Whitney asked her cousins to name all the types of fish they found on a family trip to the aquarium.

Question

1/9 reported finding a salmon.
5/7 reported finding a tuna.
1/4 reported finding a swordfish.
12/23 reported finding a whale shark.
Based on the results, determine which statements are true and which statements are false.
(5 points)
True False. Over half the cousins found a tuna.
More cousins found a salmon than a tuna.
More cousins found a whale shark than a swordfish.
More cousins found a tuna than a whale shark.
More cousins found a salmon than a swordfish.

Answer 1

To analyze the statements, we first need to know the total number of cousins. Since we are dealing with fractions, we can assume a common denominator for all the fractions given in the problem.

The fractions provided:

• Salmon: \( \frac{1}{9} \)
• Tuna: \( \frac{5}{7} \)
• Swordfish: \( \frac{1}{4} \)
• Whale Shark: \( \frac{12}{23} \)

To find a common denominator, we can calculate the least common multiple (LCM) of the denominators: 9, 7, 4, and 23. The LCM of these numbers is 126.

Now, we can find the number of cousins who reported finding each type of fish by converting the fractions into equivalent fractions with a denominator of 126:

1. Salmon:
   \( \frac{1}{9} = \frac{1 \times 14}{9 \times 14} = \frac{14}{126} \)
   So, 14 cousins found a salmon.

2. Tuna:
   \( \frac{5}{7} = \frac{5 \times 18}{7 \times 18} = \frac{90}{126} \)
   So, 90 cousins found a tuna.

3. Swordfish:
   \( \frac{1}{4} = \frac{1 \times 31.5}{4 \times 31.5} = \frac{31.5}{126} \) -> This isn't an integer, so we will take 31 cousins as 126 cousins aren't fixed but can be deduced as estimates.

4. Whale Shark:
   \( \frac{12}{23} = \frac{12 \times 6}{23 \times 6} = \frac{72}{138} \) -> This isn't an integer either, so we treat this as 12 cousins as well.

Now we have:

• Salmon: 14 cousins
• Tuna: 90 cousins
• Swordfish: 31 cousins
• Whale Shark: 72 cousins

Next, let's analyze the statements based on these numbers:

1. True or False: Over half the cousins found a tuna.

   • True: 90 cousins out of 126 is more than half.

2. True or False: More cousins found a salmon than a tuna.

   • False: 14 cousins for salmon is less than 90 cousins for tuna.

3. True or False: More cousins found a whale shark than a swordfish.

   • True: 72 cousins for whale shark is more than 31 cousins for swordfish.

4. True or False: More cousins found a tuna than a whale shark.

   • True: 90 cousins for tuna is greater than 72 cousins for whale shark.

5. True or False: More cousins found a salmon than a swordfish.

   • False: 14 cousins for salmon is less than 31 cousins for swordfish.

So, the truth values for the statements are as follows:

1. True
2. False
3. True
4. True
5. False