Question

A diver is at a certain depth in the ocean. After ascending 10 3/4 feet, the diver is now at a depth of -56 1/2 feet.
What was the diver's initial depth? Write your answer as a mixed number in simplest form.

Answer 1

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.

1. 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:
\[
-56 \frac{1}{2} = -56 - \frac{1}{2} = -\frac{112}{2} - \frac{1}{2} = -\frac{113}{2} \text{ feet}
\]

2. Let \(x\) represent the diver's initial depth. According to the problem, we have:
\[
x + \frac{43}{4} = -\frac{113}{2}
\]

3. To isolate \(x\), we need to subtract \(\frac{43}{4}\) from both sides:
\[
x = -\frac{113}{2} - \frac{43}{4}
\]

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 \(4\):
\[
-\frac{113}{2} = -\frac{113 \times 2}{2 \times 2} = -\frac{226}{4}
\]

5. Now we can perform the subtraction:
\[
x = -\frac{226}{4} - \frac{43}{4} = -\frac{226 + 43}{4} = -\frac{269}{4}
\]

6. 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.

Answer 2

ok how to solve using x+10 3/4=-56 1/2

Answer 3

To solve the equation \(x + 10 \frac{3}{4} = -56 \frac{1}{2}\), we want to isolate \(x\). Here’s a step-by-step breakdown of the solution:

1. **Convert the mixed numbers to improper fractions**:
- For \(10 \frac{3}{4}\):
\[
10 \frac{3}{4} = 10 + \frac{3}{4} = \frac{40}{4} + \frac{3}{4} = \frac{43}{4}
\]
- For \(-56 \frac{1}{2}\):
\[
-56 \frac{1}{2} = -56 - \frac{1}{2} = -\frac{112}{2} - \frac{1}{2} = -\frac{113}{2}
\]

2. **Substituting the improper fractions into the equation**:
\[
x + \frac{43}{4} = -\frac{113}{2}
\]

3. **To isolate \(x\)**, subtract \(\frac{43}{4}\) from both sides:
\[
x = -\frac{113}{2} - \frac{43}{4}
\]

4. **Find a common denominator** to perform the subtraction. The least common multiple of \(2\) and \(4\) is \(4\). Convert \(-\frac{113}{2}\) to have a denominator of \(4\):
\[
-\frac{113}{2} = -\frac{113 \times 2}{2 \times 2} = -\frac{226}{4}
\]

5. **Now substitute this back into the equation**:
\[
x = -\frac{226}{4} - \frac{43}{4}
\]

6. **Combine the fractions**:
\[
x = -\frac{226 + 43}{4} = -\frac{269}{4}
\]

7. **Convert \(-\frac{269}{4}\) to a mixed number**:
- Divide \(269\) by \(4\):
\[
269 \div 4 = 67 \quad \text{(with a remainder of 1)}
\]
- Thus, \(-\frac{269}{4} = -67 \frac{1}{4}\).

Finally, the initial depth of the diver is \(\boxed{-67 \frac{1}{4}}\) feet.