Q1. Represent the rational numbers \frac{2}{3}, -\frac{5}{4}, and 1\frac{1}{2} on a single number line.
Q2. Find three distinct rational numbers that lie strictly between -\frac{1}{2} and \frac{1}{4}.
Solution:
Step 1: Make the denominators equal.
The LCM of 4 and 2 is 4,
-\frac{1}{2} = -\frac{2}{4}
\frac{1}{4} = \frac{1}{4}
Step 2: Make the difference between numerators large enough
Multiplying numerator and denominator by 4
-\frac{2}{4} = -\frac{2 \times 4}{4 \times 4} = -\frac{8}{16}
\frac{1}{4} = \frac{1 \times 4}{4 \times 4} = \frac{4}{16}
Step 2: Pick three numerators between -8 and 4.
Final Answer: \boxed{ -\frac{5}{16}, -\frac{2}{16}, \frac{1}{16} } (Answers may vary).
Q3. Simplify the expression: \left(-\frac{1}{4}\right) + \left(\frac{5}{12}\right).
Solution:
Find the LCM of 4 and 12, which is 12.
Rewrite -\frac{1}{4} as -\frac{3}{12}.
Add the numerators: \frac{-3 + 5}{12} = \frac{2}{12}.
\frac{2}{12} = \frac{1}{6}.
Final Answer: \boxed{ \frac{1}{6} }
Q4. A tailor has 15\frac{3}{4} metres of fine silk cloth. If making one kurta requires 2\frac{1}{4} metres of silk cloth, exactly how many kurtas can he make?
Solution:
Convert to improper fractions.
Total silk cloth = 15\frac{3}{4} = \frac{63}{4}
Silk cloth per kurta = 2\frac{1}{4} = \frac{9}{4}
Divide total silk cloth by silk cloth per kurta.
\frac{63}{4} \div \frac{9}{4} = \frac{63}{4} \times \frac{4}{9} = \frac{63}{9} = 7.
Final Answer: The tailor can make exactly 7 kurtas.
Q5. Find three rational numbers between 3.1415 and 3.1416.
Solution:
Add extra zeros to the end to see the gaps more clearly.
Think of them as 3.14150 and 3.14160.
Pick any terminating decimals between these two values.
Final Answer: \boxed {3.14151, 3.14152, 3.14155} (Answers may vary).
Q6. Can you think of other way(s) to find a rational number between any two rational numbers?
Solution:
Yes! In addition to the “mean” or “average” method (\frac{a+b}{2}), you can convert the rational numbers to decimals and pick a terminating decimal between them (as we did in Q5).
You can also make their denominators very large (equivalent fractions) and pick integer numerators in between (as we did in Q2).