A measurement using a ruler marked to an

Given: The measurement using a ruler marked to an eighth of an inch is reported as 3 1/2 inches.
We need to find the range of values for the actual measurement.
To understand the range, let's consider the possible values that can be represented by the measurement 3 1/2 inches.
The fraction 1/2 represents half of an inch. Since the ruler is marked to an eighth of an inch, we can divide the half-inch into 8 equal parts.
So, each part represents 1/8 of an inch.
To find the range, we need to consider the measurement with the minimum possible value and the measurement with the maximum possible value.
Minimum Possible Value:
To find the minimum value, we need to subtract 1/8 from the reported measurement of 3 1/2 inches.
The minimum value is obtained by subtracting 1/8 from 3 1/2:
$3\frac{1}{2}-\frac{1}{8}$
To subtract fractions, we need a common denominator. The common denominator of 2 and 8 is 8.
$3\frac{1}{2}-\frac{1}{8}=\frac{7}{2}-\frac{1}{8}$
Next, we convert the whole number 3 to an improper fraction:
$\frac{7}{2}-\frac{1}{8}=\frac{7}{2}-\frac{1}{8}=\frac{28}{8}-\frac{1}{8}$
Now, we can subtract the fractions:
$\frac{28}{8}-\frac{1}{8}=\frac{28-1}{8}=\frac{27}{8}$
Therefore, the minimum possible value for the actual measurement is $\frac{27}{8}$ inches.
Maximum Possible Value:
To find the maximum value, we need to add 1/8 to the reported measurement of 3 1/2 inches.
The maximum value is obtained by adding 1/8 to 3 1/2:
$3\frac{1}{2}+\frac{1}{8}$
Following the same steps as before, we convert 3 to an improper fraction:
$\frac{7}{2}+\frac{1}{8}$
Now, we can add the fractions:
$\frac{7}{2}+\frac{1}{8}=\frac{28}{8}+\frac{1}{8}$
$\frac{28}{8}+\frac{1}{8}=\frac{28+1}{8}=\frac{29}{8}$
Therefore, the maximum possible value for the actual measurement is $\frac{29}{8}$ inches.
In conclusion, the range of values for the actual measurement, when using a ruler marked to an eighth of an inch and reporting 3 1/2 inches, is from $\frac{27}{8}$ inches to $\frac{29}{8}$ inches.