Last year adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult female cats were accompanied by a litter of kittens. The average number of kittens per litter was . What was the total number of cats and kittens received by the shelter last year?
- A)
- B)
- C)
- D)
- E)
Answer
B
Key insight
Follow the halves one at a time: 50 females, 25 litters, 25 times 4 = 100 kittens, then add the 100 adult cats.
Solution
Work through the story one step at a time and write down each number.
- Adult cats: . Half are female, so there are females.
- Half of those females came with a litter, so there were litters.
- The litters average kittens each. Some litters may be bigger and some smaller, but an average of over litters always means kittens in total.
- The question asks for all the animals, adults and kittens: .
The answer is .
Why this works
An average times the number of groups gives the total, even when the groups are different sizes. In a word problem with several steps, writing each number down as you go keeps you from stopping one step early. Before choosing an answer, reread what the question actually asks for; here it is the total of adults and kittens.
The trap
Answering with only the 100 kittens and forgetting to add the 100 adult cats.
Concise solution
Follow the chain of halves.
Adult females: half of , so .
Litters: half of the females brought one, so litters.
Kittens: an average of per litter over litters means kittens in all (the average times the number of litters is the total, whatever the individual litter sizes).
Total animals received: adults kittens .
The answer is .
Review in the AoPS books
- Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
- Introduction to Algebra, Chapter 6: Ratios and Percents
- Introduction to Algebra, Chapter 7: Proportion (§7.4)
- Prealgebra, Chapter 13: Data and Statistics (§13.1)
Techniques
Set up the equation/formula and compute; no special trick needed