CBSE Class 7 Mathematics Rational Numbers Worksheet Set 04

Fill in the blanks.

Question 1. A rational number where both numerator and denominator are positive integers is called a ________________ rational number.
Answer: positive
In simple words: When both the top and bottom numbers of a fraction are positive, the whole fraction is a positive rational number.

Exam Tip: A rational number is positive if both its numerator and denominator have the same sign (either both positive or both negative).

Question 2. The number that is neither a positive or a negative rational number is __________.
Answer: 0
In simple words: Zero is a unique number that is neither positive nor negative.

Exam Tip: Remember that 0 is a rational number because it can be written as 0/1, but it does not have a positive or negative sign.

Question 3. There are _________________number of rational numbers between two rational numbers.
Answer: infinite
In simple words: There is no limit to the number of fractions you can find between any two fractions.

Exam Tip: Between any two distinct rational numbers, there are infinitely many rational numbers.

Question 4. 6/-7 x -7/5 = __________ 
Answer: \( \frac{6}{5} \)
We can simplify the expression by canceling the common factor \( -7 \) from both the numerator and the denominator:
\[ \frac{6}{-7} \times \frac{-7}{5} = \frac{6}{5} \]
In simple words: Since -7 is on both top and bottom, they cancel out, leaving just 6 on top and 5 on the bottom.

Exam Tip: Cancel out identical terms from the numerator and denominator before multiplying to simplify your calculation quickly.

Question 5. The additive inverse of \( \frac{9}{11} \) is _________
Answer: \( \frac{-9}{11} \)
The additive inverse of any number \( x \) is \( -x \), because adding them together results in zero:
\[ \frac{9}{11} + \left( \frac{-9}{11} \right) = 0 \]
In simple words: To find the additive inverse, just change the sign from positive to negative.

Exam Tip: The sum of a number and its additive inverse is always 0.

Question 6. The product of a rational number with its reciprocal is always ___________
Answer: 1
For any non-zero rational number \( \frac{p}{q} \), its reciprocal is \( \frac{q}{p} \). Multiplying them together gives:
\[ \frac{p}{q} \times \frac{q}{p} = 1 \]
In simple words: If you multiply a fraction by its upside-down version, the answer is always 1.

Exam Tip: Reciprocal is also called the multiplicative inverse because their product is 1.

Do as directed.

Question 7. Write four more numbers in the same pattern: \( \frac{2}{5}, \frac{4}{10}, \frac{6}{15}, \frac{8}{20}. \)
Answer: \( \frac{10}{25}, \frac{12}{30}, \frac{14}{35}, \frac{16}{40} \)
The given pattern follows the rule \( \frac{2n}{5n} \) where \( n = 1, 2, 3, 4 \).
Continuing the pattern for \( n = 5, 6, 7, 8 \):
For \( n = 5 \): \( \frac{2 \times 5}{5 \times 5} = \frac{10}{25} \)
For \( n = 6 \): \( \frac{2 \times 6}{5 \times 6} = \frac{12}{30} \)
For \( n = 7 \): \( \frac{2 \times 7}{5 \times 7} = \frac{14}{35} \)
For \( n = 8 \): \( \frac{2 \times 8}{5 \times 8} = \frac{16}{40} \)
In simple words: Multiply both the top and bottom of the first fraction by 5, 6, 7, and 8 to get the next four fractions.

Exam Tip: Identify the basic fraction in the simplest form first to find the multiplier pattern easily.

Question 8. Is the number \( \frac{3}{-8} \) rational?
Answer: Yes
A number is rational if it can be expressed in the form \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \). For \( \frac{3}{-8} \), the numerator \( 3 \) and the denominator \( -8 \) are both integers, and the denominator is not zero. Thus, it is a rational number.
In simple words: Yes, it is a rational number because it is written as one integer divided by another non-zero integer.

Exam Tip: A negative sign in the denominator does not affect whether a number is rational; it just makes the overall value negative.

