Certainly! Let's apply the division algorithm to find the quotient for [tex]\( 314.944 \div 3.7 \)[/tex].
#### Step-by-Step Solution:
1. Alignment and Setup:
Write down the numbers for division. Here, we are dividing 314.944 by 3.7.
2. Convert the Divisor to an Integer:
To make division simpler, convert the divisor to a whole number by multiplying both the dividend and the divisor by 10. This gives us:
[tex]\[
314.944 \div 3.7 = 3149.44 \div 37
\][/tex]
3. Long Division:
Next, perform long division of 3149.44 by 37.
4. Division:
- 37 into 314:
- 37 goes into 314, 8 times, since [tex]\( 37 \times 8 = 296 \)[/tex] and [tex]\( 37 \times 9 = 333 \)[/tex] which is too large.
- Write down 8 as part of the quotient.
- Subtract [tex]\( 296 \)[/tex] from [tex]\( 314 \)[/tex] to get a remainder [tex]\( 18 \)[/tex].
- Bring Down the Next Digit (9):
- Now consider the next digit from the dividend, making it [tex]\( 189 \)[/tex].
- 37 goes into 189, 5 times, since [tex]\( 37 \times 5 = 185 \)[/tex] and [tex]\( 37 \times 6 = 222 \)[/tex] which is too large.
- Write down 5 as part of the quotient, after the 8.
- Subtract [tex]\( 185 \)[/tex] from [tex]\( 189 \)[/tex] to get a remainder [tex]\( 4 \)[/tex].
- Bring Down the Next Digit (4):
- Now consider the next digit from the dividend, making it [tex]\( 44 \)[/tex].
- 37 goes into 44, 1 time.
- Write down 1 as part of the quotient, after the 85.
- Subtract [tex]\( 37 \)[/tex] from [tex]\( 44 \)[/tex] to get a remainder [tex]\( 7 \)[/tex].
- Bring Down the Next Digit (4):
- Finally, consider the last digit from the dividend, making it [tex]\( 74 \)[/tex].
- 37 goes into 74, 2 times.
- Write down 2 as part of the quotient, after the 85.1.
- Subtract [tex]\( 74 \)[/tex] from [tex]\( 74 \)[/tex] to get a remainder [tex]\( 0 \)[/tex].
5. Construct the Quotient:
Combining the results of our long division steps, we get:
[tex]\[
85.12
\][/tex]
So, the quotient of [tex]\( 314.944 \div 3.7 \)[/tex] is [tex]\(\boxed{85.12}\)[/tex].
