What is the value of 0.6 + 0.3? Write your answer as a fraction in lowest terms. 0.6 is repeated.
The Correct Answer and Explanation is:
To find the value of 0.6 (repeating) + 0.3, we will first convert both numbers into fractions, then add them.
Step 1: Convert 0.6 repeating to a fraction
Let:
x = 0.666…
Multiply both sides by 10:
10x = 6.666…
Now subtract the original equation:
10x − x = 6.666… − 0.666…
9x = 6
x = 6/9
Simplify:
x = 2/3
So, 0.6 repeating = 2/3
Step 2: Convert 0.3 to a fraction
0.3 = 3/10
Step 3: Add the two fractions
We are adding:
2/3 + 3/10
To add these, find a common denominator. The least common denominator of 3 and 10 is 30.
Convert both fractions:
- 2/3=(2×10)/(3×10)=20/302/3 = (2×10)/(3×10) = 20/302/3=(2×10)/(3×10)=20/30
- 3/10=(3×3)/(10×3)=9/303/10 = (3×3)/(10×3) = 9/303/10=(3×3)/(10×3)=9/30
Now add:
20/30 + 9/30 = 29/30
Final Answer:
0.6 (repeating) + 0.3 = 29/30
Explanation
To solve the expression 0.6 repeating plus 0.3, we begin by understanding what each number represents. The number 0.6 repeating, written as 0.666…, means the digit 6 continues forever. This is a recurring decimal, and recurring decimals can be expressed exactly as fractions. The method to convert such decimals involves algebra.
We let x = 0.666… and multiply both sides by 10 to shift the decimal point. Doing this gives us 10x = 6.666…. Subtracting the original value x = 0.666… from this equation removes the repeating part, resulting in 9x = 6. Solving for x gives x = 6/9, which simplifies to 2/3.
Next, we convert 0.3 into a fraction. Since 0.3 is a terminating decimal, we can write it as 3/10. Now we want to add 2/3 + 3/10. To do this, we find a common denominator. The least common denominator of 3 and 10 is 30. We convert both fractions so they have the same denominator: 2/3 becomes 20/30, and 3/10 becomes 9/30.
Adding them gives us 20/30 + 9/30 = 29/30.
Therefore, 0.6 repeating plus 0.3 equals 29/30, which is the correct answer in lowest terms.
