Correct answer: 3/9.

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

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.

In this particular case, the integer part is zero, so the answer is .

**Method 2: algebraic**

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

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

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

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

Correct answer: 9/13.

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

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.

**Method 2: algebraic**

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

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

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

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

Correct answer: 41/99

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

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.

**Method 2: algebraic**

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

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

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

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

Correct answer: 2505/990

We can rewrite as: , where 30 is the period. This is a compound decimal.

**Step 1**: equal to x.

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

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

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

**step 3**: Subtract equation I from II.

**step 4**: Isolate the x and do the division.

Correct answer: 2025/990

We can rewrite as: , where 45 is the period.

**Step 1**: equal to x.

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

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

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

**step 3**: Subtract equation I from II.

**step 4**: Isolate the x and do the division.

Correct answer: a) 2

Doing the division, we find:

Note that the decimal can be rewritten as:

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: , where 36 is the period.

**Step 1**: equal to x.

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

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

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

**step 4**: Subtract equation I from II.

**step 5**: isolate the x.

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

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: , where 012 is the period.

**Step 1**: equal to x obtaining equation I.

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

**step 3**: Subtract equation I from II.

**step 4**: Isolate the x and do the division.

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

So just subtract 1 003 - 333 = 670.