Let’s tackle this problem step by step. The equation given is:

1 + 2 * 3 + 4 * 5 + 6 * 7 + 8 * 9 = 479

But it’s incorrect, and I need to add a pair of parentheses to make it true. Okay, first things first, I should understand what the current value of the left side is without any parentheses.

Let’s recall the order of operations: PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). So, multiplication comes before addition.

So, calculating step by step:

2 * 3 = 6

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

Now, plugging these back into the equation:

1 + 6 + 20 + 42 + 72

Adding these up:

1 + 6 = 7

7 + 20 = 27

27 + 42 = 69

69 + 72 = 141

So, without any parentheses, the left side equals 141, but it should be 479. That’s quite a difference. So, I need to add a pair of parentheses to make it equal to 479.

I need to think about where to place the parentheses to alter the order of operations to achieve the desired result.

Let me consider different possibilities:

Parenthesizing individual terms:

For example, (1 + 2) * 3 + 4 * 5 + 6 * 7 + 8 * 9

Calculate inside the parentheses first:

(1 + 2) = 3

Then, 3 * 3 = 9

Now, the equation becomes:

9 + 4 * 5 + 6 * 7 + 8 * 9

Following order of operations:

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

So, 9 + 20 + 42 + 72 = 143

Still not 479.

Parenthesizing more terms:

What if I do (1 + 2 * 3) + 4 * 5 + 6 * 7 + 8 * 9

First, calculate inside the parentheses:

1 + 2 * 3

Here, multiplication comes before addition, so 2 * 3 = 6, then 1 + 6 = 7

So, the equation becomes:

7 + 4 * 5 + 6 * 7 + 8 * 9

Again, following order of operations:

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

So, 7 + 20 + 42 + 72 = 141

Same as before.

Let’s try (1 + 2) * (3 + 4) * 5 + 6 * 7 + 8 * 9

First, calculate inside the parentheses:

(1 + 2) = 3

(3 + 4) = 7

So, 3 * 7 * 5 = 105

Now, the equation becomes:

105 + 6 * 7 + 8 * 9

Following order of operations:

6 * 7 = 42

8 * 9 = 72

So, 105 + 42 + 72 = 219

Still not 479.

Maybe trying to group more terms:

(1 + 2 * 3 + 4) * 5 + 6 * 7 + 8 * 9

First, calculate inside the parentheses:

1 + 2 * 3 + 4

Multiplication first: 2 * 3 = 6

Then, 1 + 6 + 4 = 11

So, 11 * 5 = 55

Now, the equation becomes:

55 + 6 * 7 + 8 * 9

Following order of operations:

6 * 7 = 42

8 * 9 = 72

So, 55 + 42 + 72 = 169

Getting closer, but still not 479.

Let’s try a different approach. Maybe grouping the additions together:

1 + (2 * 3 + 4 * 5 + 6 * 7 + 8 * 9)

First, calculate inside the parentheses:

2 * 3 = 6

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

So, 6 + 20 + 42 + 72 = 140

Then, 1 + 140 = 141

Same as before.

Perhaps grouping two multiplications together:

1 + 2 * (3 + 4) * 5 + 6 * 7 + 8 * 9

First, calculate inside the parentheses:

(3 + 4) = 7

So, 2 * 7 * 5 = 70

Now, the equation becomes:

1 + 70 + 6 * 7 + 8 * 9

Following order of operations:

6 * 7 = 42

8 * 9 = 72

So, 1 + 70 + 42 + 72 = 185

Still not 479.

Maybe grouping more multiplications:

1 + 2 * 3 + 4 * (5 + 6) * 7 + 8 * 9

First, calculate inside the parentheses:

(5 + 6) = 11

So, 4 * 11 * 7 = 308

Now, the equation becomes:

1 + 2 * 3 + 308 + 8 * 9

Following order of operations:

2 * 3 = 6

8 * 9 = 72

So, 1 + 6 + 308 + 72 = 387

Closer, but still not 479.

Let’s try grouping the last terms:

1 + 2 * 3 + 4 * 5 + 6 * (7 + 8) * 9

First, calculate inside the parentheses:

(7 + 8) = 15

So, 6 * 15 * 9 = 810

Now, the equation becomes:

1 + 2 * 3 + 4 * 5 + 810

Following order of operations:

2 * 3 = 6

4 * 5 = 20

So, 1 + 6 + 20 + 810 = 837

Now that’s way over 479.

Maybe I need to group terms in a way that subtraction is implied, but since there are no subtractions in the original equation, perhaps I need to consider that.

Wait, maybe the problem allows for adding parentheses to change the operations indirectly.

Alternatively, perhaps the parentheses need to group additions in a specific way.

Let me try grouping the first few terms:

(1 + 2 * 3 + 4 * 5) + 6 * 7 + 8 * 9

First, calculate inside the parentheses:

2 * 3 = 6

4 * 5 = 20

So, 1 + 6 + 20 = 27

Now, the equation becomes:

27 + 6 * 7 + 8 * 9

Following order of operations:

6 * 7 = 42

8 * 9 = 72

So, 27 + 42 + 72 = 141

Same as before.

Maybe grouping further:

(1 + 2 * 3 + 4 * 5 + 6 * 7) + 8 * 9

First, calculate inside the parentheses:

2 * 3 = 6

4 * 5 = 20

6 * 7 = 42

So, 1 + 6 + 20 + 42 = 69

Now, the equation becomes:

69 + 8 * 9

Following order of operations:

8 * 9 = 72

So, 69 + 72 = 141

Still not 479.

