← Guess-and-check a legs problem · Two-Category Counts from a Total

Guess-and-check a legs problem · 10 practice problems

3.OA.A.33.OA.D.8

Generated variants — 10

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 2 more cows

On Chloe's farm there are $6$ animals in all, some cows and some chickens. When she counts all the legs of the cows and chickens, there are $20$ legs in total. How many more cows than chickens are there?

Show solution

1 · Understandwhat's really being asked

A farm has 6 animals that are cows and chickens, with 20 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 6 animals in all (cows and chickens).
There are 20 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 6.
Total legs equal 20.

2 · Planchoose the strategy

#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 right. A short organized list makes the right guess quick to reach.

3 · Execute3 carry out the plan

1Start from all chickens

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

If all 6 animals were chickens, the legs would be 6 x 2 = 12. That is 8 fewer than 20, so some animals must be cows.

6 \times 2 = 12,\quad 20 - 12 = 8

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

2Swap chickens for cows

#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 8 legs, swap 8 / 2 = 4 chickens into cows.

8 \div 2 = 4

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

3Check and compare

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

So there are 4 cows and 2 chickens. Check: 4 x 4 + 2 x 2 = 16 + 4 = 20 legs. The number of more cows than chickens is 4 - 2 = 2.

4 \times 4 + 2 \times 2 = 20,\quad 4 - 2 = 2

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

Answer: 2 more cows

4 · Reviewdoes it hold up?

4 cows and 2 chickens make 6 animals and 16 + 4 = 20 legs, matching both conditions, so the difference of 2 is correct.

Another way: You could make a table of cow counts (0,1,2,...) with total legs and read off the row giving 20 legs, which lands on 4 cows and 2 chickens.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Computing leg totals like 6 x 2 and 4 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.

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

Variant 2 easy answer: 1 more cows

On Sophia's farm there are $7$ animals in all, some cows and some chickens. When she counts all the legs of the cows and chickens, there are $22$ legs in total. How many more cows than chickens are there?

Show solution

1 · Understandwhat's really being asked

A farm has 7 animals that are cows and chickens, with 22 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 7 animals in all (cows and chickens).
There are 22 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 7.
Total legs equal 22.

2 · Planchoose the strategy

#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 right. A short organized list makes the right guess quick to reach.

3 · Execute3 carry out the plan

1Start from all chickens

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

If all 7 animals were chickens, the legs would be 7 x 2 = 14. That is 8 fewer than 22, so some animals must be cows.

7 \times 2 = 14,\quad 22 - 14 = 8

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

2Swap chickens for cows

#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 8 legs, swap 8 / 2 = 4 chickens into cows.

8 \div 2 = 4

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

3Check and compare

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

So there are 4 cows and 3 chickens. Check: 4 x 4 + 3 x 2 = 16 + 6 = 22 legs. The number of more cows than chickens is 4 - 3 = 1.

4 \times 4 + 3 \times 2 = 22,\quad 4 - 3 = 1

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

Answer: 1 more cows

4 · Reviewdoes it hold up?

4 cows and 3 chickens make 7 animals and 16 + 6 = 22 legs, matching both conditions, so the difference of 1 is correct.

Another way: You could make a table of cow counts (0,1,2,...) with total legs and read off the row giving 22 legs, which lands on 4 cows and 3 chickens.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Computing leg totals like 7 x 2 and 4 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.

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

Variant 3 easy answer: 2 more cows

On Ethan's farm there are $8$ animals in all, some cows and some chickens. When she counts all the legs of the cows and chickens, there are $26$ legs in total. How many more cows than chickens are there?

Show solution

1 · Understandwhat's really being asked

A farm has 8 animals that are cows and chickens, with 26 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 8 animals in all (cows and chickens).
There are 26 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 8.
Total legs equal 26.

2 · Planchoose the strategy

#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 right. A short organized list makes the right guess quick to reach.

3 · Execute3 carry out the plan

1Start from all chickens

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

If all 8 animals were chickens, the legs would be 8 x 2 = 16. That is 10 fewer than 26, so some animals must be cows.

8 \times 2 = 16,\quad 26 - 16 = 10

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

2Swap chickens for cows

#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 10 legs, swap 10 / 2 = 5 chickens into cows.

10 \div 2 = 5

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

3Check and compare

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

So there are 5 cows and 3 chickens. Check: 5 x 4 + 3 x 2 = 20 + 6 = 26 legs. The number of more cows than chickens is 5 - 3 = 2.

5 \times 4 + 3 \times 2 = 26,\quad 5 - 3 = 2

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

Answer: 2 more cows

4 · Reviewdoes it hold up?

5 cows and 3 chickens make 8 animals and 20 + 6 = 26 legs, matching both conditions, so the difference of 2 is correct.

Another way: You could make a table of cow counts (0,1,2,...) with total legs and read off the row giving 26 legs, which lands on 5 cows and 3 chickens.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Computing leg totals like 8 x 2 and 5 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.

