Use casting out nines to see if these calculations may have an error. If incorrect, give the correct sum, difference, or product. (a) $$ 32,211+7514+69,338=109,163 $$ (b) $$ 8514+6872+2535+5961=23,782 $$ (c) $$ 78,604-3514=74,090 $$ (d) $$ 394 imes 582=229,408 $$ (i) Click here to view a description of the process of casting out nines. (a) Does the calculation $$ 32,211+7514+69,338=109,163 $$ have an error based on casting out nines? Select the correct choice below and fill in the answer box within your choice. A. The calculation checks using casting out nines. The remainder is $$ \square $$. B. Casting out nines shows there is an error. The given answer has a remainder of $$ \square $$, but casting out nines gives a remainder of $$ \square $$. The correct answer is $$ \square $$.

Question

Use casting out nines to see if these calculations may have an error. If incorrect, give the correct sum, difference, or product. (a) $$ 32,211+7514+69,338=109,163 $$ (b) $$ 8514+6872+2535+5961=23,782 $$ (c) $$ 78,604-3514=74,090 $$ (d) $$ 394 	imes 582=229,408 $$ (i) Click here to view a description of the process of casting out nines. (a) Does the calculation $$ 32,211+7514+69,338=109,163 $$ have an error based on casting out nines? Select the correct choice below and fill in the answer box within your choice. A. The calculation checks using casting out nines. The remainder is $$ \square $$. B. Casting out nines shows there is an error. The given answer has a remainder of $$ \square $$, but casting out nines gives a remainder of $$ \square $$. The correct answer is $$ \square $$.

UpStudy AI · Accepted Answer

We start by checking the remainders upon division by 9.

**Step 1. Calculate the remainder (casting out nines) for each addend.**

- For $$32{,}211$$:  
  Sum of digits: $$3+2+2+1+1=9$$.  
  Since $$9$$ is divisible by 9, we say $$32{,}211 \equiv 0 \pmod{9}$$.

- For $$7{,}514$$:  
  Sum of digits: $$7+5+1+4=17$$.  
  Then, $$17 \equiv 8 \pmod{9}$$.

- For $$69{,}338$$:  
  Sum of digits: $$6+9+3+3+8=29$$.  
  Then, $$29-27=2$$, so $$29 \equiv 2 \pmod{9}$$.

**Step 2. Sum the remainders of the addends.**

$$
0 + 8 + 2 = 10 \quad \text{and} \quad 10 \equiv 1 \pmod{9}.
$$

Thus, the correct result of the sum should have a remainder of $$1$$ when divided by $$9$$.

**Step 3. Check the provided sum $$109{,}163$$.**

- For $$109{,}163$$:  
  Sum of digits: $$1+0+9+1+6+3=20$$.  
  Then, $$20-18=2$$, so $$109{,}163 \equiv 2 \pmod{9}$$.

Since $$2 \neq 1$$, the provided sum fails the casting out nines test.

**Step 4. Find the correct sum.**

First, add the numbers:
$$
32{,}211 + 7{,}514 = 39{,}725.
$$
Then,
$$
39{,}725 + 69{,}338 = 109{,}063.
$$

Check $$109{,}063$$ by summing its digits:
$$
1+0+9+0+6+3=19 \quad \text{and} \quad 19-18=1,
$$
so $$109{,}063 \equiv 1 \pmod{9}$$, which agrees with our expected remainder.

**Final Answer for (a):**

B. Casting out nines shows there is an error. The given answer has a remainder of $$2$$, but casting out nines gives a remainder of $$1$$. The correct answer is $$109{,}063$$.
The calculation $$32{,}211 + 7{,}514 + 69{,}338 = 109{,}163$$ has an error based on casting out nines. The given answer has a remainder of 2, but casting out nines gives a remainder of 1. The correct sum is $$109{,}063$$.

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