Question 9. Is it correct to say that all integers are rational numbers? Justify your answer with example.
Answer: Yes, all integers are rational numbers.
Any integer \( a \) can be written as \( \frac{a}{1} \), which fits the definition of a rational number since the denominator is not zero.
For example, the integer \( 5 \) can be written as \( \frac{5}{1} \), and the integer \( -7 \) can be written as \( \frac{-7}{1} \).
In simple words: Yes, you can turn any whole number into a fraction by placing it over 1, so every integer is a rational number.

Exam Tip: Remember to use the standard definition of a rational number \( \frac{p}{q} \) with \( q = 1 \) to justify why integers are rational.

Question 10. All fractions are rational numbers. Is the converse true? Justify.
Answer: No, the converse is not true.
While every fraction is a rational number, not every rational number is a fraction. Fractions represent parts of a whole and only use positive integers (or zero) in the numerator and denominator. Rational numbers, however, can include negative integers.
For example, \( \frac{-4}{5} \) is a rational number but it is not a fraction because of the negative numerator.
In simple words: No, because rational numbers can have negative signs (like -3/5), but basic fractions are only made of positive numbers.

Exam Tip: Use negative rational numbers like \( \frac{-1}{2} \) as your counterexample when disproving this converse statement.

Question 11. List 5 rational numbers between
a) \( \frac{-4}{7} \) and \( \frac{-3}{8} \)
b) -1 and 0
Answer:
a) Between \( \frac{-4}{7} \) and \( \frac{-3}{8} \):
First, find a common denominator for the two fractions. The LCM of 7 and 8 is 56:
\[ \frac{-4}{7} = \frac{-4 \times 8}{7 \times 8} = \frac{-32}{56} \]
\[ \frac{-3}{8} = \frac{-3 \times 7}{8 \times 7} = \frac{-21}{56} \]
Now, we can select any five rational numbers with a denominator of 56 whose numerators are between \( -32 \) and \( -21 \). For example:
\[ \frac{-31}{56}, \frac{-30}{56}, \frac{-29}{56}, \frac{-28}{56}, \frac{-27}{56} \]
These simplify to:
\[ \frac{-31}{56}, \frac{-15}{28}, \frac{-29}{56}, \frac{-1}{2}, \frac{-27}{56} \]

b) Between -1 and 0:
We can write \( -1 \) as \( \frac{-6}{6} \) and \( 0 \) as \( \frac{0}{6} \).
The five rational numbers between \( \frac{-6}{6} \) and \( \frac{0}{6} \) are:
\[ \frac{-5}{6}, \frac{-4}{6}, \frac{-3}{6}, \frac{-2}{6}, \frac{-1}{6} \]
These simplify to:
\[ \frac{-5}{6}, \frac{-2}{3}, \frac{-1}{2}, \frac{-1}{3}, \frac{-1}{6} \]
In simple words: Make the denominators the same to find five fractions that lie neatly in the middle of the two numbers.

Exam Tip: If there aren't enough integers between the numerators, multiply both the numerator and denominator by a larger number to create a wider gap.

Question 12. Reduce to the standard form:
a) \( \frac{-18}{45} \)
b) \( \frac{-3}{-15} \)
c) \( \frac{28}{56} \)
d) \( \frac{45}{99} \)
Answer:
a) Find the HCF of 18 and 45, which is 9. Divide the numerator and denominator by 9:
\[ \frac{-18 \div 9}{45 \div 9} = \frac{-2}{5} \]

b) The signs cancel out first. The HCF of 3 and 15 is 3. Divide by -3:
\[ \frac{-3 \div (-3)}{-15 \div (-3)} = \frac{1}{5} \]

c) Find the HCF of 28 and 56, which is 28. Divide by 28:
\[ \frac{28 \div 28}{56 \div 28} = \frac{1}{2} \]

d) Find the HCF of 45 and 99, which is 9. Divide by 9:
\[ \frac{45 \div 9}{99 \div 9} = \frac{5}{11} \]
In simple words: Simplify each fraction by dividing the top and bottom by their largest shared factor.

