GSEB Class 7 Maths Solutions Chapter 9 સંમેય સંખ્યાઓ InText Questions

પ્રયત્ન કરો: (પાઠ્યપુસ્તક પાન નંબર. 174)

Question 1. શું \( \frac {2}{-3} \) એ સંમેય સંખ્યા છે? એના વિશે વિચાર કરો.
Answer: હા, \( \frac {2}{-3} \) એક સંમેય સંખ્યા છે. આનું કારણ છે કે 2 અને -3 બંને પૂર્ણાંકો છે, અને છેદ -3 શૂન્ય નથી.
In simple words: Yes, \( \frac {2}{-3} \) is a rational number because both numbers are integers and the bottom number is not zero.

Exam Tip: To determine if a number is rational, always check if it can be expressed as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \).

Question 2. દસ સંમેય સંખ્યાઓની યાદી બનાવો.
Answer: સંમેય સંખ્યાઓના કેટલાક ઉદાહરણોમાં \( \frac{1}{2} \), \( \frac{-3}{5} \), \( \frac{6}{7} \), \( \frac{1}{-2} \), \( \frac{-2}{5} \), \( \frac{-1}{-4} \), \( 2.5 \), \( 3\frac{1}{2} \), \( 0.09 \), અને \( 0.18 \) નો સમાવેશ થાય છે. આ બધી સંખ્યાઓને \( \frac{p}{q} \) અપૂર્ણાંક તરીકે લખી શકાય છે, જ્યાં \( q \neq 0 \) છે.
In simple words: Rational numbers are numbers that can be written as a fraction. Here are ten examples: \( \frac{1}{2}, \frac{-3}{5}, \frac{6}{7}, \frac{1}{-2}, \frac{-2}{5}, \frac{-1}{-4}, 2.5, 3\frac{1}{2}, 0.09, 0.18 \).

Exam Tip: Remember that all integers, fractions, and terminating or repeating decimals are rational numbers.

Question 1. ખાલી જગ્યા પૂરોઃ
(i) \( \frac{5}{4} = \frac{25}{16} = \frac{}{-15} \)
(ii) \( \frac{-3}{7} = \frac{9}{14} = \frac{}{-6} \)
Answer:
(i) \( \frac{5}{4} = \frac{20}{16} = \frac{25}{20} = \frac{-15}{-12} \)
(ii) \( \frac{-3}{7} = \frac{9}{-21} = \frac{-6}{14} \)
In simple words: For equivalent fractions, multiply or divide both the top and bottom numbers by the same non-zero number to fill in the missing parts correctly.

Exam Tip: When finding equivalent fractions, ensure you multiply or divide both the numerator and denominator by the exact same number to keep the value of the fraction unchanged.

પ્રયત્ન કરો: (પાઠ્યપુસ્તક પાન નંબર. 175)

Question 1. શું 5 એ ધન સંમેય સંખ્યા છે? હા, 5 એ ધન સંમેય સંખ્યા છે.
Answer: 5 ને \( \frac {5}{1} \) તરીકે લખી શકાય છે. અહીં, અંશ (5) અને છેદ (1) બંને ધન છે. તેથી, 5 એક ધન સંમેય સંખ્યા છે.
In simple words: Yes, 5 is a positive rational number because you can write it as \( \frac{5}{1} \), where both the top and bottom numbers are positive.

Exam Tip: Any positive integer can be expressed as a positive rational number by placing it over 1, e.g., \( n = \frac{n}{1} \).

Question 2. પાંચ ધન સંમેય સંખ્યાઓની યાદી બનાવો.
Answer: પાંચ ધન સંમેય સંખ્યાઓ છે: \( \frac{1}{2}, \frac{3}{7}, \frac{11}{17}, \frac{9}{23}, \frac{4}{13} \).
In simple words: Five examples of positive rational numbers are \( \frac{1}{2}, \frac{3}{7}, \frac{11}{17}, \frac{9}{23}, \frac{4}{13} \).

