GSEB Class 7 Maths Solutions Chapter 9 Rational Numbers InText Questions

Try These (Page 174)

Question 1. Is the number \( \frac {2}{-3} \) rational? Think about it.
Answer: Yes, \( \frac {2}{-3} \) is a rational number. This is because 2 and -3 are integers, and the denominator, -3, is not equal to zero.
In simple words: Yes, \( \frac {2}{-3} \) is a rational number because both the top and bottom numbers are whole numbers, and the bottom number isn't zero.

Exam Tip: Remember, a rational number can always be written as a fraction \( \frac{p}{q} \), where p and q are integers and q is not zero.

Question 2. List ten rational numbers.
Answer: The following are ten examples of rational numbers: \( \frac { 1 }{ 3 } \), \( \frac { 2 }{ -3 } \), \( \frac { 4 }{ 5 } \), \( \frac { 1 }{ -6 } \), \( \frac { -3 }{ -4 } \), 5.8, \( 2\frac { 4 }{ 5 } \), 0.93, 18, and 11.07.
In simple words: Here are ten rational numbers: one-third, two over negative three, four-fifths, one over negative six, negative three over negative four, 5.8, two and four-fifths, 0.93, eighteen, and 11.07.

Exam Tip: Rational numbers include integers, fractions, terminating decimals, and repeating decimals. Just make sure they can be written as \( \frac{p}{q} \).

Try These (Page 175)

Question 1. Fill in the boxes:
(i) \( \frac { 5 }{ 4 } = \frac { 25 }{ 16 } = \frac { -15 }{ \square } \)
(ii) \( \frac { -3 }{ 7 } = \frac { \square }{ 14 } = \frac { 9 }{ \square } = \frac { -6 }{ \square } \)
Answer:
(i) \( \frac { 5 }{ 4 } = \frac { 25 }{ 16 } = \frac { -15 }{ -12 } \)
(ii) \( \frac { -3 }{ 7 } = \frac { -6 }{ 14 } = \frac { 9 }{ -21 } = \frac { -6 }{ 14 } \)
In simple words: To find the missing numbers, multiply or divide the numerator and denominator by the same amount to keep the fraction equivalent. For example, for 5/4, to get 25, multiply 5 by 5, so you also multiply 4 by 5 to get 20, then for -15, multiply 5 by -3, so you multiply 4 by -3 to get -12.

Exam Tip: When filling boxes in equivalent fractions, make sure to apply the same operation (multiplication or division) to both the numerator and the denominator.

Try These (Page 175)

Question 1. Is 5 a positive rational number?
Answer: Yes, 5 (or \( \frac {5}{ 1 } \)) is a positive rational number. Its numerator and denominator are both positive values, which makes the whole number positive.
In simple words: Yes, 5 is a positive rational number because it can be written as 5/1, and both 5 and 1 are positive.

Exam Tip: Any positive integer can be expressed as a positive rational number by writing it with a denominator of 1.

Question 2. List five more positive rational numbers.
Answer: Here are five more positive rational numbers: \( \frac { 1 }{ 7 } \), \( \frac { 3 }{ 8 } \), \( \frac { 5 }{ 17 } \), \( \frac {2}{9} \), and \( \frac {5}{18} \).
In simple words: Five more positive rational numbers are: one-seventh, three-eighths, five-seventeenths, two-ninths, and five-eighteenths.

Exam Tip: Positive rational numbers have either both positive numerator and denominator or both negative numerator and denominator (which simplifies to a positive fraction).

Question 1. Is – 8 a negative rational number?
Answer: Yes, -8 (or \( \frac { -8 }{ 1 } \)) is a negative rational number. This is because its numerator is a negative integer, making the entire fraction negative.
In simple words: Yes, -8 is a negative rational number because the number itself is negative.

Exam Tip: A rational number is negative if either the numerator or the denominator (but not both) is a negative integer.

Question 2. List five more negative rational numbers.
Answer: The following are five negative rational numbers: \( \frac {-5}{9 } \), \( \frac { -6 }{ 11 } \), \( \frac { -3 }{ 13 } \), \( \frac {3}{-10 } \), and \( \frac {-1}{7} \).
In simple words: Here are five more negative rational numbers: negative five-ninths, negative six-elevenths, negative three-thirteenths, three over negative ten, and negative one-seventh.

Exam Tip: To create a negative rational number, ensure one part (numerator or denominator) is negative and the other is positive.

