Practice · Fixed sum: one side falls, other rises · Sensim Math

Variant 1 easy

At the store, Wujin bought snacks and juice boxes. Each snack costs $300$ won and each juice box costs $500$ won. He bought $20$ items in all and paid $7400$ won. How many snacks and how many juice boxes did Wujin buy?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Wujin bought 20 things altogether -- snacks at 300 won each and juice boxes at 500 won each -- and spent 7400 won. We need how many of each.

Givens

A snack costs 300 won; a juice box costs 500 won.
He bought 20 items in total.
He paid 7400 won.

Unknowns

How many snacks and how many juice boxes.

Constraints

The number of items is fixed, so buying one more juice box means buying one fewer snack.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #5 Look for a Pattern#2 Make a Systematic List

Start from the simplest possible basket -- all snacks -- and swap one snack for one juice box at a time. Each swap moves the bill by the same amount, so the shortfall says how many swaps are needed.

3 · Execute5 carry out the plan

1Start with an easy guess: all 20 are snacks

#6 Guess and Check 4.NBT.B.5

If every item were a snack, the bill would be 20 lots of 300.

20 \times 300 = 6000

That guess costs 6000 won.

2See how far off the guess is

#6 Guess and Check 4.NBT.B.4

The real bill was 7400, so the guess falls short.

7400 - 6000 = 1400

1400 won unaccounted for.

3Find the steady jump per swap (the pattern)

#5 Look for a Pattern 5.OA.A.2

Swapping one snack for one juice box keeps the count at 20 but raises the bill by the price difference.

500 - 300 = 200

Every swap adds 200 won.

4Count how many swaps reach the gap

#5 Look for a Pattern 4.OA.A.3

Divide the shortfall by the size of one swap.

1400 \div 200 = 7

7 juice boxes.

5Find the snacks and confirm with a quick list

#2 Make a Systematic List 4.OA.A.3

The rest of the 20 items are snacks; check the bill.

20 - 7 = 13,\quad 13 \times 300 + 7 \times 500 = 7400

13 snacks and 7 juice boxes.

Answer: 13 snacks and 7 juice boxes

4 · Reviewdoes it hold up?

The bill 7400 sits between all-snacks (6000) and all-juice (10000), so a mix was right -- and it leans towards snacks, which is what 13 to 7 shows.

Another way: Starting from all juice boxes instead costs 10000 won, which is 2600 too much; dividing that by 200 gives 13 snacks directly.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Costing the all-snacks starting guess.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Finding how far the guess falls short.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the price gap as the value of one swap.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Counting the swaps and splitting the items between the two kinds.

💡Takeaway. When the number of items is fixed, trading one for the other changes the bill by the same step every time -- so the gap counts the trades.

Variant 2 easy

At the store, Wujin bought snacks and juice boxes. Each snack costs $250$ won and each juice box costs $400$ won. He bought $25$ items in all and paid $7600$ won. How many snacks and how many juice boxes did Wujin buy?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Wujin bought 25 things altogether -- snacks at 250 won each and juice boxes at 400 won each -- and spent 7600 won. We need how many of each.

Givens

A snack costs 250 won; a juice box costs 400 won.
He bought 25 items in total.
He paid 7600 won.

Unknowns

How many snacks and how many juice boxes.

Constraints

The number of items is fixed, so buying one more juice box means buying one fewer snack.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #5 Look for a Pattern#2 Make a Systematic List

Start from the simplest possible basket -- all snacks -- and swap one snack for one juice box at a time. Each swap moves the bill by the same amount, so the shortfall says how many swaps are needed.

3 · Execute5 carry out the plan

1Start with an easy guess: all 25 are snacks

#6 Guess and Check 4.NBT.B.5

If every item were a snack, the bill would be 25 lots of 250.

25 \times 250 = 6250

That guess costs 6250 won.

2See how far off the guess is

#6 Guess and Check 4.NBT.B.4

The real bill was 7600, so the guess falls short.

7600 - 6250 = 1350

1350 won unaccounted for.

3Find the steady jump per swap (the pattern)

#5 Look for a Pattern 5.OA.A.2

Swapping one snack for one juice box keeps the count at 25 but raises the bill by the price difference.

400 - 250 = 150

Every swap adds 150 won.

4Count how many swaps reach the gap

#5 Look for a Pattern 4.OA.A.3

Divide the shortfall by the size of one swap.

1350 \div 150 = 9

9 juice boxes.

5Find the snacks and confirm with a quick list

#2 Make a Systematic List 4.OA.A.3

The rest of the 25 items are snacks; check the bill.

25 - 9 = 16,\quad 16 \times 250 + 9 \times 400 = 7600

16 snacks and 9 juice boxes.

Answer: 16 snacks and 9 juice boxes

4 · Reviewdoes it hold up?

The bill 7600 sits between all-snacks (6250) and all-juice (10000), so a mix was right -- and it leans towards snacks, which is what 16 to 9 shows.