Exam Tip: In standard form, always ensure the denominator is a positive integer and the numerator and denominator have no common factors other than 1.

Question 13. Find three equivalent fractions each for
a) \( \frac{7}{8} \)
b) \( \frac{9}{11} \)
c) \( \frac{3}{5} \)
Answer:
To find equivalent fractions, multiply both the numerator and the denominator by the same non-zero integers (such as 2, 3, and 4):
a) \( \frac{7}{8} \):
Multiply by 2: \( \frac{7 \times 2}{8 \times 2} = \frac{14}{16} \)
Multiply by 3: \( \frac{7 \times 3}{8 \times 3} = \frac{21}{24} \)
Multiply by 4: \( \frac{7 \times 4}{8 \times 4} = \frac{28}{32} \)
Equivalent fractions: \( \frac{14}{16}, \frac{21}{24}, \frac{28}{32} \)

b) \( \frac{9}{11} \):
Multiply by 2: \( \frac{9 \times 2}{11 \times 2} = \frac{18}{22} \)
Multiply by 3: \( \frac{9 \times 3}{11 \times 3} = \frac{27}{33} \)
Multiply by 4: \( \frac{9 \times 4}{11 \times 4} = \frac{36}{44} \)
Equivalent fractions: \( \frac{18}{22}, \frac{27}{33}, \frac{36}{44} \)

c) \( \frac{3}{5} \):
Multiply by 2: \( \frac{3 \times 2}{5 \times 2} = \frac{6}{10} \)
Multiply by 3: \( \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \)
Multiply by 4: \( \frac{3 \times 4}{5 \times 4} = \frac{12}{20} \)
Equivalent fractions: \( \frac{6}{10}, \frac{9}{15}, \frac{12}{20} \)
In simple words: Multiply the top and bottom of each fraction by 2, then by 3, and then by 4.

Exam Tip: You can find infinite equivalent fractions by using any integers as multipliers for both parts of the fraction.

Question 14. Write the following in descending order.
a) \( -\frac{28}{56}, -\frac{2}{3}, \frac{3}{8} \)
b) \( -\frac{8}{5}, -\frac{2}{6}, -\frac{5}{7} \)
Answer:
a) \( -\frac{28}{56}, -\frac{2}{3}, \frac{3}{8} \):
Simplify \( -\frac{28}{56} \) to \( -\frac{1}{2} \).
Convert the fractions to decimals to compare easily:
\( -\frac{1}{2} = -0.5 \)
\( -\frac{2}{3} \approx -0.67 \)
\( \frac{3}{8} = 0.375 \)
Arranging from largest to smallest:
\( 0.375 > -0.5 > -0.67 \)
Descending order: \( \frac{3}{8}, -\frac{28}{56}, -\frac{2}{3} \)

b) \( -\frac{8}{5}, -\frac{2}{6}, -\frac{5}{7} \):
Convert the fractions to decimals:
\( -\frac{8}{5} = -1.6 \)
\( -\frac{2}{6} \approx -0.33 \)
\( -\frac{5}{7} \approx -0.71 \)
Arranging from largest to smallest:
\( -0.33 > -0.71 > -1.6 \)
Descending order: \( -\frac{2}{6}, -\frac{5}{7}, -\frac{8}{5} \)
In simple words: Turn the fractions into decimal numbers to see which is largest and list them from biggest to smallest.

Exam Tip: Be careful with negative numbers; a smaller absolute value (like -0.33) is actually larger than a bigger absolute value (like -1.6).

Question 15. Represent the following rational numbers on a number line.
a) \( \frac{2}{3} \)
b) \( -\frac{4}{3} \)
c) \( -\frac{8}{5} \)
Answer:
a) \( \frac{2}{3} \): This positive number is between 0 and 1. Split the section between 0 and 1 into 3 equal parts. Point to the second mark from 0.

b) \( -\frac{4}{3} \): This is equal to \( -1\frac{1}{3} \), which is between -1 and -2. Split the section between -1 and -2 into 3 equal parts. Point to the first mark to the left of -1.

c) \( -\frac{8}{5} \): This is equal to \( -1\frac{3}{5} = -1.6 \), which lies between -1 and -2. Split the section between -1 and -2 into 5 equal parts. Point to the third mark to the left of -1.

