Question

Consider the following expressions: 6−1/4⋅28−2 1/4_______3/4+4⋅2−11. Which operator can be inserted to make the statement true?

Answer 1

To solve the expression \( 6 - \frac{1}{4} \cdot 28 - 2 \frac{1}{4} \_ \frac{3}{4} + 4 \cdot 2 - 11 \), we need to determine which operator (e.g., +, -, ×, ÷) can be inserted in place of the underscore to make the equation true.

Let's break down the expression step by step.

1. Calculate \( 6 - \frac{1}{4} \cdot 28 - 2 \frac{1}{4} \):
- First, calculate \( \frac{1}{4} \cdot 28 = 7 \).
- Now substitute it back: \( 6 - 7 - 2 \frac{1}{4} \).
- Convert \( 2 \frac{1}{4} \) to an improper fraction: \( 2 \frac{1}{4} = \frac{9}{4} \).
- Therefore, the expression becomes \( 6 - 7 - \frac{9}{4} \).

Now, calculate \( 6 - 7 = -1 \).
- So, we have \( -1 - \frac{9}{4} = -1 - 2.25 = -3.25 = -\frac{13}{4} \).

2. Now let's evaluate the right side \( \frac{3}{4} + 4 \cdot 2 - 11 \):
- First, calculate \( 4 \cdot 2 = 8 \).
- Next, substitute it back: \( \frac{3}{4} + 8 - 11 \).
- Now calculate \( 8 - 11 = -3 \).
- Therefore, the expression becomes \( \frac{3}{4} - 3 \), which equals \( \frac{3}{4} - \frac{12}{4} = -\frac{9}{4} \).

Now we have:
\[
-\frac{13}{4} \_ -\frac{9}{4}
\]

Next, we can test the operators:

1. If we test with \( \_ = < \):
\(-\frac{13}{4} < -\frac{9}{4}\), which is true.

2. If we test with \( \_ = = \):
\(-\frac{13}{4} = -\frac{9}{4}\), which is false.

3. If we test with \( \_ = > \):
\(-\frac{13}{4} > -\frac{9}{4}\), which is false.

4. If we test with \( \_ = + \):
\(-\frac{13}{4} + -\frac{9}{4} = -\frac{22}{4} = -\frac{11}{2}\), which doesn't make it true.

5. If we test with \( \_ = - \):
\(-\frac{13}{4} - -\frac{9}{4} = -\frac{13}{4} + \frac{9}{4} = -\frac{4}{4} = -1\), which doesn't make it true.

6. If we test with \( \_ = × \):
\(-\frac{13}{4} \times -\frac{9}{4} = \frac{117}{16}\), which doesn't make it true.

7. If we test with \( \_ = ÷ \):
\(-\frac{13}{4} ÷ -\frac{9}{4} = \frac{13}{9}\), which doesn't make it true.

Only with \( < \) did we find a true statement:
\[
-\frac{13}{4} < -\frac{9}{4}
\]
Thus, the operator that can be inserted to make the statement true is \( < \).