Another way: Starting from all juice boxes instead costs 10000 won, which is 2400 too much; dividing that by 150 gives 16 snacks directly.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Costing the all-snacks starting guess.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Finding how far the guess falls short.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the price gap as the value of one swap.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Counting the swaps and splitting the items between the two kinds.

💡Takeaway. When the number of items is fixed, trading one for the other changes the bill by the same step every time -- so the gap counts the trades.

Variant 3 easy

At the store, Wujin bought snacks and juice boxes. Each snack costs $500$ won and each juice box costs $700$ won. He bought $15$ items in all and paid $8300$ won. How many snacks and how many juice boxes did Wujin buy?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Wujin bought 15 things altogether -- snacks at 500 won each and juice boxes at 700 won each -- and spent 8300 won. We need how many of each.

Givens

A snack costs 500 won; a juice box costs 700 won.
He bought 15 items in total.
He paid 8300 won.

Unknowns

How many snacks and how many juice boxes.

Constraints

The number of items is fixed, so buying one more juice box means buying one fewer snack.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #5 Look for a Pattern#2 Make a Systematic List

Start from the simplest possible basket -- all snacks -- and swap one snack for one juice box at a time. Each swap moves the bill by the same amount, so the shortfall says how many swaps are needed.

3 · Execute5 carry out the plan

1Start with an easy guess: all 15 are snacks

#6 Guess and Check 4.NBT.B.5

If every item were a snack, the bill would be 15 lots of 500.

15 \times 500 = 7500

That guess costs 7500 won.

2See how far off the guess is

#6 Guess and Check 4.NBT.B.4

The real bill was 8300, so the guess falls short.

8300 - 7500 = 800

800 won unaccounted for.

3Find the steady jump per swap (the pattern)

#5 Look for a Pattern 5.OA.A.2

Swapping one snack for one juice box keeps the count at 15 but raises the bill by the price difference.

700 - 500 = 200

Every swap adds 200 won.

4Count how many swaps reach the gap

#5 Look for a Pattern 4.OA.A.3

Divide the shortfall by the size of one swap.

800 \div 200 = 4

4 juice boxes.

5Find the snacks and confirm with a quick list

#2 Make a Systematic List 4.OA.A.3

The rest of the 15 items are snacks; check the bill.

15 - 4 = 11,\quad 11 \times 500 + 4 \times 700 = 8300

11 snacks and 4 juice boxes.

Answer: 11 snacks and 4 juice boxes

4 · Reviewdoes it hold up?

The bill 8300 sits between all-snacks (7500) and all-juice (10500), so a mix was right -- and it leans towards snacks, which is what 11 to 4 shows.

Another way: Starting from all juice boxes instead costs 10500 won, which is 2200 too much; dividing that by 200 gives 11 snacks directly.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Costing the all-snacks starting guess.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Finding how far the guess falls short.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the price gap as the value of one swap.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Counting the swaps and splitting the items between the two kinds.

💡Takeaway. When the number of items is fixed, trading one for the other changes the bill by the same step every time -- so the gap counts the trades.

Variant 4 easy

At the store, Wujin bought snacks and juice boxes. Each snack costs $450$ won and each juice box costs $600$ won. He bought $18$ items in all and paid $9000$ won. How many snacks and how many juice boxes did Wujin buy?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Wujin bought 18 things altogether -- snacks at 450 won each and juice boxes at 600 won each -- and spent 9000 won. We need how many of each.

Givens

A snack costs 450 won; a juice box costs 600 won.
He bought 18 items in total.
He paid 9000 won.

Unknowns

How many snacks and how many juice boxes.

Constraints

The number of items is fixed, so buying one more juice box means buying one fewer snack.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #5 Look for a Pattern#2 Make a Systematic List

Start from the simplest possible basket -- all snacks -- and swap one snack for one juice box at a time. Each swap moves the bill by the same amount, so the shortfall says how many swaps are needed.

3 · Execute5 carry out the plan

1Start with an easy guess: all 18 are snacks

#6 Guess and Check 4.NBT.B.5

If every item were a snack, the bill would be 18 lots of 450.

18 \times 450 = 8100

That guess costs 8100 won.

2See how far off the guess is

#6 Guess and Check 4.NBT.B.4

The real bill was 9000, so the guess falls short.

9000 - 8100 = 900

900 won unaccounted for.

3Find the steady jump per swap (the pattern)

#5 Look for a Pattern 5.OA.A.2

Swapping one snack for one juice box keeps the count at 18 but raises the bill by the price difference.

600 - 450 = 150

Every swap adds 150 won.

4Count how many swaps reach the gap

#5 Look for a Pattern 4.OA.A.3

Divide the shortfall by the size of one swap.

900 \div 150 = 6

6 juice boxes.

5Find the snacks and confirm with a quick list

#2 Make a Systematic List 4.OA.A.3

The rest of the 18 items are snacks; check the bill.

18 - 6 = 12,\quad 12 \times 450 + 6 \times 600 = 9000

12 snacks and 6 juice boxes.