In simple words: Place each number on the line by identifying the two whole numbers it sits between, and then divide the gap into equal steps.

Exam Tip: Convert improper fractions to mixed numbers first to easily locate which two integers they lie between.

Question 16. Write the additive inverse and multiplicative inverse of the following: a)-\frac{2}{9} ,b)\frac{8}{7} ,c)-\frac{3}{11}
Answer:
a) \( -\frac{2}{9} \):
Additive inverse = \( \frac{2}{9} \)
Multiplicative inverse (reciprocal) = \( -\frac{9}{2} \)

b) \( \frac{8}{7} \):
Additive inverse = \( -\frac{8}{7} \)
Multiplicative inverse (reciprocal) = \( \frac{7}{8} \)

c) \( -\frac{3}{11} \):
Additive inverse = \( \frac{3}{11} \)
Multiplicative inverse (reciprocal) = \( -\frac{11}{3} \)
In simple words: Additive inverse means swapping the positive or negative sign. Multiplicative inverse means turning the fraction upside down.

Exam Tip: The multiplicative inverse of a negative number is also negative. Only the numerator and denominator switch places, the sign remains unchanged.

Question 17. Find the sum of a) \( 3\frac{1}{7} + \frac{2}{5} \) b) \( -2\frac{1}{8} + -\frac{6}{10} \) c)-\frac{7}{13} + \left(-\frac{8}{15}\right)
Answer:
a) \( 3\frac{1}{7} + \frac{2}{5} \):
Convert the mixed fraction: \( 3\frac{1}{7} = \frac{22}{7} \).
Find a common denominator for 7 and 5, which is 35:
\[ \frac{22}{7} + \frac{2}{5} = \frac{22 \times 5}{35} + \frac{2 \times 7}{35} = \frac{110 + 14}{35} = \frac{124}{35} = 3\frac{19}{35} \]

b) \( -2\frac{1}{8} + -\frac{6}{10} \):
Convert the mixed fraction: \( -2\frac{1}{8} = -\frac{17}{8} \).
Simplify \( -\frac{6}{10} = -\frac{3}{5} \).
Find a common denominator for 8 and 5, which is 40:
\[ -\frac{17}{8} + \left(-\frac{3}{5}\right) = \frac{-17 \times 5}{40} + \frac{-3 \times 8}{40} = \frac{-85 - 24}{40} = \frac{-109}{40} = -2\frac{29}{40} \]

c) \( -\frac{7}{13} + \left(-\frac{8}{15}\right) \):
Find a common denominator for 13 and 15, which is 195:
\[ -\frac{7}{13} - \frac{8}{15} = \frac{-7 \times 15}{195} - \frac{8 \times 13}{195} = \frac{-105 - 104}{195} = \frac{-209}{195} = -1\frac{14}{195} \]
In simple words: First convert mixed fractions to improper fractions, find a common denominator, and then add the numerators together.

Exam Tip: Simplify fractions before finding a common denominator to keep calculations with larger numbers easier.

Question 18. Find a) \( \frac{7}{24} - \left(\frac{19}{36}\right) \) b) \( \left( -4\frac{1}{9} \right) - 1\frac{2}{7} \) c) \( 8-2\frac{11}{14} \) d) \( \frac{15}{12}-\frac{22}{15} \)
Answer:
a) \( \frac{7}{24} - \frac{19}{36} \):
Find the LCM of 24 and 36, which is 72:
\[ \frac{7 \times 3}{72} - \frac{19 \times 2}{72} = \frac{21 - 38}{72} = \frac{-17}{72} \]

