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13 (a) (i) Write \( \frac{1}{3} \) as a recurring decimal - OCR - GCSE Maths - Question 13 - 2019 - Paper 1
Question 13
13 (a) (i) Write \( \frac{1}{3} \) as a recurring decimal. (ii) Write \( \frac{1}{30} \) as a recurring decimal. (b) Simplify fully by rationalising the denominato... show full transcript
Worked Solution & Example Answer:13 (a) (i) Write \( \frac{1}{3} \) as a recurring decimal - OCR - GCSE Maths - Question 13 - 2019 - Paper 1
Step 1
(i) Write \( \frac{1}{3} \) as a recurring decimal.
Answer
To convert ( \frac{1}{3} ) into a recurring decimal, we can perform long division. Dividing 1 by 3 results in:
- 1.000000...
- 3 goes into 10 three times (3 x 3 = 9), leaving a remainder of 1.
- Bringing down another 0 yields another 10, and the process repeats.
Thus, ( \frac{1}{3} = 0.333... ) or simply ( 0.3\overline{3} ).
Step 2
(ii) Write \( \frac{1}{30} \) as a recurring decimal.
Answer
To find ( \frac{1}{30} ), we similarly use long division:
- 1.000000...
- 30 goes into 100 three times (3 x 30 = 90), leaving a remainder of 10.
- Bringing down another 0 gives us 100 again, and it continues in this cycle.
Thus, ( \frac{1}{30} = 0.0333... ) or ( 0.0\overline{3} ).
Step 3
(b) Simplify fully by rationalising the denominator: \( \frac{20}{\sqrt{5}} \)
Answer
To simplify ( \frac{20}{\sqrt{5}} ), we multiply the numerator and the denominator by ( \sqrt{5} ):
[ \frac{20 \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} = \frac{20\sqrt{5}}{5} = 4\sqrt{5}. ]
Thus, the simplified form is ( 4\sqrt{5} ).