Exam Tip: Positive rational numbers have both their numerator and denominator (when in simplest form) having the same sign (either both positive or both negative).

પ્રયત્ન કરો: (પાઠ્યપુસ્તક પાન નંબર. 176)

Question 1. શું – 8 એ ઋણ સંમેય સંખ્યા છે? હા, – 8 એ ઋણ સંમેય સંખ્યા છે.
Answer: -8 ને \( \frac {-8}{1} \) તરીકે લખી શકાય છે. \( \frac {-8}{1} \) નો અંશ (-8) ઋણ છે, અને તેથી તે એક ઋણ સંમેય સંખ્યા છે.
In simple words: Yes, -8 is a negative rational number because it can be written as \( \frac{-8}{1} \), which has a negative number on top.

Exam Tip: A rational number is negative if its numerator and denominator have different signs when expressed as \( \frac{p}{q} \).

Question 2. પાંચ ઋણ સંમેય સંખ્યાઓની યાદી બનાવો.
Answer: પાંચ ઋણ સંમેય સંખ્યાઓ છે: \( \frac{-3}{7}, \frac{-8}{13}, \frac{-11}{17}, \frac{2}{-5}, \frac{4}{-9} \).
In simple words: Here are five negative rational numbers: \( \frac{-3}{7}, \frac{-8}{13}, \frac{-11}{17}, \frac{2}{-5}, \frac{4}{-9} \).

Exam Tip: When identifying negative rational numbers, look for one negative sign, either in the numerator or denominator. For example, \( \frac{2}{-5} \) is equivalent to \( \frac{-2}{5} \).

પ્રયત્ન કરો : (પાઠ્યપુસ્તક પાન નંબર. 176)

Question. કઈ સંખ્યાઓ ઋણ સંમેય સંખ્યાઓ છે?
(ii) \( \frac {5}{7} \)
(iii) \( \frac {3}{-5} \)
(iv) 0
(v) \( \frac {6}{11} \)
(vi) \( \frac {-2}{-9} \)
Answer:
(i) \( \frac {-2}{3} \) એ ઋણ સંમેય સંખ્યા છે.
(ii) \( \frac {5}{7} \) એ ધન સંમેય સંખ્યા છે.
(iii) \( \frac {3}{-5} \) એ ઋણ સંમેય સંખ્યા છે.
(iv) 0 એ ઋણ સંમેય સંખ્યા કે ધન સંમેય સંખ્યા નથી.
(v) \( \frac {6}{11} \) એ ધન સંમેય સંખ્યા છે.
(vi) \( \frac {-2}{-9} \) એટલે કે, \( \frac {2}{9} \) થાય, જે ધન સંમેય સંખ્યા છે.
In simple words: A rational number is negative if it has one minus sign (either top or bottom). So, \( \frac{-2}{3} \) and \( \frac{3}{-5} \) are negative. Zero is neither positive nor negative. \( \frac{-2}{-9} \) becomes \( \frac{2}{9} \), which is positive.

Exam Tip: Zero is a rational number, but it is neither positive nor negative. A fraction with two negative signs (e.g., \( \frac{-a}{-b} \)) simplifies to a positive fraction (e.g., \( \frac{a}{b} \)).

પ્રયત્ન કરો (પાઠ્યપુસ્તક પાન નંબર. 178)

Question. પ્રશ્ન (i) \( \frac {-18}{45} \)
Answer:
\( = \frac{-18 \div 9}{45 \div 9} \) [: 18 અને 45નો ગુ.સા.અ. 9 છે.]
\( = \frac {-2}{5} \)
In simple words: To simplify this fraction, we divide both the top and bottom numbers by their biggest common factor, which is 9. This gives us the simplest form.

Exam Tip: Always find the greatest common divisor (GCD) to simplify fractions to their lowest terms. This ensures no further reduction is possible.

Question. પ્રશ્ન (ii) \( \frac {-12}{18} \) નું પ્રમાણિત રૂપ મેળવો.
Answer:
\( = \frac{-12 \div 6}{18 \div 6} \) [: 12 અને 18નો ગુ.સા.અ. 6 છે.]
\( = \frac {-2}{3} \)
In simple words: To write this fraction in its standard (simplest) form, we divide both numbers by their highest shared factor, which is 6.

