To find where the first digit of the quotient will be placed for \(872 \div 30\), we can estimate how many times 30 fits into 872.
- Since 30 is close to 33 (which is 1/3 of 100), we can start estimating.
- Multiply 30 by 20, which equals 600. This is less than 872.
- Then multiply 30 by 30, which equals 900. This is too much.
- Thus, the first digit of the quotient will be placed in the tens place, which means the quotient will be somewhere between 20 and 30.
Now for the calculations:
9.) \(375 \div 40\)
First, we can simplify this problem by estimating:
- \(40\) is close to \(40\), so we can calculate how many \(40s\) fit into \(375\).
- If we estimate, we find that \(10 \times 40 = 400\) is too much, but \(9 \times 40 = 360\) is acceptable. Thus, 9 goes into the answer.
- Multiply \(40\) by \(9\): \[ 9 \times 40 = 360 \]
- Subtract \(360\) from \(375\): \[ 375 - 360 = 15 \]
- Now we have a remainder of \(15\). To express the answer in decimal form, we can add a decimal point and a zero to the \(15\), allowing us to divide again.
- Bringing down the zero, we get \(150\).
- Now divide \(150\) by \(40\): \[ 3 \times 40 = 120 \]
- Subtract again: \[ 150 - 120 = 30 \]
- To continue, we can add another zero to the remainder, and divide \(300\) by \(40\): \[ 7 \times 40 = 280 \]
- Subtract again: \[ 300 - 280 = 20 \]
- Next, we introduce another zero, getting \(200\) and divide: \[ 5 \times 40 = 200 \]
- Subtract again, we have \(0\).
So, the answer for \(375 \div 40\) is: \[ 9.375 \]
10.) \(828 \div 23\)
- Estimate how many times \(23\) fits into \(828\).
- Use \(20\) as a base for estimates. \(20 \times 30 = 600\) and \(20 \times 40 = 800\), so we start somewhere around \(30\) or \(35\).
- Trying \(35\): \[ 23 \times 35 = 805 \]
- Now trying \(36\): \[ 23 \times 36 = 828 \]
- We found it fits perfectly!
So, the answer for \(828 \div 23\) is: \[ 36 \]
In summary:
- 9.) \(375 \div 40 = 9.375\)
- 10.) \(828 \div 23 = 36\)