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Q3(ii):
Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
(ii) $0.4\overline{7}$
Solution :
Initial Setup & Variable Assignment
We are tasked with converting the mixed recurring decimal $0.4\overline{7}$ into a rational number of the form $\frac{p}{q}$, where $p, q \in \mathbb{Z}$ and $q \neq 0$.
Let the given number be represented by the variable $x$:
$x = 0.47777\dots$ (Equation 1)
[By definition of the vinculum (bar) notation, only the digit $7$ repeats infinitely, while the digit $4$ is a non-repeating decimal component].
Step 1: Isolating the Repeating Decimal Block
To eliminate the infinite repeating sequence, we must first align the decimal point immediately before the repeating block. Since there is exactly one non-repeating digit ($4$) after the decimal point, we multiply both sides of Equation 1 by $10^1$.
$10 \cdot x = 10 \cdot (0.47777\dots)$
$10x = 4.7777\dots$ (Equation 2)
[Per the properties of base-10 positional notation, multiplying by $10$ shifts the decimal point one place to the right].
Step 2: Shifting the Decimal Past the Repeating Block
Next, we need a second equation where the fractional part is identical to the fractional part of Equation 2 (which is $.7777\dots$). Since the repeating block consists of exactly one digit ($7$), we multiply Equation 2 by $10^1$.
$10 \cdot 10x = 10 \cdot (4.7777\dots)$
$100x = 47.7777\dots$ (Equation 3)
Step 3: Algebraic Elimination of the Infinite Decimal
We now subtract Equation 2 from Equation 3. Because the infinite repeating decimal tails ($.7777\dots$) are perfectly aligned, they subtract to zero, leaving only integers.
| $100x$ | $=$ | $47.7777\dots$ |
| $- \quad 10x$ | $=$ | $- \quad 4.7777\dots$ |
| $90x$ | $=$ | $43.0000\dots$ |
[By the Subtraction Property of Equality, if $a = b$ and $c = d$, then $a - c = b - d$].
Step 4: Solving for the Rational Form $\frac{p}{q}$
We now have a simple linear equation with integer coefficients:
$90x = 43$
Isolating $x$ by dividing both sides by $90$:
$x = \frac{43}{90}$
We must verify that this fraction is in its simplest form. The numerator, $43$, is a prime number. The denominator, $90$, is not a multiple of $43$ (since $43 \times 2 = 86$). Therefore, the greatest common divisor $\text{GCD}(43, 90) = 1$. The fraction is irreducible, and both $p = 43$ and $q = 90$ are integers with $q \neq 0$.
Final Solution: The mixed recurring decimal $0.4\overline{7}$ expressed in the rational form $\frac{p}{q}$ is $\frac{43}{90}$.
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.3
- Q1(i): Write the following in decimal form and say what kind of decimal expansion each has : (i) $\frac{36}{100}$
- Q1(ii): Write the following in decimal form and say what kind of decimal expansion each has : (ii) $\frac{1}{11}$
- Q1(iii): Write the following in decimal form and say what kind of decimal expansion each has : (iii) $4\frac{1}{8}$
- Q1(iv): Write the following in decimal form and say what kind of decimal expansion each has : (iv) $\frac{3}{13}$
- Q1(v): Write the following in decimal form and say what kind of decimal expansion each has : (v) $\frac{2}{11}$
- Q1(vi): Write the following in decimal form and say what kind of decimal expansion each has : (vi) $\frac{329}{400}$
- Q2: You know that $\frac{1}{7} = 0.\overline{142857}$. Can you predict what the decimal expansions of $\frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}$ are, without actually doing the long division? If so, how? [Hint : Study the remainders while finding the value of $\frac{1}{7}$ carefully.]
- Q3(i): Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (i) $0.\overline{6}$
- Q3(iii): Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (iii) $0.\overline{001}$
- Q4: Express $0.99999 ....$ in the form $\frac{p}{q}$. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.
- Q5: What can the maximum number of digits be in the repeating block of digits in the decimal expansion of $\frac{1}{17}$? Perform the division to check your answer.
- Q6: Look at several examples of rational numbers in the form $\frac{p}{q}$ ($q \neq 0$), where $p$ and $q$ are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property $q$ must satisfy?
- Q7: Write three numbers whose decimal expansions are non-terminating non-recurring.
- Q8: Find three different irrational numbers between the rational numbers $\frac{5}{7}$ and $\frac{9}{11}$.
- Q9(i): Classify the following numbers as rational or irrational : (i) $\sqrt{23}$
- Q9(ii): Classify the following numbers as rational or irrational : (ii) $\sqrt{225}$
- Q9(iii): Classify the following numbers as rational or irrational : (iii) $0.3796$
- Q9(iv): Classify the following numbers as rational or irrational : (iv) $7.478478...$
- Q9(v): Classify the following numbers as rational or irrational : (v) $1.101001000100001...$
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
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