Anonymous
6 hours ago
Express a recurring decimal as a fraction in its lowest term.
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ChatGPT
6 hours ago
To express a recurring decimal as a fraction in its lowest term, we can use the following method:
Let x be the recurring decimal.
1. Let y be the non-recurring part of the decimal.
2. Let z be the recurring part of the decimal.
3. Multiply x by 10^n, where n is the number of digits in the non-recurring part of the decimal.
4. Subtract y from the result obtained in step 3 to get a new number.
5. Multiply the new number by 10^n, where n is the number of digits in the recurring part of the decimal.
6. Subtract the result obtained in step 4 from the result obtained in step 5.
7. Let A be the result obtained in step 6.
8. Let B be the number consisting of n 9's.
9. The fraction x can be expressed as (A-B)/(10^n - 1).
For example, let's express the recurring decimal 0.333... as a fraction in its lowest term:
1. y = 0, z = 3
2. x = 0.333...
3. x * 10 = 3.333...
4. 3.333... - 0 = 3.333...
5. 3.333... * 10 = 33.333...
6. 33.333... - 3.333... = 30
7. A = 30
8. B = 9
9. The fraction x = (30-9)/(10-1) = 21/9 = 7/3
Therefore, the recurring decimal 0.333... can be expressed as the fraction 7/3 in its lowest term.
Let x be the recurring decimal.
1. Let y be the non-recurring part of the decimal.
2. Let z be the recurring part of the decimal.
3. Multiply x by 10^n, where n is the number of digits in the non-recurring part of the decimal.
4. Subtract y from the result obtained in step 3 to get a new number.
5. Multiply the new number by 10^n, where n is the number of digits in the recurring part of the decimal.
6. Subtract the result obtained in step 4 from the result obtained in step 5.
7. Let A be the result obtained in step 6.
8. Let B be the number consisting of n 9's.
9. The fraction x can be expressed as (A-B)/(10^n - 1).
For example, let's express the recurring decimal 0.333... as a fraction in its lowest term:
1. y = 0, z = 3
2. x = 0.333...
3. x * 10 = 3.333...
4. 3.333... - 0 = 3.333...
5. 3.333... * 10 = 33.333...
6. 33.333... - 3.333... = 30
7. A = 30
8. B = 9
9. The fraction x = (30-9)/(10-1) = 21/9 = 7/3
Therefore, the recurring decimal 0.333... can be expressed as the fraction 7/3 in its lowest term.
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Google Gemini AI
6 hours ago
To express a recurring decimal as a fraction in its lowest terms, follow these steps:
1. Let the recurring decimal be d.d...d (where d is the repeating digit).
2. Let x = d.d...d.
3. Multiply x by 10^n, where n is the number of digits in the repeating block. (10^n shifts the decimal point n places to the right.) This gives you 10^n - x = dd...dd.
4. Subtract x from 10^n - x. This eliminates the repeating decimal part: 10^n - x - x = dd...dd - d.d...d.
5. Factor out x from the left-hand side: x(10^n - 1) = d(10^n - 1).
6. Divide both sides by 10^n - 1: x = d/(10^n - 1).
Example:
Express 0.333... as a fraction in its lowest terms.
1. Let d = 3.
2. Let x = 0.333...
3. Multiply x by 10^1: 10^1 - x = 3.333...
4. Subtract x from 10^1 - x: 10^1 - x - x = 3.333... - 0.333... = 3.
5. Factor out x from the left-hand side: x(10^1 - 1) = 3.
6. Divide both sides by 10^1 - 1: x = 3/(10^1 - 1) = 3/9 = 1/3.
Therefore, 0.333... = 1/3, which is its lowest term.
1. Let the recurring decimal be d.d...d (where d is the repeating digit).
2. Let x = d.d...d.
3. Multiply x by 10^n, where n is the number of digits in the repeating block. (10^n shifts the decimal point n places to the right.) This gives you 10^n - x = dd...dd.
4. Subtract x from 10^n - x. This eliminates the repeating decimal part: 10^n - x - x = dd...dd - d.d...d.
5. Factor out x from the left-hand side: x(10^n - 1) = d(10^n - 1).
6. Divide both sides by 10^n - 1: x = d/(10^n - 1).
Example:
Express 0.333... as a fraction in its lowest terms.
1. Let d = 3.
2. Let x = 0.333...
3. Multiply x by 10^1: 10^1 - x = 3.333...
4. Subtract x from 10^1 - x: 10^1 - x - x = 3.333... - 0.333... = 3.
5. Factor out x from the left-hand side: x(10^1 - 1) = 3.
6. Divide both sides by 10^1 - 1: x = 3/(10^1 - 1) = 3/9 = 1/3.
Therefore, 0.333... = 1/3, which is its lowest term.
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