What is 0.46 with 6 repeating as a fraction
The Correct Answer and Explanation is:
To convert 0.4666… (where the 6 repeats) into a fraction:
Let
x = 0.4666…
Multiply both sides by 10 to move the decimal point one place right:
10x = 4.666…
Now subtract the original equation from this new equation:
10x – x = 4.666… – 0.4666…
This simplifies to:
9x = 4.2
Now divide both sides by 9:
x = 4.2 / 9
Since 4.2 is a decimal, convert it to a fraction:
4.2 = 42 / 10
So:
x = (42 / 10) ÷ 9 = 42 / (10 × 9) = 42 / 90
Now simplify 42/90 by dividing both numerator and denominator by 6:
42 ÷ 6 = 7
90 ÷ 6 = 15
Final answer: 0.4666… = 7/15
Explanation
Repeating decimals can be expressed as fractions using algebraic methods. In this case, 0.4666… has one digit repeating: the number 6. First, we assign the decimal to a variable, calling it x. That gives us:
x = 0.4666…
To eliminate the repeating part, we multiply x by 10. This moves the decimal point one place to the right, giving us:
10x = 4.666…
Now, subtracting the original x = 0.4666… from this new expression allows the repeating parts to cancel out:
10x – x = 4.666… – 0.4666…, which simplifies to:
9x = 4.2
To isolate x, divide both sides by 9:
x = 4.2 ÷ 9
Now we deal with the decimal 4.2. It can be rewritten as a fraction:
4.2 = 42/10
So:
x = (42/10) ÷ 9 = 42 / (10 × 9) = 42/90
This fraction can be simplified. The greatest common divisor of 42 and 90 is 6. Dividing both by 6 gives:
42 ÷ 6 = 7, 90 ÷ 6 = 15, so:
x = 7/15
Therefore, the repeating decimal 0.4666… is equal to the fraction 7/15.
