Guess-and-check a legs problem

3.OA.A.33.OA.D.8 · take · grade 3

Archetype: Two-Category Counts from a Total · step in a 3-type progression

On Naomi's farm there are $10$ animals in all, some cows and some chickens. When she counts all the legs of the cows and chickens, there are $32$ legs in total. How many more cows than chickens are there?

Show solution

Understand

A farm has 10 animals that are cows and chickens, with 32 legs in total. Cows have 4 legs and chickens have 2 legs. We want how many more cows there are than chickens.

Givens

There are 10 animals in all (cows and chickens).
There are 32 legs in total.
A cow has 4 legs and a chicken has 2 legs.

Unknowns

The number of cows and the number of chickens.
How many more cows than chickens there are.

Constraints

Cows plus chickens equal 10.
Total legs equal 32.

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Guess a number of cows, compute the legs with multiplication, and adjust until the total is 32. A short organized list makes the right guess quick to reach.

Execute

#6 Guess and Check 3.OA.A.3

If all 10 animals were chickens, the legs would be 10 x 2 = 20. That is 12 fewer than 32, so some animals must be cows.

10 \times 2 = 20,\quad 32 - 20 = 12

Counting all legs as 2 each is an easy starting guess, and the shortfall tells how many extra legs are needed.

#6 Guess and Check 3.OA.D.8

Each time a chicken becomes a cow, the leg count goes up by 2 (from 2 legs to 4 legs). To add the missing 12 legs, swap 12 / 2 = 6 chickens into cows.

12 \div 2 = 6

Every cow adds exactly 2 legs over a chicken, so dividing the missing 12 by 2 gives the number of cows.

#2 Make a Systematic List 3.OA.D.8

So there are 6 cows and 4 chickens. Check: 6 x 4 + 4 x 2 = 24 + 8 = 32 legs. The number of more cows than chickens is 6 - 4 = 2.

6 \times 4 + 4 \times 2 = 32,\quad 6 - 4 = 2

Verifying the legs confirms the guess, and subtracting gives how many more cows there are.

Answer: 2 more cows

Review

6 cows and 4 chickens make 10 animals and 24 + 8 = 32 legs, matching both conditions, so the difference of 2 is correct.

You could make a table of cow counts (0,1,2,...) with total legs and read off the row giving 32 legs, which lands on 6 cows and 4 chickens.

Standards · min grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Computing leg totals like 10 x 2 and 6 x 4 from the animal counts.
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the leg conditions and finding how many more cows than chickens.

💡 Guess all chickens, then trade up to cows two legs at a time -- just Grade 3 multiply-and-check!