Try These (Page 176)

Question 1. Which of these are negative rational numbers?
(i) \( \frac {-2}{ 3 } \)
(ii) \( \frac {5}{7} \)
(iii) \( \frac { 3 }{ -5 } \)
(iv) \( 0 \)
(v) \( \frac { 6 }{ 11 } \)
(vi) \( \frac {-2}{ -9 } \)
Answer:
(i) \( \frac { -2 }{ 3 } \) is a negative rational number.
(ii) \( \frac { 5 }{ 7 } \) is a positive rational number.
(iii) \( \frac { 3 }{ -5 } \) is a negative rational number.
(iv) \( 0 \) is neither a positive nor a negative rational number.
(v) \( \frac { 6 }{ 11 } \) is a positive rational number.
(vi) \( \frac { -2 }{ -9 } \) is a positive rational number.
Thus, (i) \( \frac { -2 }{ 3 } \) and (iii) \( \frac { 3 }{ -5 } \) are the negative rational numbers from the list.
In simple words: Negative rational numbers have one negative part and one positive part. Zero is special and is neither positive nor negative. If both parts are negative, the number becomes positive. So, only (i) and (iii) are negative.

Exam Tip: A rational number is negative if the signs of its numerator and denominator are different (one positive, one negative). If the signs are the same, it's a positive rational number.

Try These (Page 178)

Question 1. Find the standard form of:
(i) \( \frac { -18 }{ 45 } \)
(ii) \( \frac {-12 }{ 18 } \)
Answer:
(i) Since the HCF (Highest Common Factor) of 18 and 45 is 9.
\( \frac { -18 }{ 45 } = \frac { (-18) \div 9 }{ 45 \div 9 } = \frac { -2 }{ 5 } \)
Thus, the standard form of \( \frac { -18 }{ 45 } \) is \( \frac { -2 }{ 5 } \).
(ii) Since, the HCF of 12 and 18 is 6.
\( \frac { -12 }{ 18 } = \frac { (-12) \div 6 }{ 18 \div 6 } = \frac { -2 }{ 3 } \)
Thus, the standard form of \( \frac { -12 }{ 18 } \) is \( \frac { -2 }{ 3 } \).
In simple words: To write a fraction in standard form, divide both the top and bottom numbers by their biggest common factor until you cannot divide them anymore. Make sure the denominator is positive.

Exam Tip: The standard form of a rational number means it is in its simplest form, and its denominator is always a positive integer.

Try These (Page 181)

Question 1. Find five rational numbers between \( \frac { -5}{7} \) and \( \frac { -3 }{ 8 } \).
Answer: First, we convert the given rational numbers to have common denominators.
The LCM (Least Common Multiple) of 7 and 8 is 56.
So, \( \frac { -5 }{ 7 } = \frac { (-5) \times 8 }{ 7 \times 8 } = \frac { -40 }{ 56 } \)
And \( \frac { -3 }{ 8 } = \frac { (-3) \times 7 }{ 8 \times 7 } = \frac { -21 }{ 56 } \)
Now, we have: \( \frac { -40 }{ 56 } < \frac { -39 }{ 56 } < \frac { -38 }{ 56 } < \frac { -37 }{ 56 } < \frac { -36 }{ 56 } < \frac { -35 }{ 56 } < \frac { -21 }{ 56 } \)
Therefore, five rational numbers between \( \frac { -5 }{ 7 } \) and \( \frac { -3 }{ 8 } \) are: \( \frac { -39 }{ 56 } \), \( \frac { -38 }{ 56 } \), \( \frac { -37 }{ 56 } \), \( \frac { -36 }{ 56 } \), \( \frac { -35 }{ 56 } \).
These can also be simplified as: \( \frac { -39 }{ 56 } \), \( \frac { -19 }{ 28 } \), \( \frac { -37 }{ 56 } \), \( \frac { -9 }{ 14 } \), \( \frac { -5 }{ 8 } \).
In simple words: To find rational numbers between two fractions, first make their bottom numbers the same. Then, you can easily find numbers with numerators between the two original numerators.

Exam Tip: When finding rational numbers between two given numbers, always convert them to equivalent fractions with a common denominator first. If there aren't enough integers between the numerators, multiply both fractions by a larger common factor.