Exam Tip: The "standard form" of a rational number means its simplest form, where the numerator and denominator have no common factors other than 1, and the denominator is positive.

પ્રયત્ન કરો (પાઠ્યપુસ્તક પાન નંબર. 181)

Question 1. \( \frac {-5}{7} \) અને \( \frac {-3}{8} \) ની વચ્ચે આવતી પાંચ સંમેય સંખ્યાઓ શોધો.
Answer: આપણે પહેલા \( \frac{-5}{7} \) અને \( \frac{-3}{8} \) ને સમાન છેદવાળા અપૂર્ણાંકોમાં ફેરવીએ છીએ.
7 અને 8 નો લઘુત્તમ સામાન્ય અવયવી (લ.સા.અ.) 56 છે.
\( \frac{-5}{7} = \frac{-5 \times 8}{7 \times 8} = \frac{-40}{56} \)
\( \frac{-3}{8} = \frac{-3 \times 7}{8 \times 7} = \frac{-21}{56} \)
હવે, \( -40 < -39 < -38 < -37 < -36 < -35 < \ldots < -21 \)
તેથી, \( \frac{-39}{56} < \frac{-38}{56} < \frac{-37}{56} < \frac{-36}{56} < \frac{-35}{56} < \ldots < \frac{-21}{56} \)
આમ, \( \frac{-5}{7} \) અને \( \frac{-3}{8} \) ની વચ્ચેની પાંચ સંમેય સંખ્યાઓ \( \frac{-39}{56}, \frac{-38}{56}, \frac{-37}{56}, \frac{-36}{56} \) અને \( \frac{-35}{56} \) છે.
In simple words: To find rational numbers between two fractions, first make their bottom numbers (denominators) the same using the lowest common multiple. Then, find integers between the new top numbers. Convert these into fractions to get the numbers in between.

Exam Tip: When finding rational numbers between two given rational numbers, it is often helpful to convert them to equivalent fractions with a common denominator. If there are not enough integers between the numerators, multiply both fractions by a suitable factor (like 10/10) to create more space.

પ્રયત્ન કરો (પાઠ્યપુસ્તક પાન નંબર. 185)

Question 1. શોધોઃ
(i) \( \frac{-13}{7}+\frac{6}{7} \)
(ii) \( \frac{19}{5}+\frac{-7}{5} \)
Answer:
(i) \( \frac{-13}{7}+\frac{6}{7} = \frac{-13+6}{7} = \frac{-7}{7} = -1 \)
(ii) \( \frac{19}{5}+\frac{-7}{5} = \frac{19+(-7)}{5} = \frac{12}{5} = 2\frac{2}{5} \)
In simple words: When adding fractions that already have the same bottom number (denominator), simply add the top numbers (numerators) together and keep the bottom number the same. Then, simplify the result.

Exam Tip: Remember that adding or subtracting fractions only requires finding a common denominator when the original denominators are different. If they are the same, combine the numerators directly.

પ્રયત્ન કરો (પાઠ્યપુસ્તક પાન નંબર. 185)

Question 1. શોધોઃ
(i) \( \frac{-3}{7}+\frac{2}{3} \)
(ii) \( \frac{-5}{6}+\frac{-3}{11} \)
Answer:
(i) \( \frac{-3}{7}+\frac{2}{3} \)
અહીં 7 અને 3 નો લ.સા.અ. 21 છે.
\( \frac{-3}{7} = \frac{(-3) \times 3}{7 \times 3} = \frac{-9}{21} \)
\( \frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21} \)
હવે, \( \frac{-3}{7} + \frac{2}{3} = \frac{-9}{21} + \frac{14}{21} = \frac{-9+14}{21} = \frac{5}{21} \)
(ii) \( \frac{-5}{6}+\frac{-3}{11} \)
અહીં 6 અને 11 નો લ.સા.અ. 66 છે.
\( \frac{-5}{6} = \frac{(-5) \times 11}{6 \times 11} = \frac{-55}{66} \)
\( \frac{-3}{11} = \frac{(-3) \times 6}{11 \times 6} = \frac{-18}{66} \)
હવે, \( \frac{-5}{6} + \frac{-3}{11} = \frac{-55}{66} + \frac{-18}{66} = \frac{-55-18}{66} = \frac{-73}{66} \)
In simple words: When adding fractions with different bottom numbers, first find the lowest common multiple for the bottom numbers. Then, change each fraction to have this new common bottom number. Finally, add the top numbers and keep the common bottom number.

