Exercises on generating fraction and repeating decimal

Correct answer: 3/9.

The period, the part that repeats after the comma, is 3. Thus, the decimal can be written as: 0 comma 3 with slash superscript.

We can solve it by two methods:

Method 1: fractional

We add the whole part with a fraction, where the numerator will be the period and, in the denominator, a digit 9 for each digit different from the period.

0 space plus space 3 over 9

In this particular case, the integer part is zero, so the answer is 3 out of 9.

Method 2: algebraic

Step 1: we equate the decimal to x, obtaining equation I.

x equals 0 comma 3 with slash superscript space left parenthesis and q u atio n space I right parenthesis

Step 2: we multiply both sides of the equation by 10, obtaining equation II.

10 space. straight space x equals 10 space. space 0 comma 3 with slash superscript 10 straight x equals 3 comma 3 with slash superscript space left parenthesis and when space I I right parenthesis

Step 3: we subtract from equation II the equation I.

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Step 4: We isolate x and find the generating fraction.

x equals 3 over 9

Correct answer: 9/13.

The period, the part that repeats after the comma, is 4. Thus, the decimal can be written as: 1 comma 4 with slash superscript.

We can solve it by two methods:

Method 1: fractional

We add the whole part with a fraction, where the numerator will be the period and, in the denominator, a digit 9 for each digit different from the period.

1 space plus space 4 over 9 equals 9 over 9 plus 4 over 9 equals 13 over 9

Method 2: algebraic

Step 1: we equate the decimal to x, obtaining equation I.

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straight x equals 14 comma 4 with slash superscript space left parenthesis and when space I right parenthesis

Step 2: we multiply both sides of the equation by 10, obtaining equation II.

10 space. straight space x equals 10 space. space 1 comma 4 with slash superscript 10 straight x equals 14 comma 4 with slash superscript

Step 3: we subtract from equation II the equation I.

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Step 4: We isolate x and find the generating fraction.

straight x equals 13 over 9

Correct answer: 41/99

The period, the part that repeats after the comma, is 41. Thus, the decimal can be written as: 0 comma 41 with slash superscript.

We can solve it by two methods:

Method 1: fractional

We add the whole part with a fraction, where the numerator will be the period and, in the denominator, a digit 9 for each digit different from the period.

0 space plus space 41 over 99 equals 41 over 99

Method 2: algebraic

Step 1: we equate the decimal to x, obtaining equation I.

straight x equals 0 comma 41 with slash superscript space left parenthesis and when space I right parenthesis

Step 2: we multiply both sides of the equation by 100, obtaining equation II. (because there are two digits in the decimal).

100 space. straight space x equals 100 space. space 0 comma 41 with slash superscript 100 straight x equals 41 comma 41 with slash superscript space left parenthesis and q u a tion space I I right parenthesis

Step 3: we subtract from equation II the equation I.

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Step 4: We isolate x and find the generating fraction.

x equals 41 over 99

Correct answer: 2505/990

We can rewrite as: 2 comma 5 30 with slash superscript, where 30 is the period. This is a compound decimal.

Step 1: equal to x.

straight x equals 2 comma 5 30 with slash superscript

step 2: Multiply both sides of the equation by 10, obtaining equation I.

Since the tithe is compound, this will make it simple.

10 space. straight space x equals 10 space. space 2 comma 5 30 with slash superscript 10 straight x equals 25 comma 30 with slash superscript space left parenthesis and q u a tion space I right parenthesis

step 3: multiply equation I by 100 on both sides of the equality, obtaining equation II.

100 space. space 10 straight x equals 100 space. space 25 comma 30 with slash superscript 1 space 000 straight x equals 2 space 530 comma 30 with slash superscript

step 3: Subtract equation I from II.

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step 4: Isolate the x and do the division.

x equals numerator 2 space 505 over denominator 990 end of fraction equals 2 comma 5 30 with slash superscript space equals space 2 comma 5303030 space... space

Correct answer: 2025/990

We can rewrite as: 2 comma 0 45 with slash superscript, where 45 is the period.

