Mathematics solution NCERT
Class 9 - Chapter 3: The World of Numbers
Question 2. Find three distinct rational numbers that lie strictly between $$-\frac{1}{2}$$ and $$\frac{1}{4}$$.
Solution:
Write both rational numbers with a common denominator.
$$ -\frac{1}{2} = -\frac{4}{8}, \qquad \frac{1}{4} = \frac{2}{8} $$The rational numbers lying strictly between $$-\frac{4}{8}$$ and $$\frac{2}{8}$$ are:
$$ -\frac{3}{8}, \quad -\frac{2}{8}, \quad -\frac{1}{8} $$Simplify the fractions wherever possible.
$$ -\frac{2}{8} = -\frac{1}{4} $$Answer:
$$ \boxed{ -\frac{3}{8}, \; -\frac{1}{4}, \; -\frac{1}{8} } $$Question 3. Simplify the expression:
$$ \left(-\frac14\right)+\left(\frac{5}{12}\right) $$Solution:
LCM of 4 and 12 is 12.
$$ -\frac14 = -\frac3{12} $$Now, add the fractions.
$$ \begin{aligned} -\frac14+\frac5{12} &= -\frac3{12}+\frac5{12}\\ &= \frac{-3+5}{12}\\ &= \frac2{12}\\ &= \frac16 \end{aligned} $$Answer:
$$ \boxed{\frac16} $$Question 4. A tailor has $$15\frac34$$ metres of fine silk. If making one kurta requires $$2\frac14$$ metres of silk, exactly how many kurtas can he make?
Solution:
Convert the mixed fractions into improper fractions.
$$ 15\frac34 = \frac{63}{4} $$ $$ 2\frac14 = \frac94 $$Number of kurtas
$$ \begin{aligned} \frac{63}{4} \div \frac94 &= \frac{63}{4} \times \frac49\\ &= \frac{63}{9}\\ &= 7 \end{aligned} $$Answer:
$$ \boxed{7\text{ kurtas}} $$Question 5. Find three rational numbers between $$3.1415$$ and $$3.1416$$.
Solution:
Write both numbers up to five decimal places.
$$ 3.14150 \qquad\text{and}\qquad 3.14160 $$Three rational numbers between them are:
$$ 3.14151, \qquad 3.14152, \qquad 3.14153 $$Answer:
$$ \boxed{ 3.14151,\; 3.14152,\; 3.14153 } $$Question 6. Can you think of other way(s) to find a rational number between any two rational numbers?
Solution:
Yes. One simple method is to find the average of the two rational numbers.
If the two rational numbers are $$a$$ and $$b$$, then
$$ \frac{a+b}{2} $$is a rational number lying between them.
Repeating this process gives infinitely many rational numbers between the given rational numbers.
Answer:
$$ \boxed{ \text{The average }\frac{a+b}{2}\text{ always lies between }a\text{ and }b. } $$