Convert the following decimal numbers in the form of . (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)

Understand the Question Terminating Decimals: To convert a terminating decimal into form, write the digits as the numerator and divide by , where is the number of digits after the decimal point, then simplify to lowest terms. Repeating Decimals (Non-terminating Recurring): 1. Let equal the recurring decimal. 2. Multiply by appropriate powers of so that the recurring digits align after the decimal point. 3. Subtract the equations to eliminate the repeating part and solve for as a fraction . Part (i) Convert in the form of . Step 1 · Convert to Fraction and Simplify Since there is decimal place Dividing numerator and denominator by Answer > (i) Part (ii) Convert in the form of . Step 1 · Convert to Fraction and Simplify Write as Dividing numerator and denominator by Answer > (ii) Part (iii) Convert in the form of . Step 1 · Set Up Equations to Eliminate Repeating Digits Let Multiplying equation by Multiplying equation by Step 2 · Subtract and Solve for Subtracting equation from equation Answer > (iii) Part (iv) Convert in the form of . Step 1 · Set Up Equations to Eliminate Repeating Digits Let Multiplying equation by Multiplying equation by Step 2 · Subtract and Solve for Subtracting equation from equation Answer > (iv) Part (v) Convert in the form of . Step 1 · Set Up Equations and Solve for Let Multiplying equation by Subtracting equation from equation Answer > (v) Part (vi) Convert in the form of . Step 1 · Set Up Equations and Solve for Let Multiplying equation by Multiplying equation by Subtracting equation from equation Dividing numerator and denominator by Answer > (vi) Part (vii) Convert in the form of . Step 1 · Set Up Equations and Solve for Let Multiplying equation by Multiplying equation by Subtracting equation from equation Answer > (vii) Part (viii) Convert in the form of . Step 1 · Set Up Equations and Solve for Let Multiplying equation by Multiplying equation by Subtracting equation from equation Answer > (viii) Part (ix) Convert in the form of . Step 1 · Set Up Equations and Solve for Let Since digits are under the bar, multiply equation by Subtracting equation from equation Answer > (ix) Common Mistakes Bar Placement Confusion: Only the digits strictly under the bar repeat. For example, in , only repeats (), not . Incorrect Multiplier: Not shifting the decimal point past non-repeating digits before subtracting, which fails to cancel out the repeating fractional part. Incomplete Simplification: Forgetting to divide numerator and denominator by their greatest common divisor (e.g. leaving instead of ).

The World of Numbers | EOT

Question 3

Convert the following decimal numbers in the form of $\dfrac{p}{q}$ .

(i) $12.6$

(ii) $0.0120$

(iii) $3.05\overline{2}$

(iv) $1.2\overline{35}$

(v) $0.\overline{23}$

(vi) $2.0\overline{5}$

(vii) $2.12\overline{5}$

(viii) $3.12\overline{5}$

(ix) $2.\overline{1625}$

Solution

Understand the Question

Terminating Decimals: To convert a terminating decimal into $\dfrac{p}{q}$ form, write the digits as the numerator and divide by $10^n$ , where $n$ is the number of digits after the decimal point, then simplify to lowest terms.
Repeating Decimals (Non-terminating Recurring):
1. Let $x$ equal the recurring decimal.
2. Multiply $x$ by appropriate powers of $10$ so that the recurring digits align after the decimal point.
3. Subtract the equations to eliminate the repeating part and solve for $x$ as a fraction $\dfrac{p}{q}$ .

(i) Convert $12.6$ in the form of $\dfrac{p}{q}$ .

Step 1 · Convert to Fraction and Simplify

Since there is $1$ decimal place

12.6 = \dfrac{126}{10}

Dividing numerator and denominator by $2$

\dfrac{126 \div 2}{10 \div 2} = \dfrac{63}{5}

Answer

(i) $\dfrac{63}{5}$

(ii) Convert $0.0120$ in the form of $\dfrac{p}{q}$ .

Step 1 · Convert to Fraction and Simplify

Write $0.0120$ as $0.012$

0.012 = \dfrac{12}{1000}

Dividing numerator and denominator by $4$

\dfrac{12 \div 4}{1000 \div 4} = \dfrac{3}{250}

Answer

(ii) $\dfrac{3}{250}$

(iii) Convert $3.05\overline{2}$ in the form of $\dfrac{p}{q}$ .

Step 1 · Set Up Equations to Eliminate Repeating Digits

Let

x = 3.05222\dots \quad \dots (1)

Multiplying equation $(1)$ by $100$

100x = 305.222\dots \quad \dots (2)

Multiplying equation $(1)$ by $1000$

1000x = 3052.222\dots \quad \dots (3)

Step 2 · Subtract and Solve for $x$

Subtracting equation $(2)$ from equation $(3)$

\begin{aligned} 1000x - 100x &= 3052.222\dots - 305.222\dots \\ 900x &= 2747 \\ x &= \dfrac{2747}{900} \end{aligned}

Answer

(iii) $\dfrac{2747}{900}$

(iv) Convert $1.2\overline{35}$ in the form of $\dfrac{p}{q}$ .

