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 remainin
The Correct Answer and Explanation is:
The method of casting out nines is a simple way to check calculations by reducing numbers modulo 9 (i.e., finding the remainder when dividing by 9) and performing the operation again with these reduced values. If the new computation matches, the original computation is likely correct. If the reduced computation doesn’t match, there was an error in the original computation. Here’s how it works for each case:
1. 99 + 27 = 326
- Step 1: Find the remainders (modulo 9) of each number.
- 99 ÷ 9 = 11 remainder 0 (99 ≡ 0 mod 9)
- 27 ÷ 9 = 3 remainder 0 (27 ≡ 0 mod 9)
- 326 ÷ 9 = 36 remainder 2 (326 ≡ 2 mod 9)
- Step 2: Perform the operation on the remainders.
- 0 + 0 = 0, and check if it equals the remainder of 326, which is 2. The remainders do not match, so the original equation is incorrect.
2. 10,199 – 10,178
- Step 1: Find the remainders (modulo 9) of each number.
- 10,199 ÷ 9 = 1,133 remainder 2 (10,199 ≡ 2 mod 9)
- 10,178 ÷ 9 = 1,130 remainder 8 (10,178 ≡ 8 mod 9)
- Step 2: Perform the operation on the remainders.
- 2 – 8 = -6, which is equivalent to 3 modulo 9 (since -6 + 9 = 3).
- The result of the subtraction on the left side is 3, but the right side of the equation (10,199 – 10,178) should give a remainder of 0. Therefore, the computation is wrong.
3. 99 * 23 = 2278
- Step 1: Find the remainders (modulo 9) of each number.
- 99 ÷ 9 = 11 remainder 0 (99 ≡ 0 mod 9)
- 23 ÷ 9 = 2 remainder 5 (23 ≡ 5 mod 9)
- 2278 ÷ 9 = 252 remainder 4 (2278 ≡ 4 mod 9)
- Step 2: Perform the operation on the remainders.
- 0 * 5 = 0, and check if it equals the remainder of 2278, which is 4. The remainders do not match, so the computation is incorrect.
Conclusion:
In all three examples, the casting out nines method reveals that the computations are incorrect because the remainders do not match. This method is a quick and reliable way to spot errors without going through the entire calculation again.
