AMC 10 · 2012 · #5

Grade 3 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

multi-digit-arithmeticfraction-multiplication identify-subproblems ↑ Prerequisites: multi-digit-arithmetic

📏 Medium solution 💡 2 insights

Problem

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?

Pick an answer.

(A)

150

(B)

200

(C)

250

(D)

300

(E)

400

AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

View mode:

Toolkit + CCSS Solution

Understand

Restated: A shelter took in 100 adult cats. Half are female. Half of those females each brought a litter, and each litter has 4 kittens on average. Find the total number of cats plus kittens.

Givens: 100 adult cats total; Half of the adult cats are female; Half of the female cats came with a litter; Average litter size is 4 kittens

Unknowns: Total number of cats and kittens received

Understand

Restated: A shelter took in 100 adult cats. Half are female. Half of those females each brought a litter, and each litter has 4 kittens on average. Find the total number of cats plus kittens.

Givens: 100 adult cats total; Half of the adult cats are female; Half of the female cats came with a litter; Average litter size is 4 kittens

Plan

Primary tool: #7 Identify Subproblems

Secondary: #8 Analyze the Units

The total is built from a chain of smaller counts: females, then litters, then kittens, then the grand total. Solving each small count in order and keeping cats and kittens as separate groups avoids the classic slip of reporting only the kittens.

Execute — Answer: B

#7 Identify Subproblems 3.OA.A.3 Step 1

Count the female cats

Half of the 100 adult cats are female, so divide 100 by 2 to get the number of female cats.

$$100 \div 2 = 50$$

💡 "Half of" a group just means split it into two equal parts and take one.

#7 Identify Subproblems 3.OA.A.3 Step 2

Count the litters

Half of the 50 female cats each came with a litter, so divide 50 by 2 to get the number of litters.

$$50 \div 2 = 25$$

💡 One litter per female that had kittens, so the number of litters equals that half.

#7 Identify Subproblems 3.OA.A.3 Step 3

Count the kittens

Each of the 25 litters has 4 kittens on average, so multiply 25 by 4 to get the total kittens.

$$25 \times 4 = 100$$

💡 Equal-sized groups repeated means multiply the number of groups by the group size.

#8 Analyze the Units 3.NBT.A.2 Step 4

Add cats and kittens

Add the 100 adult cats to the 100 kittens.
Keeping the two groups distinct until the end gives the total, which is 200, choice (B).

$$100 + 100 = 200$$

💡 The question asks for everything received, so the adults and the new kittens both belong in the sum.

[1] #7 3.OA.A.3 Half of the 100 adult cats are female, so divide 100 by 2 to get the number of f

[2] #7 3.OA.A.3 Half of the 50 female cats each came with a litter, so divide 50 by 2 to get the

[3] #7 3.OA.A.3 Each of the 25 litters has 4 kittens on average, so multiply 25 by 4 to get the

[4] #8 3.NBT.A.2 Add the 100 adult cats to the 100 kittens. Keeping the two groups distinct until

Review

Reasonableness: The kittens (100) came out equal to the adult cats (100), so a total near double the adults makes sense; 200 sits right in the middle of the answer choices, and it is more than 100 because kittens were added. Answers like 150 would mean forgetting some kittens, and 100 would mean forgetting the adults entirely.

Alternative: Chain the fractions in one line: kittens = 100 \times \tfrac12 \times \tfrac12 \times 4 = 100, then add the 100 adults for 200. This confirms the step-by-step count with a single expression.

CCSS standards used (min grade 3)

3.OA.A.3 Solve multiplication and division word problems within 100 (Halving the cat counts and multiplying litters by the average litter size)
3.NBT.A.2 Fluently add and subtract within 1000 (Adding the adult cats and the kittens for the grand total)

⭐ Break a word problem into small counts, solve them in order, and add every group the question asks about.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Count the female cats

Count the litters

Count the kittens

Add cats and kittens

Review

CCSS standards used (min grade 3)

More like this