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.