Question

5+4\cdot2+6-2\cdot2-1

Insert parentheses in Expression 2 so that it has a value of 19. Then show why your expression has a
value of 19.

Answer 1

To insert parentheses in the expression \( 5 + 4 \cdot 2 + 6 - 2 \cdot 2 - 1 \) so that it equals 19, we can proceed with the following arrangement:

\[
5 + (4 \cdot 2) + 6 - (2 \cdot 2) - 1
\]

Now let's calculate the value of this expression step by step:

1. Compute \( 4 \cdot 2 \):
\[
4 \cdot 2 = 8
\]

2. Compute \( 2 \cdot 2 \):
\[
2 \cdot 2 = 4
\]

3. Substitute these values back into the expression:
\[
5 + 8 + 6 - 4 - 1
\]

4. Now perform the addition and subtraction in order:
- Start with \( 5 + 8 \):
\[
5 + 8 = 13
\]
- Then add \( 6 \):
\[
13 + 6 = 19
\]
- Next, subtract \( 4 \):
\[
19 - 4 = 15
\]
- Finally, subtract \( 1 \):
\[
15 - 1 = 14
\]

Oh, I realize I misunderstood your requirement. The expression must equal 19. Let's try again by inserting different parentheses.

After examining how to reach a sum of 19, here’s an alternative arrangement:

Revising to:
\[
(5 + 4) \cdot 2 + 6 - 2 \cdot 2 - 1
\]

Calculating this gives:

1. Calculate \( 5 + 4 \):
\[
5 + 4 = 9
\]

2. Now multiply by \( 2 \):
\[
9 \cdot 2 = 18
\]

3. Substitute this value back into the expression:
\[
18 + 6 - 4 - 1
\]

4. Now perform the addition and subtraction:
- Start with \( 18 + 6 \):
\[
18 + 6 = 24
\]
- Next, subtract \( 4 \):
\[
24 - 4 = 20
\]
- Finally, subtract \( 1 \):
\[
20 - 1 = 19
\]

Thus, the arrangement \( (5 + 4) \cdot 2 + 6 - 2 \cdot 2 - 1 \) indeed evaluates to \( 19 \).