Question

Are these ratios equivalent? 6 crossword puzzles : 26 word searches 17 crossword puzzles : 47 word searches

Answer 1

To determine if the ratios \(6 \text{ crossword puzzles} : 26 \text{ word searches}\) and \(17 \text{ crossword puzzles} : 47 \text{ word searches}\) are equivalent, we can simplify each ratio.

1. **First ratio: \(6 \text{ crossword puzzles} : 26 \text{ word searches}\)**

We can simplify this ratio by dividing both parts by 2:
\[
\frac{6}{26} = \frac{6 \div 2}{26 \div 2} = \frac{3}{13}
\]
So, the simplified first ratio is \(3 : 13\).

2. **Second ratio: \(17 \text{ crossword puzzles} : 47 \text{ word searches}\)**

Now, let's simplify this ratio. We check if both numbers have a common divisor:
They do not share any common factors (17 is a prime number).
Thus, the second ratio remains \(17 : 47\) and cannot be simplified further.

3. **Comparing the simplified ratios**

Now we compare \(3 : 13\) and \(17 : 47\).
We can check if they are equivalent by cross-multiplying:
\[
3 \times 47 = 141
\]
\[
13 \times 17 = 221
\]

Since \(141 \neq 221\), the ratios \(6 \text{ crossword puzzles} : 26 \text{ word searches}\) and \(17 \text{ crossword puzzles} : 47 \text{ word searches}\) are **not equivalent**.