Casting out nines” computation check method involving remainders. This method involves dividing each number on both sides of the equation by 9, replacing each number with its remainder after being divided by 9, and redoing the calculations. If the original computation was correct, then the new computation will be correct. If the new computation is wrong, then the original computation was wrong. Use the method of casting out nines to show that the following computations are wrong: 99 + 27 = 326 10,199 – 10,178 99 * 23 = 2278 The sum of the remainders when each number on the left side of the equation is divided by 9 is: 8 parts remaining
The Correct Answer and Explanation is:
The “casting out nines” method involves using the remainder (or “digit sum”) of each number when divided by 9. Here’s a step-by-step breakdown of how to apply it to check the validity of the computations provided.
1. 99 + 27 = 326
- Step 1: Find the remainders when each number is divided by 9:
- 99÷9=1199 \div 9 = 1199÷9=11 remainder 0.
- 27÷9=327 \div 9 = 327÷9=3 remainder 0.
- 326÷9=36326 \div 9 = 36326÷9=36 remainder 1.
- Step 2: Add the remainders on the left side:
- 0+0=00 + 0 = 00+0=0.
- Step 3: Compare it to the remainder of the right side:
- The remainder of 326 is 1.
Since 0 ≠ 1, the equation is incorrect.
2. 10,199 – 10,178
- Step 1: Find the remainders when each number is divided by 9:
- 10,199÷9=1,13310,199 \div 9 = 1,13310,199÷9=1,133 remainder 2.
- 10,178÷9=1,13110,178 \div 9 = 1,13110,178÷9=1,131 remainder 8.
- Step 2: Subtract the remainders on the left side:
- 2−8=−62 – 8 = -62−8=−6 (which is equivalent to 3, since we’re dealing with remainders modulo 9).
- Step 3: Find the remainder of the result of the subtraction:
- 10,199−10,178=2110,199 – 10,178 = 2110,199−10,178=21, and 21÷9=221 \div 9 = 221÷9=2 remainder 3.
Since the remainder on both sides is 3, this computation is correct.
3. 99 * 23 = 2,278
- Step 1: Find the remainders when each number is divided by 9:
- 99÷9=1199 \div 9 = 1199÷9=11 remainder 0.
- 23÷9=223 \div 9 = 223÷9=2 remainder 5.
- 2,278÷9=2532,278 \div 9 = 2532,278÷9=253 remainder 1.
- Step 2: Multiply the remainders on the left side:
- 0∗5=00 * 5 = 00∗5=0.
- Step 3: Compare it to the remainder of the right side:
- The remainder of 2,278 is 1.
Since 0 ≠ 1, the equation is incorrect.
Conclusion
Using the casting out nines method, we find that:
- The first equation 99+27=32699 + 27 = 32699+27=326 is incorrect.
- The second equation 10,199−10,178=2110,199 – 10,178 = 2110,199−10,178=21 is correct.
- The third equation 99∗23=2,27899 * 23 = 2,27899∗23=2,278 is incorrect.
