To solve \( 1,175 \div 5 \) using partial quotients, we can start by determining an appropriate power of 10 to use for our initial subtraction. The key is to find a power of 10 times 5 that is close to, but not greater than, 1,175.
First, let's identify some powers of 10 multiplied by 5:
- \( 5 \times 10^0 = 5 \)
- \( 5 \times 10^1 = 50 \)
- \( 5 \times 10^2 = 500 \)
- \( 5 \times 10^3 = 5000 \) (too high)
Now, the highest power of 10 multiplied by 5 that we can use without exceeding 1,175 is \( 5 \times 10^2 = 500 \).
To use partial quotients, we can subtract \( 500 \) from \( 1,175 \):
- Subtract \( 500 \): \[ 1,175 - 500 = 675 \]
Next, we can repeat the process with \( 675 \). We can again check the multiples of 5:
- \( 5 \times 10^2 = 500 \) is still valid.
- Subtract \( 500 \) again: \[ 675 - 500 = 175 \]
Now, we do the same with \( 175 \):
- The next power we can use is again \( 5 \times 10^1 = 50 \).
- Subtract \( 50 \): \[ 175 - 50 = 125 \]
Continuing with \( 125 \):
- Subtract \( 50 \) once more: \[ 125 - 50 = 75 \]
Next, with \( 75 \):
- Subtract \( 50 \): \[ 75 - 50 = 25 \]
Finally, for \( 25 \):
- Subtract \( 25 \): \[ 25 - 25 = 0 \]
Now, we can combine the quotients we derived from our subtractions:
- From \( 1,175 - 500 \) we added \( 2 \) to our quotient (for \( 500 \)).
- From \( 175 - 50 \) we added \( 3 \) to our quotient (for \( 50 \)).
- The last \( 25 \) accounted for another \( 1 \).
Thus, we have \( 2 + 3 + 1 = 6 \).
So, our final answer, represented as \( 1,175 \div 5 = 235 \). This means when subtracting using partial quotients, we begin with \( 500 \) and then proceed to \( 50 \) and \( 25 \) based on proximity to the remaining value.