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(a) (i) Complete these equivalent fractions.
$\frac{1}{2} = \frac{5}{10}$
$\frac{1}{5} = \frac{2}{10}$
$\frac{7}{20} = \frac{\phantom{00}}{100}$
$\frac{1}{25} = \frac{\phantom{00}}{100}$
$\frac{3}{500} = \frac{\phantom{000}}{1000}$
(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.
$10 = \phantom{\quad} = 2 \times 5$
$100 = 10 \times 10 = 2 \times 5 \times \phantom{\quad} \times \phantom{\quad}$
$1000 = 10 \times 10 \times 10 = 2 \times 5 \times \phantom{\quad} \times \phantom{\quad} \times \phantom{\quad}$
(iii) Complete the table.
\[\begin{array}{|c|c|}\hline\text{Fraction} & \frac{1}{2} & \frac{1}{5} & \frac{7}{20} & \frac{1}{25} & \frac{3}{500} \\ \hline\text{Decimal} & 0.5 & 0.2 & \phantom{\quad} & \phantom{\quad} & \phantom{\quad} \\ \hline\end{array}\]
(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.
$20 = 2 \times 2 \times 5$
$25 = 5 \times 5$
$50 = 2 \times \phantom{\quad} \times 5$
$100 = \phantom{\quad} \times \phantom{\quad} \times 5 \times 5$
$500 = 2 \times 2 \times \phantom{\quad} \times \phantom{\quad} \times 5 \times \phantom{\quad}$
(ii) Use your answers to part (i) to help you complete the table.
\[\begin{array}{|c|c|c|c|c|}\hline\text{Fraction} & \text{Decimal} & \text{Number of decimal places} & \text{Denominator written as a product of primes using powers} & \text{Larger power} \\ \hline\frac{1}{20} & 0.05 & 2 & 2^{2} \times 5 & 2 \\ \frac{7}{25} & 0.28 & 2 & 5^{2} & 2 \\ \frac{9}{50} & 0.18 & 2 & \phantom{\quad} & \phantom{\quad} \\ \frac{19}{100} & 0.19 & \phantom{\quad} & \phantom{\quad} & 2 \\ \frac{13}{200} & 0.065 & 3 & 2^{3} \times 5^{2} & 3 \\ \frac{11}{500} & 0.022 & \phantom{\quad} & \phantom{\quad} & \phantom{\quad} \\\frac{17}{5000} & 0.0034 & \phantom{\quad} & 2^{3} \times 5^{4} & 4 \\ \hline\end{array}\]
(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.
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(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.
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546
537
593
This question is about repeating decimals.
The number of digits in the repeating pattern is called the repeat length.
Example
$$\frac{1}{13} = 0.076923 \overline{076923} \ldots = 0. \overline{076923}$$ This is a repeating decimal with a repeat length of 6.
(a) (i) Complete these equivalent fractions.
$$\frac{1}{3} = \frac{1}{9} \quad \frac{1}{11} = \frac{1}{99} \quad \frac{1}{37} = \frac{1}{999} \quad \frac{1}{111} = \frac{1}{999} \quad \frac{1}{41} = \frac{1}{99999} \quad \frac{1}{7} = \frac{1}{999999}$$
(ii) Complete the table.
\[\begin{array}{|c|c|c|c|c|c|c|}\hline\text{Fraction} & \frac{1}{3} & \frac{1}{11} & \frac{1}{37} & \frac{1}{111} & \frac{1}{41} & \frac{1}{7} \\\hline\text{Decimal} & 0.\overline{3} & 0.\overline{09} & 0.\overline{027} & & & 0.\overline{142857} \\\hline\text{Repeat length} & 1 & 2 & 3 & & 5 & 6 \\\hline\end{array}\]
(iii) Use your answers to part (i) and part (ii) to help you complete the table.
\[\begin{array}{|c|c|c|c|c|}\hline\text{Fraction} & \text{Decimal} & \text{Repeat length} & \text{Denominator of equivalent fraction} \\\hline\frac{1}{3} & 0.\overline{3} & 1 & 9 = 10^1 - 1 \\\hline\frac{1}{11} & 0.\overline{09} & 2 & 99 = 10^2 - 1 \\\hline\frac{1}{37} & 0.\overline{027} & 3 & 999 = \\\hline\frac{1}{111} & & 3 & 999 = \\\hline\frac{1}{41} & & 5 & 99999 = \\\hline\frac{1}{7} & 0.\overline{142857} & 6 & 999999 = \\\hline\end{array}\]
(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.
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(b) (i) $$\frac{1}{407} = \frac{1}{11 \times 37} = \frac{1}{11} \times \frac{1}{37}$$ 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 $\frac{1}{259}$.
518
558
535
(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.
[Table_1]
| Fraction | Decimal | Number of non-repeating decimal places | Repeat length | Denominator written as a product of primes using powers |
|-------------|------------------|----------------------------------------|---------------|----------------------------------------------------------|
| $\frac{1}{6}$ | 0.1\overline{6} | 1 | 1 | 2 \times 3 |
| $\frac{1}{12}$ | 0.08\overline{3} | 2 | 1 | |
| $\frac{7}{75}$ | | | | |
| $\frac{11}{24}$ | | 3 | | |
| $\frac{317}{600}$ | 0.528 \overline{3} | | | 2^{3} \times 5^{2} \times 3 |
| $\frac{1}{1320}$ | 0.000 \overline{75}$| 3 | 2 | 2^{3} \times 5 \times 11 \times 3 |
| $\frac{50001}{101750}$ | 0.491 \overline{410319}$| 3 | 6 | 2 \times 5^{3} \times 11 \times 37 |
(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.
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562
559
518
