The World of Numbers • Class 9 Mathematics • Ganita Manjari
Practice Mode: पहले question खुद solve करें। समझ न आए तो 🇮🇳 हिंदी में समझें दबाएँ। फिर Solution देखें से diagram और complete working check करें.
Topic: Rational Numbers on the Number Line, rational numbers between two numbers, simplification and a real-life word problem.
Question 1
Representing Rational Numbers on a Number Line
Represent the rational numbers 2/3, −5/4 and 1 1/2 on a single number line.
हिंदी में समझें:Convert the mixed number: 1 1/2 = 3/2.
For −5/4, divide the interval from −2 to −1 into 4 equal parts and mark the first point to the left of −1.
For 2/3, divide the interval from 0 to 1 into 3 equal parts and mark the second point from 0.
For 3/2, divide the interval from 1 to 2 into 2 equal parts and mark the midpoint.
Hence the order is −5/4 < 2/3 < 3/2.
Step-by-Step Solution:
Diagram: Visual representation को देखकर number placement और fraction calculation समझें।
Convert the mixed number: 1 1/2 = 3/2.
For −5/4, divide the interval from −2 to −1 into 4 equal parts and mark the first point to the left of −1.
For 2/3, divide the interval from 0 to 1 into 3 equal parts and mark the second point from 0.
For 3/2, divide the interval from 1 to 2 into 2 equal parts and mark the midpoint.
Hence the order is −5/4 < 2/3 < 3/2.
✅ Final Answer: −5/4, 2/3 and 3/2 are correctly represented on the same number line.
Exam Tip: Number-line questions में intervals बराबर रखें और “strictly between” का मतलब endpoints को include न करना याद रखें।
Question 2
Three Rational Numbers Between Two Numbers
Find three distinct rational numbers that lie strictly between −1/2 and 1/4.
हिंदी में समझें:Convert to a common denominator:
−1/2 = −4/8 and 1/4 = 2/8.
Numbers strictly between −4/8 and 2/8 include −3/8, −2/8 and −1/8.
Taking three distinct values and simplifying −2/8 gives −1/4.
Step-by-Step Solution:
Diagram: Visual representation को देखकर number placement और fraction calculation समझें।
Convert to a common denominator:
−1/2 = −4/8 and 1/4 = 2/8.
Numbers strictly between −4/8 and 2/8 include −3/8, −2/8 and −1/8.
Taking three distinct values and simplifying −2/8 gives −1/4.
✅ Final Answer: −3/8, −1/4 and 1/8 are three valid answers.
Exam Tip: Number-line questions में intervals बराबर रखें और “strictly between” का मतलब endpoints को include न करना याद रखें।
Question 3
Simplifying a Rational Expression
Simplify: (−1/4) + (5/12).
हिंदी में समझें:LCM of 4 and 12 is 12.
−1/4 = −3/12.
Therefore −3/12 + 5/12 = 2/12 = 1/6.
Step-by-Step Solution:
Diagram: Visual representation को देखकर number placement और fraction calculation समझें।
LCM of 4 and 12 is 12.
−1/4 = −3/12.
Therefore −3/12 + 5/12 = 2/12 = 1/6.
✅ Final Answer: 1/6
Exam Tip: Number-line questions में intervals बराबर रखें और “strictly between” का मतलब endpoints को include न करना याद रखें।
Question 4
Silk for Kurtas
A tailor has 15 3/4 metres of fine silk. If making one kurta requires 2 1/4 metres, how many kurtas can he make?
हिंदी में समझें:Convert mixed numbers:
15 3/4 = 63/4 and 2 1/4 = 9/4.
Number of kurtas = (63/4) ÷ (9/4)
= (63/4) × (4/9)
= 63/9
= 7.
Thus the tailor can make 7 complete kurtas.
Step-by-Step Solution:
Diagram: Visual representation को देखकर number placement और fraction calculation समझें।
Convert mixed numbers:
15 3/4 = 63/4 and 2 1/4 = 9/4.
Number of kurtas = (63/4) ÷ (9/4)
= (63/4) × (4/9)
= 63/9
= 7.
Thus the tailor can make 7 complete kurtas.
✅ Final Answer: 7 kurtas
Exam Tip: Number-line questions में intervals बराबर रखें और “strictly between” का मतलब endpoints को include न करना याद रखें।
Question 5
Rational Numbers Between Close Decimals
Find three rational numbers between 3.1415 and 3.1416.
हिंदी में समझें:Both endpoints are terminating decimals, hence rational. We can write more decimal places without changing their size: 3.14150 and 3.14160.
Now choose values strictly between them, for example 3.14151, 3.14155 and 3.14159. Each is a terminating decimal, so each is rational and lies strictly between the given numbers.
Step-by-Step Solution:
Diagram: Visual representation को देखकर number placement और fraction calculation समझें।
Both endpoints are terminating decimals, hence rational. We can write more decimal places without changing their size: 3.14150 and 3.14160.
Now choose values strictly between them, for example 3.14151, 3.14155 and 3.14159. Each is a terminating decimal, so each is rational and lies strictly between the given numbers.
✅ Final Answer: 3.14151, 3.14155 and 3.14159
Exam Tip: Number-line questions में intervals बराबर रखें और “strictly between” का मतलब endpoints को include न करना याद रखें।
Question 6
Another Way to Find a Rational Number Between Two Numbers
Can you think of another way to find a rational number between two rational numbers? Explain your method.
हिंदी में समझें:Let a and b be two rational numbers with a < b.
Take their average: (a+b)/2.
Because a < b, adding a to both sides gives 2a < a+b. Dividing by 2 gives a < (a+b)/2.
Also a+b < 2b, so (a+b)/2 < b.
Therefore (a+b)/2 lies strictly between a and b and is rational.
Step-by-Step Solution:
Diagram: Visual representation को देखकर number placement और fraction calculation समझें।
Let a and b be two rational numbers with a < b.
Take their average: (a+b)/2.
Because a < b, adding a to both sides gives 2a < a+b. Dividing by 2 gives a < (a+b)/2.
Also a+b < 2b, so (a+b)/2 < b.
Therefore (a+b)/2 lies strictly between a and b and is rational.
✅ Final Answer: (a+b)/2 is a rational number strictly between a and b.
Exam Tip: Number-line questions में intervals बराबर रखें और “strictly between” का मतलब endpoints को include न करना याद रखें।