b) \( \left(-4\frac{1}{9}\right) - 1\frac{2}{7} \):
Convert the mixed fractions: \( -4\frac{1}{9} = -\frac{37}{9} \) and \( 1\frac{2}{7} = \frac{9}{7} \).
Find the LCM of 9 and 7, which is 63:
\[ -\frac{37}{9} - \frac{9}{7} = \frac{-37 \times 7}{63} - \frac{9 \times 9}{63} = \frac{-259 - 81}{63} = \frac{-340}{63} = -5\frac{25}{63} \]

c) \( 8 - 2\frac{11}{14} \):
Convert the mixed fraction: \( 2\frac{11}{14} = \frac{39}{14} \).
\[ \frac{8}{1} - \frac{39}{14} = \frac{8 \times 14 - 39}{14} = \frac{112 - 39}{14} = \frac{73}{14} = 5\frac{3}{14} \]

d) \( \frac{15}{12} - \frac{22}{15} \):
Simplify \( \frac{15}{12} \) to \( \frac{5}{4} \).
Find the LCM of 4 and 15, which is 60:
\[ \frac{5}{4} - \frac{22}{15} = \frac{5 \times 15 - 22 \times 4}{60} = \frac{75 - 88}{60} = \frac{-13}{60} \]
In simple words: Find a common denominator to subtract the top numbers, and simplify mixed fractions first.

Exam Tip: Reduce any fraction to its lowest terms before subtracting to keep LCM calculations simple.

Question 19. Find the value of : a) \( -4 \div \frac{8}{7} \) , b)\( \left(-\frac{3}{5}\right) \div 7 \) , c) \( \left(-\frac{8}{11}\right) \div \frac{16}{22} \) , d) \( \frac{3}{13} \div -\frac{9}{2} \)
Answer:
To divide by a fraction, multiply by its reciprocal:
a) \[ -4 \times \frac{7}{8} = \frac{-28}{8} = -\frac{7}{2} = -3\frac{1}{2} \]

b) \[ -\frac{3}{5} \times \frac{1}{7} = -\frac{3}{35} \]

c) \[ -\frac{8}{11} \times \frac{22}{16} = -\frac{8 \times 22}{11 \times 16} = -\frac{1 \times 2}{1 \times 2} = -1 \]

d) \[ \frac{3}{13} \times \left(-\frac{2}{9}\right) = \frac{3 \times -2}{13 \times 9} = -\frac{2}{117} \]
In simple words: When dividing fractions, flip the second fraction upside down and multiply instead of dividing.

Exam Tip: Be sure to write the integer as a fraction (like 7 as 7/1) before finding its reciprocal to avoid simple division mistakes.

Question 20. Find the product : a) \( \frac{9}{2} \times \left(-\frac{7}{4}\right) \) , b) \( \left(-\frac{3}{11}\right) \times \left(\frac{-5}{2}\right) \) , c) \( \frac{7}{15} \times \left(\frac{9}{-28}\right) \) , d) \( \left(-\frac{4}{9}\right) \times \frac{11}{12} \)
Answer:
To find the product, multiply numerators together and denominators together:
a) \[ \frac{9 \times (-7)}{2 \times 4} = -\frac{63}{8} = -7\frac{7}{8} \]

b) \[ \frac{(-3) \times (-5)}{11 \times 2} = \frac{15}{22} \]

c) Simplify the common factors first (7 divides -28 to -4, and 3 divides 9 and 15 to 3 and 5):
\[ \frac{7}{15} \times \frac{9}{-28} = \frac{1}{5} \times \frac{3}{-4} = -\frac{3}{20} \]

d) Simplify common factors first (4 divides 12 to 3):
\[ \frac{-4}{9} \times \frac{11}{12} = \frac{-1 \times 11}{9 \times 3} = -\frac{11}{27} \]
In simple words: Multiply the top numbers together and multiply the bottom numbers together, simplifying whenever possible.

Exam Tip: Simplify diagonally across the multiplication sign first to keep your final numbers small and easy to handle.

