To find the diver's initial depth, we start with the information provided: after ascending \(10 \frac{3}{4}\) feet, the diver reaches a depth of \(-56 \frac{1}{2}\) feet.
-
Convert the mixed numbers to improper fractions for easier calculations:
- The diver ascends \(10 \frac{3}{4}\) feet: \[ 10 \frac{3}{4} = 10 + \frac{3}{4} = \frac{40}{4} + \frac{3}{4} = \frac{43}{4} \text{ feet} \]
- The current depth of the diver is \(-56 \frac{1}{2}\) feet: \[ -56 \frac{1}{2} = -56 - \frac{1}{2} = -\frac{112}{2} - \frac{1}{2} = -\frac{113}{2} \text{ feet} \]
-
Let \(x\) represent the diver's initial depth. According to the problem, we have: \[ x + \frac{43}{4} = -\frac{113}{2} \]
-
To isolate \(x\), we need to subtract \(\frac{43}{4}\) from both sides: \[ x = -\frac{113}{2} - \frac{43}{4} \]
-
To perform this subtraction, we need a common denominator. The least common multiple of \(2\) and \(4\) is \(4\). Convert \(-\frac{113}{2}\) to have a denominator of \(4\): \[ -\frac{113}{2} = -\frac{113 \times 2}{2 \times 2} = -\frac{226}{4} \]
-
Now we can perform the subtraction: \[ x = -\frac{226}{4} - \frac{43}{4} = -\frac{226 + 43}{4} = -\frac{269}{4} \]
-
To express \(-\frac{269}{4}\) as a mixed number, we divide \(269\) by \(4\): \[ 269 \div 4 = 67 \quad \text{(since } 4 \times 67 = 268 \text{)} \]
- The remainder is \(269 - 268 = 1\). Thus, \[ -\frac{269}{4} = -67 \frac{1}{4} \]
So, the diver's initial depth is \(\boxed{-67 \frac{1}{4}}\) feet.
1 answer