \neq 0 \).
In simple words: If you have a fraction with 1 on top and 'x' on the bottom, its reciprocal is just 'x'.

Exam Tip: When finding the reciprocal of a reciprocal, you get back the original number. The condition \( x \neq 0 \) is crucial because division by zero is undefined.

Question 11. નીચેની ખાલી જગ્યા પૂરોઃ
(v) બે સંમેય સંખ્યાનો ગુણાકાર હંમેશાં .......... જ હોય.
Answer:
(v) બે સંમેય સંખ્યાનો ગુણાકાર હંમેશાં સંમેય સંખ્યા જ હોય.
In simple words: If you multiply two numbers that can be written as fractions, your answer will always be another number that can also be written as a fraction.

Exam Tip: This property is called the closure property of rational numbers under multiplication. Rational numbers are closed under addition, subtraction, and multiplication.

Question 11. નીચેની ખાલી જગ્યા પૂરોઃ
(vi) ધન સંમેય સંખ્યાની વ્યસ્ત સંખ્યા .......... હોય.
Answer:
(vi) ધન સંમેય સંખ્યાની વ્યસ્ત સંખ્યા ધન હોય.
In simple words: If you have a positive number that can be written as a fraction, its reciprocal will also be a positive number.

Exam Tip: The reciprocal of a positive number is always positive, and the reciprocal of a negative number is always negative. The sign does not change when finding the reciprocal.