Step 1 · Set Up Equations to Eliminate Repeating Digits

Let

x = 1.2353535\dots \quad \dots (1)

Multiplying equation $(1)$ by $10$

10x = 12.3535\dots \quad \dots (2)

Multiplying equation $(1)$ by $1000$

1000x = 1235.3535\dots \quad \dots (3)

Step 2 · Subtract and Solve for $x$

Subtracting equation $(2)$ from equation $(3)$

\begin{aligned} 1000x - 100x &= 1235.3535\dots - 12.3535\dots \\ 990x &= 1223 \\ x &= \dfrac{1223}{990} \end{aligned}

Answer

(iv) $\dfrac{1223}{990}$

(v) Convert $0.\overline{23}$ in the form of $\dfrac{p}{q}$ .

Step 1 · Set Up Equations and Solve for $x$

Let

x = 0.232323\dots \quad \dots (1)

Multiplying equation $(1)$ by $100$

100x = 23.2323\dots \quad \dots (2)

Subtracting equation $(1)$ from equation $(2)$

\begin{aligned} 100x - x &= 23.2323\dots - 0.2323\dots \\ 99x &= 23 \\ x &= \dfrac{23}{99} \end{aligned}

Answer

(v) $\dfrac{23}{99}$

(vi) Convert $2.0\overline{5}$ in the form of $\dfrac{p}{q}$ .

Step 1 · Set Up Equations and Solve for $x$

Let

x = 2.0555\dots \quad \dots (1)

Multiplying equation $(1)$ by $10$

10x = 20.555\dots \quad \dots (2)

Multiplying equation $(1)$ by $100$

100x = 205.555\dots \quad \dots (3)

Subtracting equation $(2)$ from equation $(3)$

\begin{aligned} 100x - 10x &= 205.555\dots - 20.555\dots \\ 90x &= 185 \\ x &= \dfrac{185}{90} \end{aligned}

Dividing numerator and denominator by $5$

x = \dfrac{185 \div 5}{90 \div 5} = \dfrac{37}{18}

Answer

(vi) $\dfrac{37}{18}$

(vii) Convert $2.12\overline{5}$ in the form of $\dfrac{p}{q}$ .

Step 1 · Set Up Equations and Solve for $x$

Let

x = 2.12555\dots \quad \dots (1)

Multiplying equation $(1)$ by $100$

100x = 212.555\dots \quad \dots (2)

Multiplying equation $(1)$ by $1000$

1000x = 2125.555\dots \quad \dots (3)

Subtracting equation $(2)$ from equation $(3)$

\begin{aligned} 1000x - 100x &= 2125.555\dots - 212.555\dots \\ 900x &= 1913 \\ x &= \dfrac{1913}{900} \end{aligned}

Answer

(vii) $\dfrac{1913}{900}$

(viii) Convert $3.12\overline{5}$ in the form of $\dfrac{p}{q}$ .

Step 1 · Set Up Equations and Solve for $x$

Let

x = 3.12555\dots \quad \dots (1)

Multiplying equation $(1)$ by $100$

100x = 312.555\dots \quad \dots (2)

Multiplying equation $(1)$ by $1000$

1000x = 3125.555\dots \quad \dots (3)

Subtracting equation $(2)$ from equation $(3)$

\begin{aligned} 1000x - 100x &= 3125.555\dots - 312.555\dots \\ 900x &= 2813 \\ x &= \dfrac{2813}{900} \end{aligned}

Answer

(viii) $\dfrac{2813}{900}$

(ix) Convert $2.\overline{1625}$ in the form of $\dfrac{p}{q}$ .

Step 1 · Set Up Equations and Solve for $x$

Let

x = 2.16251625\dots \quad \dots (1)

Since $4$ digits are under the bar, multiply equation $(1)$ by $10000$

10000x = 21625.1625\dots \quad \dots (2)

Subtracting equation $(1)$ from equation $(2)$

\begin{aligned} 10000x - x &= 21625.1625\dots - 2.1625\dots \\ 9999x &= 21623 \\ x &= \dfrac{21623}{9999} \end{aligned}

Answer

(ix) $\dfrac{21623}{9999}$

Common Mistakes

Bar Placement Confusion: Only the digits strictly under the bar repeat. For example, in $3.05\overline{2}$ , only $2$ repeats ( $3.05222\dots$ ), not $052$ .
Incorrect Multiplier: Not shifting the decimal point past non-repeating digits before subtracting, which fails to cancel out the repeating fractional part.
Incomplete Simplification: Forgetting to divide numerator and denominator by their greatest common divisor (e.g. leaving $\dfrac{185}{90}$ instead of $\dfrac{37}{18}$ ).

Question 3

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