Question

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Solve this with an explanation/ clear working out, please, using the following rule : |

A fraction in its simplest form is not equal to a terminating decimal if its denominator contains any prime factors other than 2 or 5.

Answer 1

Answer:

1/2, 1/4, 1/5, 1/8, 1/10, 1/16, and 1/20

Step-by-step explanation:

To determine the fractions of the form 1/n that are equivalent to terminating decimals within the range of n from 2 to 20, we need to consider the prime factors of each denominator.

1. Prime factors of a number are the prime numbers that multiply together to give the original number.

2. A fraction in its simplest form will result in a terminating decimal if its denominator contains only 2s and/or 5s as prime factors.

3. Any other prime factors in the denominator will result in a repeating decimal instead of a terminating one.

Let's evaluate each denominator from 2 to 20:

- For n = 2, the denominator only has the prime factor 2, which is acceptable. So, 1/2 is equivalent to a terminating decimal.

- For n = 3, the prime factor of the denominator is 3, which is not 2 or 5. Therefore, 1/3 will result in a repeating decimal.

- For n = 4, the prime factors of the denominator are 2^2, which are acceptable. So, 1/4 is equivalent to a terminating decimal.

- For n = 5, the prime factor of the denominator is 5, which is acceptable. Therefore, 1/5 is equivalent to a terminating decimal.

Continuing this evaluation for each value of n up to 20:

- Fractions 1/2, 1/4, 1/5, 1/8, 1/10, 1/16, and 1/20 have denominators containing only 2s and/or 5s, making them equivalent to terminating decimals.

- Fractions 1/3, 1/6, 1/7, 1/9, 1/11, 1/12, 1/13, 1/14, 1/15, 1/17, 1/18, 1/19 do not have denominators containing only 2s and/or 5s, resulting in repeating decimals.

Therefore, out of the fractions of the form 1/n where n ranges from 2 to 20, there are 7 fractions that are equivalent to terminating decimals.

Answer 2

Answer:

7

Step-by-step explanation:

A terminating decimal is one that ends after a finite number of decimal places.

For a fraction to result in a terminating decimal when expressed in decimal form, its denominator should only contain factors of 2 and/or 5.

To determine how many fractions in the form 1/n, where n is an integer and 2 ≤ n ≤ 20, are equivalent to terminating decimals, we need to express each value of n as the product of its prime factors.

We can immediately discount 3, 7, 11, 13, 17 and 19, since they are prime numbers and their only factors are 1 and themselves. Therefore, we have:

2 = 2

4 = 2 × 2

5 = 5

6 = 2 × 3

8 = 2 × 2 × 2

9 = 3 × 3

10 = 2 × 5

12 = 2 × 2 × 3

14 = 2 × 7

15 = 3 × 5

16 = 2 × 2 × 2 × 2

18 = 2 × 3 × 3

20 = 2 × 2 × 5

So, the values of n that only contain factors of 2 and/or 5 are:

2, 4, 8, 10, 16 and 20.

This means that the number of fractions of the form 1/n that are equivalent to terminating decimals is 7.