Step 1: equal to x.

straight x equals 2 comma 0 45 with slash superscript

step 2: multiply both sides of the equation by 10, obtaining equation I.

Since the tithe is compound, this will make it simple.

10 space. straight space x equals 10 space. space 2 comma 0 45 with slash superscript 10 straight x equals 20 comma 45 with slash superscript space left parenthesis and q u a tion space I right parenthesis

step 3: multiply equation I by 100 on both sides of the equality, obtaining equation II.

100 space. space 10 straight x equals 100 space. space 20 comma 45 with slash superscript space 1 space 000 straight x equals 2 space 045 comma 45 with slash superscript space left parenthesis and what space I I right parenthesis

step 3: Subtract equation I from II.

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step 4: Isolate the x and do the division.

x equals numerator 2 space 025 over denominator 990 end of fraction equals 2 comma 0 45 with slash superscript space equals space 2 comma 0454545 space...

Correct answer: a) 2

Doing the division, we find:

numerator 22 space 229 over denominator 27 space 027 end of fraction equals 0 comma 822473 822473 822473 822473 space... space

Note that the decimal can be rewritten as: 0 comma 822473 with slash superscript

The period repeats every 6 digits, and the nearest integer multiple of the 50th decimal place will be:

6 x 8 = 48

Thus, the last digit 3 of the period will occupy the 48th decimal place. Therefore, in the next repetition, the first digit 2 will occupy the 50th position.

Correct answer: b) 89

It is necessary to determine the generating fraction and, after, simplify and add numerator and denominator.

We can rewrite as: 0 comma 011 36 with slash superscript, where 36 is the period.

Step 1: equal to x.

straight x equals 0 comma 011 36 with slash superscript

step 2: multiply both sides of the equation by 1000, obtaining equation I.

Since the tithe is compound, this will make it simple.

1000 space. straight space x equals 1000 space. space 0 comma 011 36 with slash superscript 1000 straight x equals 11 comma 36 with slash superscript space left parenthesis and q u a tion space I right parenthesis

step 3: multiply equation I by 100 on both sides of the equality, obtaining equation II.

100 space. space 1000 straight x equals 100 space. space 11 comma 36 with slash superscript space 100 space 000 straight x equals 1136 comma 36 with slash superscript space left parenthesis and q u a tion space I I right parenthesis

step 4: Subtract equation I from II.

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step 5: isolate the x.

x equals 1125 over 99000

Once the generating fraction is determined, we must simplify it. Dividing numerator and denominator by 25, by 9, and again by 9.

1125 over 99000 equals numerator 45 over denominator 3960 end of fraction equals 9 over 792 equals 1 over 88

So just add 1 + 88 = 89.

Correct answer: a) 670

It is necessary to determine the generating fraction and, after, simplify and subtract the numerator and denominator.

We can rewrite as: 3 comma 012 with slash superscript, where 012 is the period.

Step 1: equal to x obtaining equation I.

straight x equals 3 comma 012 with slash superscript space left parenthesis and q u a tion space I right parenthesis

step 2: multiply both sides of the equation by 1000, obtaining equation II.

1 space 000 space. straight space x equals 1 space 000 space. space 3 comma 012 with slash superscript 1 space 000 straight x equals 3 space 012 comma 012 with slash superscript space left parenthesis and what space I I right parenthesis

step 3: Subtract equation I from II.

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step 4: Isolate the x and do the division.

x equals numerator 3 space 009 over denominator 999 end of fraction equals 3 comma 012 with slash superscript

Once the generating fraction is determined, we must simplify it. Dividing numerator and denominator by 3.

numerator 3 space 009 over denominator 999 end of fraction equals numerator 1 space 003 over denominator 333 space end of fraction

So just subtract 1 003 - 333 = 670.

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