Question 21. The sum of two numbers is \( -\frac{11}{12} \). One of them is \( \frac{9}{2} \). Find the other.
Answer: \( -\frac{65}{12} \) (or \( -5\frac{5}{12} \))
Let the other number be \( x \).
According to the question:
\[ x + \frac{9}{2} = -\frac{11}{12} \]
Subtract \( \frac{9}{2} \) from both sides:
\[ x = -\frac{11}{12} - \frac{9}{2} \]
Find a common denominator of 12:
\[ x = \frac{-11 - 9 \times 6}{12} = \frac{-11 - 54}{12} = -\frac{65}{12} = -5\frac{5}{12} \]
The other number is \( -5\frac{5}{12} \).
In simple words: To find the missing number, subtract the given number from the total sum.

Exam Tip: Double-check your final answer by adding it back to the given number to see if you get the original sum.

Question 22. The product of two numbers is \( \left(-\frac{6}{7}\right) \). One of them is \( \frac{3}{4} \). Find the other.
Answer: \( -\frac{8}{7} \) (or \( -1\frac{1}{7} \))
Let the other number be \( y \).
According to the question:
\[ y \times \frac{3}{4} = -\frac{6}{7} \]
To solve for \( y \), divide both sides by \( \frac{3}{4} \), which is equivalent to multiplying by its reciprocal \( \frac{4}{3} \):
\[ y = -\frac{6}{7} \times \frac{4}{3} = \frac{-2 \times 4}{7 \times 1} = -\frac{8}{7} = -1\frac{1}{7} \]
The other number is \( -1\frac{1}{7} \).
In simple words: Divide the final product by the given number to find the missing multiplier.

Exam Tip: Be sure to simplify common factors (like dividing 6 and 3 by 3) before multiplying to speed up your calculation.

Question 23. Simplify: \( 5\frac{5}{6} + \left(-2\frac{1}{3}\right) + 1\frac{5}{6} \)
Answer: \( 5\frac{1}{3} \)
First, convert all mixed fractions into improper fractions:
\[ 5\frac{5}{6} = \frac{35}{6} \]
\[ -2\frac{1}{3} = -\frac{7}{3} = -\frac{14}{6} \]
\[ 1\frac{5}{6} = \frac{11}{6} \]
Now, sum the fractions with the common denominator of 6:
\[ \frac{35 - 14 + 11}{6} = \frac{32}{6} = \frac{16}{3} = 5\frac{1}{3} \]
The simplified answer is \( 5\frac{1}{3} \).
In simple words: Turn the mixed numbers into improper fractions, make their bottom numbers match, and then combine the top numbers.

Exam Tip: Convert your final improper fraction back into a mixed fraction if the original numbers in the question are mixed fractions.

Question 24. Compare: a) \( -3\frac{4}{5} \) and \( -3\frac{1}{5} \) b) \( -\frac{7}{5} \) and \( -\frac{5}{8} \)
Answer:
a) \( -3\frac{4}{5} \) and \( -3\frac{1}{5} \):
Convert the mixed fractions:
\( -3\frac{4}{5} = -\frac{19}{5} = -3.8 \)
\( -3\frac{1}{5} = -\frac{16}{5} = -3.2 \)
Since \( -3.2 \) is greater than \( -3.8 \), we have:
\[ -3\frac{4}{5} < -3\frac{1}{5} \]

b) \( -\frac{7}{5} \) and \( -\frac{5}{8} \):
Convert the fractions to decimals to compare:
\( -\frac{7}{5} = -1.4 \)
\( -\frac{5}{8} = -0.625 \)
Since \( -0.625 \) is greater than \( -1.4 \), we have:
\[ -\frac{7}{5} < -\frac{5}{8} \]
In simple words: Turn the fractions into decimals to compare them, remembering that negative numbers closer to zero are larger.

Exam Tip: Always write the correct comparison sign (\( < \), \( > \), or \( = \)) clearly between the two given numbers.