Exam Tip: Always find the least common multiple (LCM) of the denominators to simplify calculations when adding or subtracting fractions with different denominators. This keeps the numbers smaller and easier to manage.

પ્રયત્ન કરો (પાઠ્યપુસ્તક પાન નંબર. 186)

Question 1. \( \frac{-3}{9}, \frac{-9}{11} \) અને \( \frac {5}{7} \) નો વિરોધી ઘટક શું થશે?
Answer:
\( \frac {-3}{9} \) નો વિરોધી ઘટક \( \frac {3}{9} \) છે.
\( \frac {-9}{11} \) નો વિરોધી ઘટક \( \frac {9}{11} \) છે.
\( \frac {5}{7} \) નો વિરોધી ઘટક \( \frac {-5}{7} \) છે.
In simple words: The additive inverse of a rational number is the number that, when added to it, gives zero. You simply change the sign of the number. If it's positive, it becomes negative; if negative, it becomes positive.

Exam Tip: The additive inverse of a number 'a' is '-a'. For a fraction \( \frac{p}{q} \), its additive inverse is \( -\frac{p}{q} \) or \( \frac{-p}{q} \) or \( \frac{p}{-q} \).

પ્રયત્ન કરો (પાઠ્યપુસ્તક પાન નંબર. 187)

Question 1. શોધોઃ
(i) \( \frac{7}{9}-\frac{2}{5} \)
(ii) \( 2 \frac{1}{5}-\frac{-1}{3} \)
Answer:
(i) \( \frac{7}{9}-\frac{2}{5} \)
9 અને 5 નો લ.સા.અ. 45 છે.
\( \frac{7}{9} - \frac{2}{5} = \frac{(7 \times 5)-(2 \times 9)}{45} \)
\( = \frac{35-18}{45} = \frac{17}{45} \)
(ii) \( 2\frac{1}{5}-\frac{-1}{3} \)
5 અને 3 નો લ.સા.અ. 15 છે.
\( 2\frac{1}{5} - \frac{-1}{3} = \frac{11}{5} + \frac{1}{3} \)
\( = \frac{(11 \times 3) + (1 \times 5)}{15} \)
\( = \frac{33+5}{15} = \frac{38}{15} = 2\frac{8}{15} \)
In simple words: To subtract fractions, first ensure they have the same bottom number. If not, find the lowest common multiple and change the fractions accordingly. Then, subtract the top numbers and simplify the result. Remember that subtracting a negative number is the same as adding a positive number.

Exam Tip: Convert mixed numbers to improper fractions before performing addition or subtraction. Also, \( A - (-B) \) always simplifies to \( A + B \).

પ્રયત્ન કરો: (પાઠ્યપુસ્તક પાન નંબર. 188)

Question 1. જવાબ શો આવી શકે?
Answer: બે સંમેય સંખ્યાઓનો ગુણાકાર હંમેશા સંમેય સંખ્યા જ હોય ​​છે. તેથી, ગુણાકારનો જવાબ પણ એક સંમેય સંખ્યા જ આવશે.
In simple words: When you multiply two rational numbers, the answer will always be another rational number. It's a fundamental property of rational numbers.

Exam Tip: The set of rational numbers is closed under multiplication, meaning the product of any two rational numbers is always a rational number.

