Exam-Style Questions.

1.	GCSE Higher

Prove algebraically that \(0.2\dot6\) can be written as \( \dfrac{4}{15}\)

Worked Solution

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2.	GCSE Higher

Use algebra to prove that \(0.3\dot1\dot8 \times 0.\dot8\) is equal to \( \frac{28}{99} \).

Worked Solution

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3.	GCSE Higher

(a) Prove that the recurring decimal \(0.\dot2 \dot1\) has the value \(\frac{7}{33}\)

(b) The value of \(x\) is given as:

$$x=\frac{1}{5^{120}\times2^{123}}$$

Show that, when \(x\) is written as a terminating decimal, there are 120 zeros after the decimal point.

(c) The reciprocal of any prime number \(p\) (where \(p\) is neither 2 nor 5) when written as a decimal, is always a recurring decimal.

A theorem in mathematics states:

The period of a recurring decimal is the least value of \(n\) for which \(p\) is a factor of \(10^n – 1\)

Marilou tests this theorem for the reciprocal of eleven.

She uses her calculator to show that 11 is a factor of \(10^2 – 1\) then makes this statement:

"The period of the recurring decimal is 2 because 11 is a factor of \(10^2-1\). This shows the theorem to be true in this case."

Explain why Marilou's statement is incomplete.

Worked Solution

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4.	GCSE Higher

(a) Use algebra to show that the recurring decimal \(0.\dot2 \dot4\) can be written as \( \frac{8}{33}\).

(b) Find the fraction, in its lowest terms, equal to the recurring decimal \(0.5 \dot2 \dot4\).

Worked Solution

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