Answer :
To determine which expression is equivalent to [tex]\(\frac{1}{3}+\frac{2}{9}+\frac{1}{10}\)[/tex], we need to follow the steps to add these fractions. Here is a detailed step-by-step solution.
### Step 1: Find a Common Denominator
First, observe the denominators of the fractions: 3, 9, and 10. To add these fractions, we need a common denominator, which is the least common multiple (LCM) of these numbers. The LCM of 3, 9, and 10 is:
[tex]\[3 \times 9 \times 10 = 270\][/tex]
So, the common denominator is 270.
### Step 2: Convert Each Fraction
Next, we will express each fraction with the common denominator of 270.
1. Convert [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ \frac{1}{3} = \frac{1 \times 90}{3 \times 90} = \frac{90}{270} \][/tex]
2. Convert [tex]\(\frac{2}{9}\)[/tex]:
[tex]\[ \frac{2}{9} = \frac{2 \times 30}{9 \times 30} = \frac{60}{270} \][/tex]
3. Convert [tex]\(\frac{1}{10}\)[/tex]:
[tex]\[ \frac{1}{10} = \frac{1 \times 27}{10 \times 27} = \frac{27}{270} \][/tex]
### Step 3: Add the Numerators
Now, add the converted fractions by summing the numerators:
[tex]\[ \frac{90}{270} + \frac{60}{270} + \frac{27}{270} = \frac{90 + 60 + 27}{270} = \frac{177}{270} \][/tex]
### Step 4: Simplify the Fraction
The fraction [tex]\(\frac{177}{270}\)[/tex] cannot be simplified further since 177 and 270 do not share any common factors other than 1. Therefore, the simplified fraction remains [tex]\(\frac{177}{270}\)[/tex].
However, it must be noted that the fraction results approximately in decimal form as:
[tex]\[ \frac{177}{270} \approx 0.2851851851851852 \][/tex]
### Conclusion
Given the detailed steps, let's compare this result to the options provided:
A. [tex]\(\frac{1+2+1}{3+9+10} = \frac{4}{22}\)[/tex] (This is incorrect as it results in 0.181)
B. [tex]\(\frac{30+20+9}{90} = \frac{59}{90}\)[/tex] (This is incorrect as it results in 0.655)
C. [tex]\(\frac{1+2+1}{90} = \frac{4}{90}\)[/tex] (This is incorrect as it results in 0.044)
D. [tex]\(\frac{1 \times 2 \times 1}{3 \times 9 \times 10} = \frac{2}{270}\)[/tex] (This is incorrect as it results in 0.0074)
E. [tex]\(\frac{1 \times 2 \times 1}{90} = \frac{2}{90}\)[/tex] (This is incorrect as it results in 0.022)
Clearly, none of the options directly match the simplified fractional form, but considering approximate equivalence in decimal form, the expression corresponding to our calculation which maintains accuracy is:
\[
\boxed{\frac{77}{270} \approx 0.2851851851851852}\
Checking all the given expressions none of them matches the desired result accurately. The best approach would be a revision or re-interpretation of answers if expressions undergo manipulation.
### Step 1: Find a Common Denominator
First, observe the denominators of the fractions: 3, 9, and 10. To add these fractions, we need a common denominator, which is the least common multiple (LCM) of these numbers. The LCM of 3, 9, and 10 is:
[tex]\[3 \times 9 \times 10 = 270\][/tex]
So, the common denominator is 270.
### Step 2: Convert Each Fraction
Next, we will express each fraction with the common denominator of 270.
1. Convert [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ \frac{1}{3} = \frac{1 \times 90}{3 \times 90} = \frac{90}{270} \][/tex]
2. Convert [tex]\(\frac{2}{9}\)[/tex]:
[tex]\[ \frac{2}{9} = \frac{2 \times 30}{9 \times 30} = \frac{60}{270} \][/tex]
3. Convert [tex]\(\frac{1}{10}\)[/tex]:
[tex]\[ \frac{1}{10} = \frac{1 \times 27}{10 \times 27} = \frac{27}{270} \][/tex]
### Step 3: Add the Numerators
Now, add the converted fractions by summing the numerators:
[tex]\[ \frac{90}{270} + \frac{60}{270} + \frac{27}{270} = \frac{90 + 60 + 27}{270} = \frac{177}{270} \][/tex]
### Step 4: Simplify the Fraction
The fraction [tex]\(\frac{177}{270}\)[/tex] cannot be simplified further since 177 and 270 do not share any common factors other than 1. Therefore, the simplified fraction remains [tex]\(\frac{177}{270}\)[/tex].
However, it must be noted that the fraction results approximately in decimal form as:
[tex]\[ \frac{177}{270} \approx 0.2851851851851852 \][/tex]
### Conclusion
Given the detailed steps, let's compare this result to the options provided:
A. [tex]\(\frac{1+2+1}{3+9+10} = \frac{4}{22}\)[/tex] (This is incorrect as it results in 0.181)
B. [tex]\(\frac{30+20+9}{90} = \frac{59}{90}\)[/tex] (This is incorrect as it results in 0.655)
C. [tex]\(\frac{1+2+1}{90} = \frac{4}{90}\)[/tex] (This is incorrect as it results in 0.044)
D. [tex]\(\frac{1 \times 2 \times 1}{3 \times 9 \times 10} = \frac{2}{270}\)[/tex] (This is incorrect as it results in 0.0074)
E. [tex]\(\frac{1 \times 2 \times 1}{90} = \frac{2}{90}\)[/tex] (This is incorrect as it results in 0.022)
Clearly, none of the options directly match the simplified fractional form, but considering approximate equivalence in decimal form, the expression corresponding to our calculation which maintains accuracy is:
\[
\boxed{\frac{77}{270} \approx 0.2851851851851852}\
Checking all the given expressions none of them matches the desired result accurately. The best approach would be a revision or re-interpretation of answers if expressions undergo manipulation.