Question. પ્રશ્ન 1. \( \frac {-3}{5} \times 7 \)
Answer:
\( = \frac{(-3) \times 7}{5} = \frac {-21}{5} = -4\frac {1}{5} \)
In simple words: To multiply a fraction by a whole number, you just multiply the top number (numerator) of the fraction by the whole number. The bottom number (denominator) stays the same.

Exam Tip: Remember that any whole number 'n' can be written as \( \frac{n}{1} \), making it easier to visualize multiplication with fractions: \( \frac{a}{b} \times n = \frac{a}{b} \times \frac{n}{1} = \frac{a \times n}{b \times 1} \).

Question. પ્રશ્ન 2. \( \frac {-6}{5} \times (-2) \)
Answer:
\( = \frac{-6 \times (-2)}{5} = \frac {12}{5} = 2\frac {2}{5} \)
In simple words: When multiplying a fraction by a negative whole number, multiply the top number by the whole number. Remember that a negative number multiplied by a negative number gives a positive result.

Exam Tip: Pay close attention to the signs when multiplying. A negative times a negative yields a positive, while a negative times a positive yields a negative.

પ્રયત્ન કરો (પાઠ્યપુસ્તક પાન નંબર. 188)

Question 1. શોધોઃ \( \frac{-3}{4} \times \frac{1}{7} \)
Answer:
\( = \frac{(-3) \times 1}{4 \times 7}=\frac{-3}{28} \)
In simple words: To multiply two fractions, multiply the top numbers together and the bottom numbers together.

Exam Tip: Always simplify fractions before multiplying if possible, as it makes the numbers smaller and reduces the chance of errors in calculation.

Question. પ્રશ્ન 2. \( \frac{2}{3} \times \frac{-5}{9} \)
Answer:
\( = \frac{2 \times(-5)}{3 \times 9}=\frac{-10}{27} \)
In simple words: Multiply the numerators (top numbers) together and the denominators (bottom numbers) together. Remember that a positive number multiplied by a negative number results in a negative number.

Exam Tip: When multiplying fractions, there is no need to find a common denominator. Simply multiply straight across (numerator by numerator, denominator by denominator).

પ્રયત્ન કરો (પાઠ્યપુસ્તક પાન નંબર. 189)

Question 1. \( \frac {-6}{11} \) અને \( \frac {-8}{5} \) ની વ્યસ્ત સંખ્યા શું થશે?
Answer:
\( \frac {-6}{11} \) ની વ્યસ્ત સંખ્યા \( \frac {-11}{6} \) છે.
અને \( \frac {-8}{5} \) ની વ્યસ્ત સંખ્યા \( \frac {-5}{8} \) છે.
In simple words: To find the reciprocal (or multiplicative inverse) of a fraction, simply flip the fraction upside down. The numerator becomes the denominator, and the denominator becomes the numerator.

Exam Tip: The product of a rational number and its reciprocal is always 1. For example, \( \frac{a}{b} \times \frac{b}{a} = 1 \).

પ્રયત્ન કરો (પાઠ્યપુસ્તક પાન નંબર. 190)

Question 1. શોધોઃ
(i) \( \frac{2}{3} \times \frac{-7}{8} \)
(ii) \( \frac{-6}{7} \times \frac{5}{7} \)
Answer:
(i) \( \frac{2}{3} \times \frac{-7}{8} \)
\( = \frac{2 \times (-7)}{3 \times 8} \)
\( = \frac{-14}{24} \)
\( = \frac{-14 \div 2}{24 \div 2} \)
\( = \frac{-7}{12} \)
(ii) \( \frac{-6}{7} \times \frac{5}{7} \)
\( = \frac{(-6) \times 5}{7 \times 7} \)
\( = \frac{-30}{49} \)
In simple words: To multiply fractions, multiply the top numbers together and the bottom numbers together. Always simplify the final fraction to its simplest form by dividing both the numerator and denominator by their greatest common factor.

Exam Tip: Before multiplying, look for common factors between any numerator and any denominator to cross-cancel and simplify the calculation earlier. This can help prevent dealing with larger numbers.