Try These (Page 185)

Question 1. Find:
(i) \( \frac { -13 }{7} + \frac { 6 }{ 7 } \)
(ii) \( \frac { 19 }{5} + \frac { -7 }{ 5 } \)
Answer:
(i) \( \frac { -13 }{ 7 } + \frac { 6 }{ 7 } = \frac { -13+6 }{ 7 } = \frac { -7 }{ 7 } = -1 \)
Thus, \( \frac { -13 }{ 7 } + \frac { 6 }{ 7 } = -1 \).
(ii) \( \frac { 19 }{ 5 } + \frac { -7 }{ 5 } = \frac { 19+(-7) }{ 5 } = \frac { 12 }{ 5 } \)
Thus, \( \frac { 19 }{ 5 } + \frac { -7 }{ 5 } = \frac { 12 }{ 5 } = 2\frac{2}{5} \).
In simple words: When adding fractions with the same bottom number, simply add the top numbers and keep the bottom number the same. Then simplify the answer if possible.

Exam Tip: For fractions with the same denominator, remember to just add or subtract the numerators and keep the denominator unchanged. Simplify the result if it's an improper fraction.

Question 2. Find:
(i) \( \frac { -3 }{ 7 } + \frac { 2 }{ 3 } \)
(ii) \( \frac { -5 }{ 6 } + \frac { -3 }{ 11 } \)
Answer:
(i) For \( \frac { -3 }{ 7 } + \frac { 2 }{ 3 } \):
The LCM (Least Common Multiple) of 7 and 3 is 21.
So, \( \frac { -3 }{ 7 } = \frac { (-3) \times 3 }{ 7 \times 3 } = \frac { -9 }{ 21 } \)
And \( \frac { 2 }{ 3 } = \frac { 2 \times 7 }{ 3 \times 7 } = \frac { 14 }{ 21 } \)
Therefore, \( \frac { -3 }{ 7 } + \frac { 2 }{ 3 } = \frac { -9 }{ 21 } + \frac { 14 }{ 21 } = \frac { -9+14 }{ 21 } = \frac { 5 }{ 21 } \).
(ii) For \( \frac { -5 }{ 6 } + \frac { -3 }{ 11 } \):
The LCM of 6 and 11 is 66.
So, \( \frac { -5 }{ 6 } = \frac { (-5) \times 11 }{ 6 \times 11 } = \frac { -55 }{ 66 } \)
And \( \frac { -3 }{ 11 } = \frac { (-3) \times 6 }{ 11 \times 6 } = \frac { -18 }{ 66 } \)
Now, \( \frac { -5 }{ 6 } + \frac { -3 }{ 11 } = \frac { -55 }{ 66 } + \frac { -18 }{ 66 } = \frac { (-55)+(-18) }{ 66 } = \frac { -73 }{ 66 } \).
In simple words: When adding fractions with different bottom numbers, first find the smallest common bottom number. Change both fractions to use this common bottom number, then add the top numbers.

Exam Tip: To add or subtract rational numbers with different denominators, always find their LCM to get a common denominator before performing the operation.

Try These (Page 186)

Question 1. What will be the additive inverse of \( \frac { -3 }{ 9 } \)? \( \frac { -9 }{ 11 } \)? \( \frac { 5 }{ 7 } \)?
Answer:
The additive inverse of \( \frac { -3 }{ 9 } \) is \( \frac { 3 }{ 9 } \).
The additive inverse of \( \frac { -9 }{ 11 } \) is \( \frac { 9 }{ 11 } \).
The additive inverse of \( \frac { 5 }{ 7 } \) is \( \frac { -5 }{ 7 } \).
In simple words: The additive inverse of a number is simply that number with its sign changed. If it's negative, it becomes positive; if it's positive, it becomes negative.

Exam Tip: The additive inverse of any rational number \( \frac{p}{q} \) is \( -\frac{p}{q} \), such that their sum is always zero.

Try These (Page 187)

