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13 May 2025, 21:19
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Cats: 36
Dogs: 44
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
35%
(medium)
Question Stats:
71% (02:01) correct
29%
(02:25)
wrong
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A pet store is home to only cats and dogs, and the average weight of each animal at the store is 42 pounds. If the average weight of cats is 20 pounds and the average weight of dogs is 60 pounds, which of the following could be the number of cats and the number of dogs at the store (select one number in each column)?
| Cats | Dogs | |
| 11 | ||
| 16 | ||
| 22 | ||
| 27 | ||
| 36 | ||
| 44 |
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Official Answer
Cats: 36
Dogs: 44
14 May 2025, 01:11
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The store has only cats and dogs, with an overall average weight of 42 pounds, cats averaging 20 pounds, and dogs averaging 60 pounds.
Using the mixture method, we set up a weighted average to find the ratio of cats to dogs.
42
/----\
/------ \
/----------\
Cats--------Dogs
20-----------60
Calculate the differences to find the ratio:
\(\text{Distance from cats to mixture} = 42 - 20 = 22\)
\(\text{Distance from dogs to mixture} = 60 - 42 = 18\)
The ratio of dogs to cats is the inverse of these differences:
\(\text{Dogs : Cats} = 22 : 18 = 11 : 9\)
The total number of animals must be a multiple of \(11 + 9 = 20\), and the number of dogs and cats must match the ratio \(11 : 9\) and be in the provided options.
Other pairs don’t yield the 11:9 ratio or fit the options. For example, 22 cats and 27 dogs give a total of 49, not a multiple of 20, and the ratio 27:22 ≠ 11:9.
Answer: 36 cats, 44 dogs
Using the mixture method, we set up a weighted average to find the ratio of cats to dogs.
42
/----\
/------ \
/----------\
Cats--------Dogs
20-----------60
Calculate the differences to find the ratio:
\(\text{Distance from cats to mixture} = 42 - 20 = 22\)
\(\text{Distance from dogs to mixture} = 60 - 42 = 18\)
The ratio of dogs to cats is the inverse of these differences:
\(\text{Dogs : Cats} = 22 : 18 = 11 : 9\)
The total number of animals must be a multiple of \(11 + 9 = 20\), and the number of dogs and cats must match the ratio \(11 : 9\) and be in the provided options.
| Cats | Dogs | Total Animals | Ratio Check |
| 36 | 44 | 36 + 44 = 80 | 80 ÷ 20 = 4; Cats: 4 × 9 = 36, Dogs: 4 × 11 = 44. Matches ratio and options. |
Other pairs don’t yield the 11:9 ratio or fit the options. For example, 22 cats and 27 dogs give a total of 49, not a multiple of 20, and the ratio 27:22 ≠ 11:9.
Answer: 36 cats, 44 dogs
12 Jul 2025, 11:23
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1. The question asks us to find out a working pair of the number of cats and dogs.
2. Let these two values be x and y, respectively. In addition, the weights of the cats and dogs are \(c_1, c_2, ..., c_x\) and \(d_1, d_2, ..., d_y\), respectively.
3. Now, using the formula for averages, we can write down the following equations:
- \(42 = \frac{c_1 + c_2 + ... + c_x + d_1 + d_2 + ... + d_y}{x + y}\).
- \(20 = \frac{c_1 + c_2 + ... + c_x}{x} \rightarrow 20x = c_1 + c_2 + ... + c_x\).
- \(60 = \frac{d_1 + d_2 + ... + d_y}{y} \rightarrow 60y = d_1 + d_2 + ... + d_y\).
4. The first equation can be rewritten, \(42 = \frac{c_1 + c_2 + ... + c_x + d_1 + d_2 + ... + d_y}{x + y} = \frac{20x + 60y}{x + y} \rightarrow 42x + 42y = 20x + 60y \rightarrow 22x = 18y \rightarrow x = \frac{9}{11} * y\).
5. In order for x to be an integer, y must be divisible by 11, which leaves 3 options to check:
- y = 11. \(x = \frac{9}{11} * 11 = 9\), which doesn't work.
- y = 22. \(x = \frac{9}{11} * 22 = 18\), which doesn't work.
- y = 44. \(x = \frac{9}{11} * 44 = 36\), which works.
6. Our answer will be: Cats - 36 and Dogs - 44.
2. Let these two values be x and y, respectively. In addition, the weights of the cats and dogs are \(c_1, c_2, ..., c_x\) and \(d_1, d_2, ..., d_y\), respectively.
3. Now, using the formula for averages, we can write down the following equations:
- \(42 = \frac{c_1 + c_2 + ... + c_x + d_1 + d_2 + ... + d_y}{x + y}\).
- \(20 = \frac{c_1 + c_2 + ... + c_x}{x} \rightarrow 20x = c_1 + c_2 + ... + c_x\).
- \(60 = \frac{d_1 + d_2 + ... + d_y}{y} \rightarrow 60y = d_1 + d_2 + ... + d_y\).
4. The first equation can be rewritten, \(42 = \frac{c_1 + c_2 + ... + c_x + d_1 + d_2 + ... + d_y}{x + y} = \frac{20x + 60y}{x + y} \rightarrow 42x + 42y = 20x + 60y \rightarrow 22x = 18y \rightarrow x = \frac{9}{11} * y\).
5. In order for x to be an integer, y must be divisible by 11, which leaves 3 options to check:
- y = 11. \(x = \frac{9}{11} * 11 = 9\), which doesn't work.
- y = 22. \(x = \frac{9}{11} * 22 = 18\), which doesn't work.
- y = 44. \(x = \frac{9}{11} * 44 = 36\), which works.
6. Our answer will be: Cats - 36 and Dogs - 44.
