2012 AMC 10A Problem 5

P1-10~2 minPrint

Linear Equations & Word Problems Ratios, Percents & Averages Set up the equation/formula and compute; no special trick needed

Last year $100$ 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 $4$ . What was the total number of cats and kittens received by the shelter last year?

Show solution without answering

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: $100$ . Half are female, so there are $50$ females.
Half of those $50$ females came with a litter, so there were $25$ litters.
The litters average $4$ kittens each. Some litters may be bigger and some smaller, but an average of $4$ over $25$ litters always means $25 \times 4 = 100$ kittens in total.
The question asks for all the animals, adults and kittens: $100 + 100 = 200$ .

The answer is $\boxed{\textbf{(B)}\ 200}$ .

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.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Follow the chain of halves.

Adult females: half of $100$ , so $50$ .

Litters: half of the $50$ females brought one, so $25$ litters.

Kittens: an average of $4$ per litter over $25$ litters means $25 \cdot 4 = 100$ kittens in all (the average times the number of litters is the total, whatever the individual litter sizes).

Total animals received: $100$ adults $+ 100$ kittens $= 200$ .

The answer is $\boxed{\textbf{(B)}\ 200}$ .

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)