Question 1. Find:
(i) \( \frac {7}{9 } – \frac { 2 }{ 5 } \)
(ii) \( 2\frac { 1 }{ 5 } – \frac { -1 }{ 3 } \)
Answer:
(i) We have \( \frac {7}{9 } – \frac { 2 }{ 5 } \).
The LCM of 9 and 5 is 45.
\( \frac {7}{9 } – \frac { 2 }{ 5 } = \frac { (7 \times 5) - (2 \times 9) }{ 45 } = \frac { 35 - 18 }{ 45 } = \frac { 17 }{ 45 } \).
(ii) We have \( 2\frac { 1 }{ 5 } – \frac { -1 }{ 3 } \). First, convert the mixed number: \( 2\frac { 1 }{ 5 } = \frac { (2 \times 5) + 1 }{ 5 } = \frac { 11 }{ 5 } \).
Now the expression is \( \frac { 11 }{ 5 } – \frac { -1 }{ 3 } \).
The LCM of 5 and 3 is 15.
\( \frac { 11 }{ 5 } – \frac { -1 }{ 3 } = \frac { (11 \times 3) - ((-1) \times 5) }{ 15 } = \frac { 33 - (-5) }{ 15 } = \frac { 33+5 }{ 15 } = \frac { 38 }{ 15 } \).
This can also be written as a mixed number: \( 2\frac{8}{15} \).
In simple words: To subtract fractions, first get a common bottom number. If you have a mixed number, change it to an improper fraction first. Then, subtract the top numbers.

Exam Tip: When dealing with subtraction of rational numbers, especially with negative signs, be very careful with the double negative, which becomes a positive (e.g., \( -(-1) = +1 \)).

Try These (Page 188)

Question 1. What will be:
(i) \( \frac { -3 }{ 5 } \times 7 \)?
(ii) \( \frac {-6}{ 5 } \times (-2) \)?
Answer:
(i) \( \frac { -3 }{ 5 } \times 7 = \frac { (-3) \times 7 }{ 5 } = \frac { -21 }{ 5 } \).
(ii) \( \frac { -6 }{ 5 } \times (-2) = \frac { (-6) \times (-2) }{ 5 } = \frac { 12 }{ 5 } \).
In simple words: To multiply a fraction by a whole number, just multiply the top number (numerator) of the fraction by the whole number. Keep the bottom number (denominator) the same. Remember that multiplying two negative numbers gives a positive result.

Exam Tip: When multiplying a rational number by an integer, remember to treat the integer as a fraction with a denominator of 1 (e.g., \( 7 = \frac{7}{1} \)).

Try These (Page 188)

Question 1. Find:
(i) \( \frac { -3 }{ 4 } \times \frac {1}{7} \)
(ii) \( \frac { 2 }{ 3 } \times \frac { -5 }{ 9 } \)
Answer:
(i) \( \frac { -3 }{ 4 } \times \frac { 1 }{ 7 } = \frac { (-3) \times 1 }{ 4 \times 7 } = \frac { -3 }{ 28 } \).
(ii) \( \frac { 2 }{ 3 } \times \frac { -5 }{ 9 } = \frac { 2 \times (-5) }{ 3 \times 9 } = \frac { -10 }{ 27 } \).
In simple words: To multiply fractions, you simply multiply the top numbers together and then multiply the bottom numbers together.

Exam Tip: Always look for opportunities to simplify fractions diagonally or vertically before multiplying to make calculations easier.

Try These (Page 189)

Question 1. What will be the reciprocal of \( \frac {-6}{11} \) and \( \frac { -8 }{ 5 } \)?
Answer:
The reciprocal of \( \frac {-6}{11} \) is \( \frac { 11 }{ -6 } \).
The reciprocal of \( \frac { -8 }{ 5 } \) is \( \frac { -5 }{ 8 } \).
In simple words: The reciprocal of a fraction is found by flipping the fraction upside down. The numerator becomes the denominator, and the denominator becomes the numerator.

Exam Tip: When finding the reciprocal, only flip the fraction; do not change the sign. The product of a rational number and its reciprocal is always 1.

Try These (Page 190)

Question 1. Find:
(i) \( \frac {2}{ 3 } \times \frac { -7 }{ 8 } \)
(ii) \( \frac { -6 }{ 7 } \times \frac { 5 }{ 7 } \)
Answer:
(i) \( \frac { 2 }{ 3 } \times \frac { -7 }{ 8 } = \frac { 2 \times (-7) }{ 3 \times 8 } = \frac { -14 }{ 24 } \)
Simplifying by dividing both by 2, we get \( \frac { -7 }{ 12 } \).
(ii) \( \frac { -6 }{ 7 } \times \frac { 5 }{ 7 } = \frac { (-6) \times 5 }{ 7 \times 7 } = \frac { -30 }{ 49 } \).
In simple words: To multiply fractions, just multiply the top numbers together and multiply the bottom numbers together. Then, simplify your answer if you can.

Exam Tip: Always remember to simplify the resulting fraction to its lowest terms after performing multiplication, by dividing both the numerator and denominator by their greatest common divisor.