Convert 0.72¯¯¯¯¯ to a rational number in simplest form.
Which of the following is a rational number between 5 and 514?(1 point)
Responses
26−−√
start root 26 end root
517
5 Start Fraction 1 over 7 end fraction
525
5 Start Fraction 2 over 5 end fraction
513
To convert a recurring decimal into a rational number, we need to form an equation.
Let x = 0.72¯¯¯¯¯.
Multiply both sides of the equation by 100 to remove the recurring part:
100x = 72.72¯¯¯¯¯
Now, subtract the original equation from the second equation:
100x - x = 72.72¯¯¯¯¯ - 0.72¯¯¯¯¯
Simplifying the right side:
99x = 72
Divide both sides of the equation by 99:
x = 72/99
To simplify 72/99, we can find the greatest common divisor (GCD) of the numerator and the denominator, which is 9.
Dividing both the numerator and denominator by 9:
72/99 = 8/11
Therefore, the rational number equivalent to 0.72¯¯¯¯¯ is 8/11 in simplest form.
To convert the repeating decimal 0.72¯¯¯¯¯ to a rational number in simplest form, we can use the method of converting repeating decimals to fractions.
Step 1: Assign a variable to the repeating decimal
Let x = 0.72¯¯¯¯¯
Step 2: Multiply both sides of the equation by a power of 10 to remove the decimal places from the repeating part.
10x = 7.2¯¯¯¯¯
Step 3: Subtract the original equation from the new equation to eliminate the repeating part.
10x - x = 7.2¯¯¯¯¯ - 0.72¯¯¯¯¯
9x = 7.2
Step 4: Solve for x by dividing both sides of the equation by 9.
x = 7.2 / 9
x = 0.8
So, the rational number equivalent of the repeating decimal 0.72¯¯¯¯¯ is 0.8 in simplest form.
To convert 0.72¯¯¯¯¯ to a rational number in simplest form, let's call it x.
To convert the repeating decimal to a fraction, we can use the method of algebraic manipulation.
Let's multiply both sides of the equation x = 0.72¯¯¯¯¯ by 100, which moves the decimal point two places to the right: 100x = 72.72¯¯¯¯¯
Now, subtract the original equation x = 0.72¯¯¯¯¯ from the second equation 100x = 72.72¯¯¯¯¯:
100x - x = 72.72¯¯¯¯¯ - 0.72¯¯¯¯¯
99x = 72
To simplify the fraction, divide both sides of the equation by 99:
99x / 99 = 72 / 99
x = 72/99
To simplify the fraction further, we can find the greatest common divisor (GCD) between 72 and 99, which is 9:
72/9 = 8
99/9 = 11
So, the simplified fraction is 8/11.
Therefore, 0.72¯¯¯¯¯ is equivalent to the rational number 8/11 in simplest form.