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(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} $
(a) (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 = $ $ = 2 \times 5 $
$100 = 10 \times 10 = 2 \times 5 \times \text{______} \times \text{______}$
$1000 = 10 \times 10 \times 10 = 2 \times 5 \times \text{______} \times \text{______} \times \text{______}$
(a) (iii) Complete the table.
| Fraction | $\frac{1}{2}$ | $\frac{1}{5}$ | $\frac{7}{20}$ | $\frac{1}{25}$ | $\frac{3}{500}$ |
| -------- | ------- | ------- | -------- | ------- | -------- |
| Decimal | 0.5 | 0.2 | | | |
(a) (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.
$20 = 2 \times 2 \times 5$
$25 = 5 \times 5$
$50 = 2 \times \text{______} \times 5$
$100 = \text{______} \times 5 \times 5$
$500 = 2 \times 2 \times \text{______} \times \text{______} \times \text{______}$
(b) (ii) Use your answers to part (i) to help you complete the table.
| Fraction | Decimal | Number of decimal places | Denominator written as a product of primes using powers | Larger power |
| -------- | ------- | ------------------------ | ---------------------------------------------------- | ------------ |
| $\frac{1}{20}$ | 0.05 | 2 | $2^{2} \times 5$ | 2 |
| $\frac{7}{25}$ | 0.28 | 2 | $5^{2}$ | 2 |
| $\frac{9}{50}$ | 0.18 | 2 | | 2 |
| $\frac{19}{100}$ | 0.19 | | | |
| $\frac{13}{200}$ | 0.065 | 3 | $2^{3} \times 5^{2}$ | 3 |
| $\frac{11}{500}$ | 0.022 | | | |
| $\frac{17}{5000}$ | 0.0034| | $2^{3} \times 5^{4}$ | 4 |
(b) (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.
(b) (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.
575
586
560
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.
$\frac{1}{3} = \frac{1}{9} \quad \frac{1}{11} = \frac{1}{99} \quad \frac{1}{37} = \frac{1}{999} \quad \frac{1}{111} = \frac{1}{999} \quad \frac{1}{41} = \frac{1}{99\,999} \quad \frac{1}{7} = \frac{1}{999\,999}$
(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 $\frac{1}{259}$.
...............................................................
566
576
578
Some decimals have non-repeating decimal parts followed by repeating decimal parts.
Example
$$0.6\overline{5} = 0.65555\ldots$$
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.
[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.
554
588
558
