Problem 83 - Introductory Algebra for College Students

Question

Insert either \(<,>,\) or \(=\) in the shaded area to make a true statement. $$\frac{30}{40}-\frac{3}{4} \square \frac{14}{15} \cdot \frac{15}{14}$$

Step-by-Step Solution

Answer

The inequality symbol that makes the statement true is \(<\). Hence, the solution for \(\frac{30}{40} - \frac{3}{4} \square \frac{14}{15} \cdot \frac{15}{14}\) is \(\frac{30}{40} - \frac{3}{4} < \frac{14}{15} \cdot \frac{15}{14}\).

Step 1: Calculate the given fractions

First, we simplify the fractions on both sides of the shaded square. We have \(\frac{30}{40} - \frac{3}{4}\) on one side, and \(\frac{14}{15} \cdot \frac{15}{14}\) on the other side. The fraction \(\frac{30}{40}\) simplifies to \(\frac{3}{4}\), so the left side becomes \(\frac{3}{4} - \frac{3}{4}\). The fraction \(\frac{14}{15} \cdot \frac{15}{14}\) simplifies to 1.

Step 2: Subtract fractions

Next, we perform subtraction on the left side. It gives us \(\frac{3}{4} - \frac{3}{4} = 0\).

Step 3: Compare the fractions

Now we compare 0 and 1 using an inequality or equality sign. Since 0 is less than 1, we insert \(<\) in the shaded area to enforce a true statement.

Step 4: Re-check the result

Finally, we ensure our solution is correct. This can be done by replacing the shaded square in the original statement with the derived sign and check that the overall statement is true. For this case, the final check ensures that indeed 0 is less than 1.