A submarine hovers at 66 and two-thirds yards below sea level. If it ascends 24 and StartFraction 1 over 8 EndFraction yards and then descends 78 and three-fourths yards, what is the submarine’s new position, in yards, with respect to sea level?

Question

Negative 169 and StartFraction 13 over 24 EndFraction
Negative 121 and StartFraction 7 over 24 EndFraction
Negative 12 and StartFraction 1 over 24 EndFraction
12 and StartFraction 1 over 24 EndFraction

Answer 1

To find the submarine's new position with respect to sea level, we need to perform the operations step-by-step.

1. The submarine starts at \( -66 \frac{2}{3} \) yards (below sea level).

\[ -66 \frac{2}{3} = -66 - \frac{2}{3} = -66.6667 \text{ (in decimal for easier calculations)} \]

2. It ascends \( 24 \frac{1}{8} \) yards (which means we are adding a positive value).

\[ 24 \frac{1}{8} = 24 + \frac{1}{8} = 24.125 \]

So, we perform the addition:

\[ -66 \frac{2}{3} + 24 \frac{1}{8} \]

Converting the mixed numbers to improper fractions:

\[ -66 \frac{2}{3} = -\frac{200}{3} \] \[ 24 \frac{1}{8} = \frac{193}{8} \]

To add these, we need a common denominator. The least common multiple of 3 and 8 is 24.

Now we convert both fractions:

\[ -\frac{200}{3} = -\frac{200 \times 8}{3 \times 8} = -\frac{1600}{24} \] \[ \frac{193}{8} = \frac{193 \times 3}{8 \times 3} = \frac{579}{24} \]

Now we can add the two fractions:

\[ -\frac{1600}{24} + \frac{579}{24} = \frac{-1600 + 579}{24} = \frac{-1021}{24} \]

After the ascent, the submarine is at:

\[ -\frac{1021}{24} \text{ yards} \]

3. Now the submarine descends \( 78 \frac{3}{4} \) yards (which means we subtract a positive value).

\[ 78 \frac{3}{4} = 78 + \frac{3}{4} = 78.75 \]

To calculate the new position:

\[ -\frac{1021}{24} - 78 \frac{3}{4} \]

Converting \( 78 \frac{3}{4} \) to an improper fraction:

\[ 78 \frac{3}{4} = \frac{78 \times 4 + 3}{4} = \frac{312 + 3}{4} = \frac{315}{4} \]

Now, convert \( \frac{315}{4} \) to a fraction with a denominator of 24:

\[ \frac{315}{4} = \frac{315 \times 6}{4 \times 6} = \frac{1890}{24} \]

Now subtract this from the position after ascending:

\[ -\frac{1021}{24} - \frac{1890}{24} = \frac{-1021 - 1890}{24} = \frac{-2911}{24} \]

Thus, the submarine's new position with respect to sea level is:

\[ -\frac{2911}{24} \text{ yards} \]

To express this as a mixed number:

\[ -\frac{2911}{24} = -121 \frac{7}{24} \]

So the submarine's new position is:

\[ \text{Negative } 121 \frac{7}{24} \]

The correct answer is:

Negative 121 and StartFraction 7 over 24 EndFraction