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2012 AMC 10A Problem 5

2012 AMC 10A Problem 5AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 1 of 5 · about 2 min · position P1-10

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?

A)
$150$
B)
$200$
C)
$250$
D)
$300$
E)
$400$

Answer

Key insight

Half of 100 is 50 females, half of those is 25 litters, and 25 litters averaging 4 kittens is 100 kittens; add the 100 adults.

Solution

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 $\boxed{\textbf{(B)}\ 200}$ .

Why this works

"Average per litter" times "number of litters" recovers the total number of kittens, even though individual litters may differ. The problem is a sequence of simple proportional steps; writing each intermediate count down prevents losing track of what the final question asks for.

The trap

Reporting only the 100 kittens, or forgetting that the 100 adult cats must be included in the total.

Common mistakes

Reporting only the 100 kittens, or forgetting that the 100 adult cats must be included in the total.
Using all $50$ females as litter-bearers, giving $200$ kittens and a total of $300$ .

Techniques

Set up the equation/formula and compute; no special trick needed