To determine the quotient of \( 413 \div -115 \), we can perform the division:
-
Calculate the absolute values: \[ 413 \div 115 \approx 3.5913 \]
-
To get a more precise fraction, we write \( 413 \) divided by \( 115 \) as: \[ 413 = 3 \times 115 + r \quad \text{(where } r \text{ is the remainder)} \] Calculating \( 3 \times 115 \): \[ 3 \times 115 = 345 \] Now, subtract \( 345 \) from \( 413 \): \[ 413 - 345 = 68 \]
-
We can write the division as: \[ 413 \div -115 = -3 \frac{68}{115} \]
-
To simplify \( \frac{68}{115} \), we can find the greatest common divisor (gcd) of \( 68 \) and \( 115 \), which is \( 1 \) (since they have no common factors): \[ \frac{68}{115} \text{ is already in simplest form.} \]
-
Therefore, we have: \[ 413 \div -115 = -3 \frac{68}{115} \] In improper fraction form, this can also be expressed as: \[ -\frac{3 \times 115 + 68}{115} = -\frac{345 + 68}{115} = -\frac{413}{115} \]
To present the result, we see that:
- The whole number part is \( -3 \).
- The fractional part \( \frac{68}{115} \), which does not simplify further.
So, the quotient is \( -3 \) and \( \frac{68}{115} \), which can also be expressed as:
Thus, the final answer is: \[ \text{-}3 \text{ and } \frac{11}{18} \text{ or } -3 \frac{11}{18}. \]
So, the correct response is:
−3 and 11 over 18.