Let’s try grouping differently:

1 + 2 * (3 + 4 * 5 + 6 * 7 + 8 * 9)

First, calculate inside the parentheses:

3 + 4 * 5 + 6 * 7 + 8 * 9

Following order of operations inside:

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

So, 3 + 20 + 42 + 72 = 137

Now, 2 * 137 = 274

Then, the equation becomes:

1 + 274 = 275

Still not 479.

Maybe trying to group the additions and multiplications in a different way:

1 + (2 * 3 + 4 * 5 + 6 * 7 + 8 * 9)

Wait, I did this earlier and got 141.

Hmm.

Let me think differently. Maybe I need to group terms to create a larger multiplication.

For example:

1 + 2 * (3 + 4 * 5 + 6 * 7 + 8 * 9)

First, calculate inside the parentheses:

3 + 4 * 5 + 6 * 7 + 8 * 9

Following order of operations:

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

So, 3 + 20 + 42 + 72 = 137

Now, 2 * 137 = 274

Then, 1 + 274 = 275

Still not 479.

Let’s try another approach:

(1 + 2) * (3 + 4) * (5 + 6) * (7 + 8) * 9

Wait, that’s adding multiple sets of parentheses, which is not allowed since I can only add one pair.

Back to the original problem: only one pair of parentheses can be added.

Maybe I need to group a subtraction or division, but those operations aren’t present in the original equation.

Wait, perhaps I need to consider that adding parentheses can create a negative term, but again, there are no subtractions in the original equation.

Alternatively, maybe the parentheses are meant to group additions in a specific way to create a larger number.

Let me try grouping more terms together:

1 + 2 * (3 + 4 * 5 + 6 * 7 + 8 * 9)

First, inside the parentheses:

3 + 4 * 5 + 6 * 7 + 8 * 9

Following order of operations:

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

So, 3 + 20 + 42 + 72 = 137

Now, 2 * 137 = 274

Then, 1 + 274 = 275

Same as before.

Maybe trying to group the last few terms:

1 + 2 * 3 + 4 * 5 + 6 * (7 + 8 * 9)

First, inside the parentheses:

7 + 8 * 9

Following order of operations:

8 * 9 = 72

Then, 7 + 72 = 79

Now, 6 * 79 = 474

So, the equation becomes:

1 + 2 * 3 + 4 * 5 + 474

Following order of operations:

2 * 3 = 6

4 * 5 = 20

So, 1 + 6 + 20 + 474 = 501

Close to 479, but still not there.

Maybe adjusting the grouping slightly:

1 + 2 * 3 + 4 * (5 + 6 * 7 + 8 * 9)

First, inside the parentheses:

5 + 6 * 7 + 8 * 9

Following order of operations:

6 * 7 = 42

8 * 9 = 72

So, 5 + 42 + 72 = 119

Now, 4 * 119 = 476

Then, the equation becomes:

1 + 2 * 3 + 476

Following order of operations:

2 * 3 = 6

So, 1 + 6 + 476 = 483

Still not 479.

Let’s try grouping the first part differently:

(1 + 2 * 3 + 4) * (5 + 6 * 7 + 8 * 9)

First, calculate inside the first set of parentheses:

1 + 2 * 3 + 4

Following order of operations:

2 * 3 = 6

Then, 1 + 6 + 4 = 11

Now, the second set of parentheses:

5 + 6 * 7 + 8 * 9

Following order of operations:

6 * 7 = 42

8 * 9 = 72

So, 5 + 42 + 72 = 119

Now, 11 * 119 = 1309

Way over 479.

Maybe trying to group only two terms:

1 + 2 * (3 + 4) * 5 + 6 * 7 + 8 * 9

Wait, I did this earlier and got 185.

Hmm.

Let me think differently. Maybe I need to group terms to create a division or subtraction, but since those operations aren’t present, perhaps I need to consider that the parentheses can be used to create a fraction or some other operation, but that might be reaching too far.

Alternatively, maybe the problem allows for adding a pair of parentheses around the entire equation, but that doesn’t make sense.

Wait, perhaps the parentheses are meant to group additions in a specific way to create a larger number.

Let me try grouping more terms together for multiplication.

For example:

1 + 2 * (3 + 4 * 5 + 6) * 7 + 8 * 9

First, inside the parentheses:

3 + 4 * 5 + 6

Following order of operations:

4 * 5 = 20

So, 3 + 20 + 6 = 29

Now, 2 * 29 * 7 = 406

Then, the equation becomes:

1 + 406 + 8 * 9

Following order of operations:

8 * 9 = 72

So, 1 + 406 + 72 = 479

Wait a minute, that’s exactly 479!

Let me double-check:

First, inside the parentheses: 3 + 4 * 5 + 6

4 * 5 = 20

Then, 3 + 20 + 6 = 29

Then, 2 * 29 * 7 = 2 * 203 = 406

Then, 1 + 406 = 407

Plus 8 * 9 = 72

So, 407 + 72 = 479

Perfect! So, by adding parentheses around (3 + 4 * 5 + 6), and multiplying that sum by 2 and 7, we get the desired result.

So, the corrected equation is:

1 + 2 * (3 + 4 * 5 + 6) * 7 + 8 * 9 = 479

Final Answer

$$ \boxed{1 + 2 \times (3 + 4 \times 5 + 6) \times 7 + 8 \times 9 = 479} $$