Question

1. higher order thinking deondra measures an object as $2\frac{3}{4}$ inches to the nearest quarter inch. abdul measures the same object as $2\frac{1}{2}$ inches to the nearest half inch. can they both be correct? explain.

Explanation:

Step1: Simplify Deondra's measurement

$2\frac{3}{4} = 2.75$ inches

Step2: Define quarter-inch rounding range

A measurement rounded to the nearest quarter inch as $2\frac{3}{4}$ inches falls in:
$2\frac{3}{4} - \frac{1}{8} \leq \text{actual length} < 2\frac{3}{4} + \frac{1}{8}$
$\frac{22}{8} - \frac{1}{8} \leq \text{actual length} < \frac{22}{8} + \frac{1}{8}$
$2\frac{5}{8} \leq \text{actual length} < 2\frac{7}{8}$ inches

Step3: Define half-inch rounding range

A measurement rounded to the nearest half inch as $2\frac{1}{2}$ inches falls in:
$2\frac{1}{2} - \frac{1}{4} \leq \text{actual length} < 2\frac{1}{2} + \frac{1}{4}$
$2\frac{1}{4} \leq \text{actual length} < 2\frac{3}{4}$ inches

Step4: Find overlapping range

The intersection of the two ranges is:
$2\frac{5}{8} \leq \text{actual length} < 2\frac{3}{4}$ inches
This range is non-empty, so an actual length in this interval would round to both measurements.

Answer:

Yes, they can both be correct. Any object with an actual length between $2\frac{5}{8}$ inches (inclusive) and $2\frac{3}{4}$ inches (exclusive) will round to $2\frac{3}{4}$ inches when measured to the nearest quarter inch, and round to $2\frac{1}{2}$ inches when measured to the nearest half inch.