Solved on Feb 14, 2024

Find the fraction equivalent of the repeating decimal $0.\overline{074}$ .

STEP 1

Assumptions
1. We are given a repeating decimal $0.\overline{074}$ .
2. We need to express this repeating decimal as a fraction in simplest form.

STEP 2

Let's denote the repeating decimal by $x$ .
$x = 0.\overline{074}$

STEP 3

To convert the repeating decimal into a fraction, we can multiply $x$ by a power of 10 that shifts the decimal point to the right so that the repeating part aligns with the original decimal.
Since the repeating part is three digits long, we multiply by $10^3 = 1000$ .
$1000x = 74.\overline{074}$

STEP 4

Now, we subtract the original number $x$ from $1000x$ to eliminate the repeating part.
$1000x - x = 74.\overline{074} - 0.\overline{074}$

STEP 5

Perform the subtraction on the left-hand side of the equation.
$999x = 74.\overline{074} - 0.\overline{074}$

STEP 6

Perform the subtraction on the right-hand side of the equation.
Since the repeating parts cancel out, we are left with:
$999x = 74$

STEP 7

Now, we solve for $x$ by dividing both sides of the equation by 999.
$x = \frac{74}{999}$

STEP 8

We need to check if the fraction can be simplified by finding the greatest common divisor (GCD) of the numerator and the denominator.

STEP 9

The GCD of 74 and 999 is 1, as 74 is a product of prime numbers (2 and 37) and neither of these prime factors is a divisor of 999.

STEP 10

Since the GCD is 1, the fraction is already in its simplest form.
$x = \frac{74}{999}$
Therefore, $0.\overline{074}$ expressed as a fraction in simplest form is $\frac{74}{999}$ .

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Find the fraction equivalent of the repeating decimal $0.\overline{074}$ .

STEP 1

Assumptions
1. We are given a repeating decimal $0.\overline{074}$ .
2. We need to express this repeating decimal as a fraction in simplest form.

STEP 2

Let's denote the repeating decimal by $x$ .
$x = 0.\overline{074}$

STEP 3

To convert the repeating decimal into a fraction, we can multiply $x$ by a power of 10 that shifts the decimal point to the right so that the repeating part aligns with the original decimal.
Since the repeating part is three digits long, we multiply by $10^3 = 1000$ .
$1000x = 74.\overline{074}$

STEP 4

Now, we subtract the original number $x$ from $1000x$ to eliminate the repeating part.
$1000x - x = 74.\overline{074} - 0.\overline{074}$

STEP 5

Perform the subtraction on the left-hand side of the equation.
$999x = 74.\overline{074} - 0.\overline{074}$

STEP 6

Perform the subtraction on the right-hand side of the equation.
Since the repeating parts cancel out, we are left with:
$999x = 74$

STEP 7

Now, we solve for $x$ by dividing both sides of the equation by 999.
$x = \frac{74}{999}$

STEP 8

We need to check if the fraction can be simplified by finding the greatest common divisor (GCD) of the numerator and the denominator.

STEP 9

The GCD of 74 and 999 is 1, as 74 is a product of prime numbers (2 and 37) and neither of these prime factors is a divisor of 999.

STEP 10

Since the GCD is 1, the fraction is already in its simplest form.
$x = \frac{74}{999}$
Therefore, $0.\overline{074}$ expressed as a fraction in simplest form is $\frac{74}{999}$ .

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Find the fraction equivalent of the repeating decimal 0.074‾0.\overline{074}0.074.

STEP 1

Assumptions1. We are given a repeating decimal 0.074‾0.\overline{074}0.074.2. We need to express this repeating decimal as a fraction in simplest form.

STEP 2

Let's denote the repeating decimal by xxx.x=0.074‾x = 0.\overline{074}x=0.074

STEP 3

STEP 4

Now, we subtract the original number xxx from 1000x1000x1000x to eliminate the repeating part.1000x−x=74.074‾−0.074‾1000x - x = 74.\overline{074} - 0.\overline{074}1000x−x=74.074−0.074

STEP 5

Perform the subtraction on the left-hand side of the equation.999x=74.074‾−0.074‾999x = 74.\overline{074} - 0.\overline{074}999x=74.074−0.074

STEP 6

Perform the subtraction on the right-hand side of the equation.Since the repeating parts cancel out, we are left with:999x=74999x = 74999x=74

STEP 7

Now, we solve for xxx by dividing both sides of the equation by 999.x=74999x = \frac{74}{999}x=99974​

STEP 8

We need to check if the fraction can be simplified by finding the greatest common divisor (GCD) of the numerator and the denominator.

STEP 9

The GCD of 74 and 999 is 1, as 74 is a product of prime numbers (2 and 37) and neither of these prime factors is a divisor of 999.

STEP 10

Since the GCD is 1, the fraction is already in its simplest form.x=74999x = \frac{74}{999}x=99974​Therefore, 0.074‾0.\overline{074}0.074 expressed as a fraction in simplest form is 74999\frac{74}{999}99974​.

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Find the fraction equivalent of the repeating decimal $0.\overline{074}$ .

Assumptions
1. We are given a repeating decimal $0.\overline{074}$ .
2. We need to express this repeating decimal as a fraction in simplest form.

Let's denote the repeating decimal by $x$ .
$x = 0.\overline{074}$

Now, we subtract the original number $x$ from $1000x$ to eliminate the repeating part.
$1000x - x = 74.\overline{074} - 0.\overline{074}$

Perform the subtraction on the left-hand side of the equation.
$999x = 74.\overline{074} - 0.\overline{074}$

Perform the subtraction on the right-hand side of the equation.
Since the repeating parts cancel out, we are left with:
$999x = 74$

Now, we solve for $x$ by dividing both sides of the equation by 999.
$x = \frac{74}{999}$

Since the GCD is 1, the fraction is already in its simplest form.
$x = \frac{74}{999}$
Therefore, $0.\overline{074}$ expressed as a fraction in simplest form is $\frac{74}{999}$ .