Ravi Kant Kumar

Convert 0.4̅ into Fraction (p/q) Express the Decimal \(0.\overline{4}\) in the Form \( \frac{p}{q} \) Question: Express \(0.\overline{4}\) in the form \( \frac{p}{q} \). Solution: Let \[ x = 0.\overline{4} \] Multiply both sides by 10: \[ 10x = 4.\overline{4} \] Subtract the first equation from the second: \[ 10x – x = 4.\overline{4} – […]

Express Decimals into Fractions (p/q) Express Each of the Following Decimals in the Form p/q Question: Express each of the following decimals in the form \( \frac{p}{q} \): (i) 0.39 (ii) 0.750 (iii) 2.15 Solution: (i) Convert 0.39 into p/q form \[ 0.39 = \frac{39}{100} \] This is already in simplest form. Answer: \( \frac{39}{100}

Terminating Decimal Condition 📘 Question Look at several examples of rational numbers \( \frac{p}{q} \) (where \(q \ne 0\), and \(p, q\) have no common factors except 1) that have terminating decimal representations. What property must \(q\) satisfy? ✏️ Explanation Step 1: Observe examples \(\frac{1}{2} = 0.5\) \(\frac{3}{4} = 0.75\) \(\frac{7}{5} = 1.4\) \(\frac{9}{20} =

Convert Rational Numbers to Recurring Decimals 📘 Question Express the following rational numbers as decimals: (i) \( \frac{2}{3} \) (ii) \( -\frac{4}{9} \) (iii) \( -\frac{2}{15} \) (iv) \( -\frac{22}{13} \) (v) \( \frac{437}{999} \) ✏️ Step-by-Step Solution (i) \( \frac{2}{3} \) \[ \frac{2}{3} = 0.\overline{6} \] — (ii) \( -\frac{4}{9} \) \[ -\frac{4}{9} =

Rational Numbers Between 3/5 and 4/5 📘 Question Find five rational numbers between \( \frac{3}{5} \) and \( \frac{4}{5} \). ✏️ Step-by-Step Solution Step 1: Convert to equivalent fractions \[ \frac{3}{5} = \frac{18}{30}, \quad \frac{4}{5} = \frac{24}{30} \] — Step 2: Choose numbers in between \[ \frac{19}{30}, \frac{20}{30}, \frac{21}{30}, \frac{22}{30}, \frac{23}{30} \] — Step 3:

Convert Rational Numbers to Decimals 📘 Question Express the following rational numbers as decimals: (i) \( \frac{42}{100} \) (ii) \( \frac{327}{500} \) (iii) \( \frac{15}{4} \) ✏️ Step-by-Step Solution (i) \( \frac{42}{100} \) Divide numerator by denominator: \[ \frac{42}{100} = 0.42 \] — (ii) \( \frac{327}{500} \) Convert denominator to 1000: \[ \frac{327}{500} = \frac{654}{1000}

Find Rational Numbers Between 3 and 4 📘 Question Find six rational numbers between 3 and 4. ✏️ Step-by-Step Solution Step 1: Convert into fractions \[ 3 = \frac{21}{7}, \quad 4 = \frac{28}{7} \] — Step 2: Choose numbers in between \[ \frac{22}{7}, \frac{23}{7}, \frac{24}{7}, \frac{25}{7}, \frac{26}{7}, \frac{27}{7} \] — ✅ Final Answer \[ \boxed{\frac{22}{7},

Find Rational Numbers Between 1 and 2 📘 Question Find five rational numbers between 1 and 2. ✏️ Step-by-Step Solution Step 1: Convert into fractions \[ 1 = \frac{6}{6}, \quad 2 = \frac{12}{6} \] — Step 2: Choose numbers in between \[ \frac{7}{6}, \frac{8}{6}, \frac{9}{6}, \frac{10}{6}, \frac{11}{6} \] — Step 3: Simplify (optional) \[ \frac{7}{6},

Is 0 a Rational Number? 📘 Question Is zero a rational number? Can you write it in the form \( \frac{p}{q} \), where \(p\) and \(q\) are integers and \(q \ne 0\)? ✏️ Solution Step 1: Definition A rational number is any number that can be written as: \[ \frac{p}{q}, \quad \text{where } p, q

Identify Diagonal Matrix 📘 Question The matrix \[ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{bmatrix} \] is a: (a) identity matrix (b) symmetric matrix (c) skew-symmetric matrix (d) diagonal matrix ✏️ Step-by-Step Solution Step 1: Observe matrix All non-diagonal elements are