Can you find the quotients of , and ?

Understand the Question To find the quotients of and , perform long division to express them as decimals. Both divisions yield non-terminating repeating (recurring) decimals, represented using bar notation over the repeating digit block. Part (i) Find the quotient of . Step 1 · Divide 10 by 9 Divide by : Adding a decimal point and bringing down zeros repeatedly gives a remainder of at each step: Answer > (i) Part (ii) Find the quotient of . Step 1 · Divide 100 by 11 Divide by (): Adding a decimal point and bringing down gives , which is smaller than , giving quotient digit and remainder . Bringing down another gives , continuing the cycle: Answer > (ii) Common Mistakes Missing Zero in Quotient: Forgetting to place a in the quotient when after bringing down the first zero, mistakenly writing instead of . Incorrect Bar Placement: Placing the bar over only one digit (e.g., ) instead of over both repeating digits ().

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Question 13

Can you find the quotients of $10 \div 9$ , and $100 \div 11$ ?

Solution

Understand the Question

To find the quotients of $10 \div 9$ and $100 \div 11$ , perform long division to express them as decimals.
Both divisions yield non-terminating repeating (recurring) decimals, represented using bar notation over the repeating digit block.

(i) Find the quotient of $10 \div 9$ .

Step 1 · Divide 10 by 9

Diagram 1

Divide $10$ by $9$ : $10 \div 9 = 1 \text{ with remainder } 1$

Adding a decimal point and bringing down zeros repeatedly gives a remainder of $1$ at each step: $10 \div 9 = 1.111\dots = 1.\overline{1}$

Answer

(i) $1.\overline{1}$

(ii) Find the quotient of $100 \div 11$ .

Step 1 · Divide 100 by 11

Diagram 2

Divide $100$ by $11$ ( $11 \times 9 = 99$ ): $100 \div 11 = 9 \text{ with remainder } 1$

Adding a decimal point and bringing down $0$ gives $10$ , which is smaller than $11$ , giving quotient digit $0$ and remainder $10$ . Bringing down another $0$ gives $100$ , continuing the cycle: $100 \div 11 = 9.0909\dots = 9.\overline{09}$

Answer

(ii) $9.\overline{09}$

Common Mistakes

Missing Zero in Quotient: Forgetting to place a $0$ in the quotient when $10 < 11$ after bringing down the first zero, mistakenly writing $9.9\dots$ instead of $9.0909\dots$ .
Incorrect Bar Placement: Placing the bar over only one digit (e.g., $9.0\overline{9}$ ) instead of over both repeating digits ( $9.\overline{09}$ ).

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