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

Variant 4 medium answer: 1 more cows

On Mason's farm there are $9$ animals in all, some cows and some chickens. When she counts all the legs of the cows and chickens, there are $28$ legs in total. How many more cows than chickens are there?

Show solution

1 · Understandwhat's really being asked

A farm has 9 animals that are cows and chickens, with 28 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 9 animals in all (cows and chickens).
There are 28 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 9.
Total legs equal 28.

2 · Planchoose the strategy

#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 right. A short organized list makes the right guess quick to reach.

3 · Execute3 carry out the plan

1Start from all chickens

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

If all 9 animals were chickens, the legs would be 9 x 2 = 18. That is 10 fewer than 28, so some animals must be cows.

9 \times 2 = 18,\quad 28 - 18 = 10

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

2Swap chickens for cows

#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 10 legs, swap 10 / 2 = 5 chickens into cows.

10 \div 2 = 5

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

3Check and compare

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

So there are 5 cows and 4 chickens. Check: 5 x 4 + 4 x 2 = 20 + 8 = 28 legs. The number of more cows than chickens is 5 - 4 = 1.

5 \times 4 + 4 \times 2 = 28,\quad 5 - 4 = 1

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

Answer: 1 more cows

4 · Reviewdoes it hold up?

5 cows and 4 chickens make 9 animals and 20 + 8 = 28 legs, matching both conditions, so the difference of 1 is correct.

Another way: You could make a table of cow counts (0,1,2,...) with total legs and read off the row giving 28 legs, which lands on 5 cows and 4 chickens.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Computing leg totals like 9 x 2 and 5 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.

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

Variant 5 medium answer: 2 more cows

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

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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 right. A short organized list makes the right guess quick to reach.

3 · Execute3 carry out the plan

1Start from all chickens

#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.

2Swap chickens for cows

#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 legs by 2 gives the number of cows.

3Check and compare

#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

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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.

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

Variant 6 medium answer: 1 more cows

On Amelia's farm there are $11$ animals in all, some cows and some chickens. When she counts all the legs of the cows and chickens, there are $34$ legs in total. How many more cows than chickens are there?

Show solution

1 · Understandwhat's really being asked

A farm has 11 animals that are cows and chickens, with 34 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 11 animals in all (cows and chickens).
There are 34 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 11.
Total legs equal 34.

2 · Planchoose the strategy

#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 right. A short organized list makes the right guess quick to reach.

3 · Execute3 carry out the plan

1Start from all chickens

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

If all 11 animals were chickens, the legs would be 11 x 2 = 22. That is 12 fewer than 34, so some animals must be cows.

11 \times 2 = 22,\quad 34 - 22 = 12

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

2Swap chickens for cows

#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 legs by 2 gives the number of cows.

3Check and compare

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

So there are 6 cows and 5 chickens. Check: 6 x 4 + 5 x 2 = 24 + 10 = 34 legs. The number of more cows than chickens is 6 - 5 = 1.

6 \times 4 + 5 \times 2 = 34,\quad 6 - 5 = 1

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

Answer: 1 more cows

4 · Reviewdoes it hold up?

6 cows and 5 chickens make 11 animals and 24 + 10 = 34 legs, matching both conditions, so the difference of 1 is correct.

Another way: You could make a table of cow counts (0,1,2,...) with total legs and read off the row giving 34 legs, which lands on 6 cows and 5 chickens.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Computing leg totals like 11 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.

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

Variant 7 medium answer: 2 more cows

On Olivia's farm there are $12$ animals in all, some cows and some chickens. When she counts all the legs of the cows and chickens, there are $38$ legs in total. How many more cows than chickens are there?

Show solution

1 · Understandwhat's really being asked

A farm has 12 animals that are cows and chickens, with 38 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 12 animals in all (cows and chickens).
There are 38 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 12.
Total legs equal 38.

2 · Planchoose the strategy

#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 right. A short organized list makes the right guess quick to reach.

3 · Execute3 carry out the plan

1Start from all chickens

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

If all 12 animals were chickens, the legs would be 12 x 2 = 24. That is 14 fewer than 38, so some animals must be cows.

12 \times 2 = 24,\quad 38 - 24 = 14

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

2Swap chickens for cows

#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 14 legs, swap 14 / 2 = 7 chickens into cows.

14 \div 2 = 7

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

3Check and compare

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

So there are 7 cows and 5 chickens. Check: 7 x 4 + 5 x 2 = 28 + 10 = 38 legs. The number of more cows than chickens is 7 - 5 = 2.

7 \times 4 + 5 \times 2 = 38,\quad 7 - 5 = 2

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

Answer: 2 more cows

4 · Reviewdoes it hold up?

7 cows and 5 chickens make 12 animals and 28 + 10 = 38 legs, matching both conditions, so the difference of 2 is correct.

