Find the sum using the commutative and associative property of addition: `199+343+217+101`

Options

A. `880`
B. `860`
C. `870`
D. `850`

Answer

To find the sum of the numbers \(199 + 343 + 217 + 101\) using the commutative and associative properties of addition, we can follow these steps:

### Step 1: Use the Commutative Property
The commutative property of addition states that the order in which we add numbers does not change the sum.

We can rearrange the numbers:
\[ 199 + 343 + 217 + 101 = 101 + 217 + 343 + 199 \]

### Step 2: Group and Add the First Pair
Now, let's add the first two numbers from the rearranged equation:
\[ 101 + 217 = 318 \]

### Step 3: Add the Next Number
Next, we add \(343\) to \(318\):
\[ 318 + 343 = 661 \]

### Step 4: Add the Last Number
Finally, we add \(199\) to \(661\):
\[ 661 + 199 = 860 \]

### Conclusion Using Commutative Property
Thus, the sum \(199 + 343 + 217 + 101 = 860\).

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### Step 5: Use the Associative Property
The associative property of addition states that the way in which numbers are grouped does not change the sum.

We can group the numbers as follows:
\[ (199 + 343) + (217 + 101) \]

### Step 6: Add the First Group
First, we calculate:
\[ 199 + 343 = 542 \]

### Step 7: Add the Second Group
Next, we calculate:
\[ 217 + 101 = 318 \]

### Step 8: Add the Results of Both Groups
Now, we add the results of both groups:
\[ 542 + 318 = 860 \]

### Conclusion Using Associative Property
Thus, using the associative property, we also find that \(199 + 343 + 217 + 101 = 860\).

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