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This question is about terminating decimals.
(a) (i) Complete these equivalent fractions.
\[ \frac{1}{2} = \frac{5}{10} \quad \frac{1}{5} = \frac{2}{10} \quad \frac{7}{20} = \frac{\text{........}}{100} \quad \frac{1}{25} = \frac{\text{........}}{100} \quad \frac{3}{500} = \frac{\text{........}}{1000} \]
(ii) The denominators of the equivalent fractions in part (i) are 10, 100, and 1000.
The smallest prime number is 2.
Put a prime number in each box to complete these statements.
\[ 10 = \quad = 2 \times 5 \]
\[ 100 = 10 \times 10 = 2 \times 5 \times \text{\_} \times \text{\_} \]
\[ 1000 = 10 \times 10 \times 10 = 2 \times 5 \times \text{\_} \times \text{\_} \times \text{\_} \]
(iii) Complete the table.
\[ \begin{array}{c|c|c|c|c} \text{Fraction} & \frac{1}{2} & \frac{1}{5} & \frac{7}{20} & \frac{1}{25} & \frac{3}{500} \\ \hline \text{Decimal} & 0.5 & 0.2 & \text{\_} & \text{\_} & \text{\_} \end{array} \]
(iv) Write down a different fraction with a numerator of 1 and a denominator between 30 and 99 which can be written as a terminating decimal.
(b) (i) Put a prime number in each box to complete these statements.
\[ 20 = 2 \times 2 \times 5 \]
\[ 25 = 5 \times 5 \]
\[ 50 = 2 \times \text{\_} \times 5 \]
\[ 100 = \text{\_} \times \text{\_} \times 5 \times 5 \]
\[ 500 = 2 \times 2 \times \text{\_} \times \text{\_} \times \text{\_} \]
(ii) Use your answers to part (i) to help you complete the table.
\[ \begin{array}{c|c|c|c|c} \text{Fraction} & \text{Decimal} & \text{Number of decimal places} & \text{Denominator as product of primes} & \text{Larger power} \\ \hline \frac{1}{20} & 0.05 & 2 & 2^2 \times 5 & 2 \\ \frac{7}{25} & 0.28 & 2 & 5^2 & 2 \\ \frac{9}{50} & 0.18 & 2 & \text{\_} & 2 \\ \frac{19}{100} & 0.19 & \text{\_} & \text{\_} & \text{\_} \\ \frac{13}{200} & 0.065 & 3 & 2^3 \times 5^2 & 3 \\ \frac{11}{500} & 0.022 & 2 & \text{\_} & \text{\_} \\ \frac{17}{5000} & 0.0034 & \text{\_} & 2^3 \times 5^4 & 4 \end{array} \]
(iii) A fraction has a numerator of 1 and a denominator of $2^{14} \times 5^7$.
Write down the number of decimal places in the decimal form of this fraction.
\[ \text{.....................} \]
(iv) The denominator of a fraction that can be written as a terminating decimal only has one or two possible prime factors.
Write down these prime factors.
\[ \text{..................... and .....................} \]
519
504
594
This question is about repeating decimals.
The number of digits in the repeating pattern is called the repeat length.
Example
$$\frac{1}{13} = 0.076923\, 076923\, 076923\ldots = \underline{0.076\, 923}$$ This is a repeating decimal with a repeat length of 6.
(a) (i) Complete these equivalent fractions.
$$\frac{1}{3} = \frac{1}{9} \quad \frac{1}{11} = \frac{1}{99} \quad \frac{1}{37} = \frac{1}{999}$$
$$\frac{1}{111} = \frac{1}{999} \quad \frac{1}{41} = \frac{1}{99\,999} \quad \frac{1}{7} = \frac{1}{999\,999}$$
(ii) Complete the table.
[Table_1]
(iii) Use your answers to part (i) and part (ii) to help you complete the table.
[Table_2]
(iv) Give an example of a fraction with a numerator of 1 which can be written as a repeating decimal with a repeat length of 9.
(v) A repeating decimal has a repeat length of \( k \).
Write down an expression, in terms of \( k \), for the denominator of this fraction.
(b) (i) $$\frac{1}{407} = \frac{1}{11\times 37} = \frac{1}{11}\times \frac{1}{37}$$
$$\frac{1}{407}$$ is changed to its decimal form.
Show that this has a repeat length that is equal to the lowest common multiple (LCM) of the repeat lengths of the decimal forms of $$\frac{1}{11}$$ and $$\frac{1}{37}$$.
(ii) Show how the lowest common multiple (LCM) of the repeat lengths of $$\frac{1}{7}$$ and $$\frac{1}{37}$$ gives the repeat length of $$\frac{1}{259}$$.
599
581
501
Some decimals have non-repeating decimal parts followed by repeating decimal parts.
Example
0.65\overline{5} = 0.65555\ldots In this decimal, the 6 does not repeat but the 5 does.
(a) Show that adding the decimal forms of $\frac{1}{5}$ and $\frac{1}{3}$ gives a decimal of this type.
(b) Complete the table.
\begin{array}{|c|c|c|c|c|}\hline \text{Fraction} & \text{Decimal} & \text{Number of non-} \\ & & \text{repeating decimal places} & \text{Repeat length} & \text{Denominator written as} \\ & & & & \text{a product of primes using powers} \\ \hline \frac{1}{6} & 0.1\overline{6} & 1 & 1 & 2 \times 3 \\ \hline \frac{1}{12} & 0.08\overline{3} & 2 & 1 & \\ \hline \frac{7}{75} & & & 3 & \\ \hline \frac{11}{24} & & 3 & & \\ \hline \frac{317}{600} & 0.528 \overline{3} & & & 2^{3} \times 5^{2} \times 3 \\ \hline \frac{1}{1320} & 0.000 \overline{75} & 3 & 2 & 2^{3} \times 5 \times 11 \times 3 \\ \hline \frac{50001}{101750} & 0.491 \overline{41031} & 3 & 6 & 2 \times 5^{3} \times 11 \times 37 \\ \hline \end{array}
(c) A fraction of the form $\frac{1}{2^{a} \times 5^{b} \times c \times d}$ where $a$ and $b$ are positive integers and $c$ and $d$ are different primes is changed to its decimal form. Using your answers to question 1(b) and question 2(b), explain how to find the number of non-repeating decimal places and the repeat length.
530
544
546