Another way: You could make a table of cow counts (0,1,2,...) with total legs and read off the row giving 38 legs, which lands on 7 cows and 5 chickens.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Computing leg totals like 12 x 2 and 7 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.

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

Variant 8 hard answer: 2 more cows

On Henry's farm there are $14$ animals in all, some cows and some chickens. When she counts all the legs of the cows and chickens, there are $44$ legs in total. How many more cows than chickens are there?

Show solution

1 · Understandwhat's really being asked

A farm has 14 animals that are cows and chickens, with 44 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 14 animals in all (cows and chickens).
There are 44 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 14.
Total legs equal 44.

2 · Planchoose the strategy

#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 right. A short organized list makes the right guess quick to reach.

3 · Execute3 carry out the plan

1Start from all chickens

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

If all 14 animals were chickens, the legs would be 14 x 2 = 28. That is 16 fewer than 44, so some animals must be cows.

14 \times 2 = 28,\quad 44 - 28 = 16

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

2Swap chickens for cows

#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 16 legs, swap 16 / 2 = 8 chickens into cows.

16 \div 2 = 8

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

3Check and compare

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

So there are 8 cows and 6 chickens. Check: 8 x 4 + 6 x 2 = 32 + 12 = 44 legs. The number of more cows than chickens is 8 - 6 = 2.

8 \times 4 + 6 \times 2 = 44,\quad 8 - 6 = 2

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

Answer: 2 more cows

4 · Reviewdoes it hold up?

8 cows and 6 chickens make 14 animals and 32 + 12 = 44 legs, matching both conditions, so the difference of 2 is correct.

Another way: You could make a table of cow counts (0,1,2,...) with total legs and read off the row giving 44 legs, which lands on 8 cows and 6 chickens.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Computing leg totals like 14 x 2 and 8 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.

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

Variant 9 hard answer: 1 more cows

On Lucas's farm there are $15$ animals in all, some cows and some chickens. When she counts all the legs of the cows and chickens, there are $46$ legs in total. How many more cows than chickens are there?

Show solution

1 · Understandwhat's really being asked

A farm has 15 animals that are cows and chickens, with 46 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 15 animals in all (cows and chickens).
There are 46 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 15.
Total legs equal 46.

2 · Planchoose the strategy

#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 right. A short organized list makes the right guess quick to reach.

3 · Execute3 carry out the plan

1Start from all chickens

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

If all 15 animals were chickens, the legs would be 15 x 2 = 30. That is 16 fewer than 46, so some animals must be cows.

15 \times 2 = 30,\quad 46 - 30 = 16

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

2Swap chickens for cows

#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 16 legs, swap 16 / 2 = 8 chickens into cows.

16 \div 2 = 8

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

3Check and compare

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

So there are 8 cows and 7 chickens. Check: 8 x 4 + 7 x 2 = 32 + 14 = 46 legs. The number of more cows than chickens is 8 - 7 = 1.

8 \times 4 + 7 \times 2 = 46,\quad 8 - 7 = 1

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

Answer: 1 more cows

4 · Reviewdoes it hold up?

8 cows and 7 chickens make 15 animals and 32 + 14 = 46 legs, matching both conditions, so the difference of 1 is correct.

Another way: You could make a table of cow counts (0,1,2,...) with total legs and read off the row giving 46 legs, which lands on 8 cows and 7 chickens.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Computing leg totals like 15 x 2 and 8 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.

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

Variant 10 hard answer: 4 more cows

On Isaac's farm there are $20$ animals in all, some cows and some chickens. When she counts all the legs of the cows and chickens, there are $64$ legs in total. How many more cows than chickens are there?

Show solution

1 · Understandwhat's really being asked

A farm has 20 animals that are cows and chickens, with 64 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 20 animals in all (cows and chickens).
There are 64 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 20.
Total legs equal 64.

2 · Planchoose the strategy

#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 right. A short organized list makes the right guess quick to reach.

3 · Execute3 carry out the plan

1Start from all chickens

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

If all 20 animals were chickens, the legs would be 20 x 2 = 40. That is 24 fewer than 64, so some animals must be cows.

20 \times 2 = 40,\quad 64 - 40 = 24

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

2Swap chickens for cows

#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 24 legs, swap 24 / 2 = 12 chickens into cows.

24 \div 2 = 12

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

3Check and compare

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

So there are 12 cows and 8 chickens. Check: 12 x 4 + 8 x 2 = 48 + 16 = 64 legs. The number of more cows than chickens is 12 - 8 = 4.

12 \times 4 + 8 \times 2 = 64,\quad 12 - 8 = 4

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

Answer: 4 more cows

4 · Reviewdoes it hold up?

12 cows and 8 chickens make 20 animals and 48 + 16 = 64 legs, matching both conditions, so the difference of 4 is correct.

Another way: You could make a table of cow counts (0,1,2,...) with total legs and read off the row giving 64 legs, which lands on 12 cows and 8 chickens.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Computing leg totals like 20 x 2 and 12 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.

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