Answer: 12 snacks and 6 juice boxes

4 · Reviewdoes it hold up?

The bill 9000 sits between all-snacks (8100) and all-juice (10800), so a mix was right -- and it leans towards snacks, which is what 12 to 6 shows.

Another way: Starting from all juice boxes instead costs 10800 won, which is 1800 too much; dividing that by 150 gives 12 snacks directly.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Costing the all-snacks starting guess.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Finding how far the guess falls short.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the price gap as the value of one swap.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Counting the swaps and splitting the items between the two kinds.

💡Takeaway. When the number of items is fixed, trading one for the other changes the bill by the same step every time -- so the gap counts the trades.

Variant 5 medium

At the store, Wujin bought snacks and juice boxes. Each snack costs $350$ won and each juice box costs $550$ won. He bought $22$ items in all and paid $9900$ won. How many snacks and how many juice boxes did Wujin buy?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Wujin bought 22 things altogether -- snacks at 350 won each and juice boxes at 550 won each -- and spent 9900 won. We need how many of each.

Givens

A snack costs 350 won; a juice box costs 550 won.
He bought 22 items in total.
He paid 9900 won.

Unknowns

How many snacks and how many juice boxes.

Constraints

The number of items is fixed, so buying one more juice box means buying one fewer snack.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #5 Look for a Pattern#2 Make a Systematic List

Start from the simplest possible basket -- all snacks -- and swap one snack for one juice box at a time. Each swap moves the bill by the same amount, so the shortfall says how many swaps are needed.

3 · Execute5 carry out the plan

1Start with an easy guess: all 22 are snacks

#6 Guess and Check 4.NBT.B.5

If every item were a snack, the bill would be 22 lots of 350.

22 \times 350 = 7700

That guess costs 7700 won.

2See how far off the guess is

#6 Guess and Check 4.NBT.B.4

The real bill was 9900, so the guess falls short.

9900 - 7700 = 2200

2200 won unaccounted for.

3Find the steady jump per swap (the pattern)

#5 Look for a Pattern 5.OA.A.2

Swapping one snack for one juice box keeps the count at 22 but raises the bill by the price difference.

550 - 350 = 200

Every swap adds 200 won.

4Count how many swaps reach the gap

#5 Look for a Pattern 4.OA.A.3

Divide the shortfall by the size of one swap.

2200 \div 200 = 11

11 juice boxes.

5Find the snacks and confirm with a quick list

#2 Make a Systematic List 4.OA.A.3

The rest of the 22 items are snacks; check the bill.

22 - 11 = 11,\quad 11 \times 350 + 11 \times 550 = 9900

11 snacks and 11 juice boxes.

Answer: 11 snacks and 11 juice boxes

4 · Reviewdoes it hold up?

The bill 9900 sits between all-snacks (7700) and all-juice (12100), so a mix was right -- and it leans towards snacks, which is what 11 to 11 shows.

Another way: Starting from all juice boxes instead costs 12100 won, which is 2200 too much; dividing that by 200 gives 11 snacks directly.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Costing the all-snacks starting guess.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Finding how far the guess falls short.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the price gap as the value of one swap.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Counting the swaps and splitting the items between the two kinds.

💡Takeaway. When the number of items is fixed, trading one for the other changes the bill by the same step every time -- so the gap counts the trades.

Variant 6 medium

At the store, Wujin bought snacks and juice boxes. Each snack costs $900$ won and each juice box costs $1500$ won. He bought $10$ items in all and paid $10800$ won. How many snacks and how many juice boxes did Wujin buy?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Wujin bought 10 things altogether -- snacks at 900 won each and juice boxes at 1500 won each -- and spent 10800 won. We need how many of each.

Givens

A snack costs 900 won; a juice box costs 1500 won.
He bought 10 items in total.
He paid 10800 won.

Unknowns

How many snacks and how many juice boxes.

Constraints

The number of items is fixed, so buying one more juice box means buying one fewer snack.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #5 Look for a Pattern#2 Make a Systematic List

Start from the simplest possible basket -- all snacks -- and swap one snack for one juice box at a time. Each swap moves the bill by the same amount, so the shortfall says how many swaps are needed.

3 · Execute5 carry out the plan

1Start with an easy guess: all 10 are snacks

#6 Guess and Check 4.NBT.B.5

If every item were a snack, the bill would be 10 lots of 900.

10 \times 900 = 9000

That guess costs 9000 won.

2See how far off the guess is

#6 Guess and Check 4.NBT.B.4

The real bill was 10800, so the guess falls short.

10800 - 9000 = 1800

1800 won unaccounted for.

3Find the steady jump per swap (the pattern)

#5 Look for a Pattern 5.OA.A.2

Swapping one snack for one juice box keeps the count at 10 but raises the bill by the price difference.

1500 - 900 = 600

Every swap adds 600 won.

4Count how many swaps reach the gap

#5 Look for a Pattern 4.OA.A.3

Divide the shortfall by the size of one swap.

1800 \div 600 = 3

3 juice boxes.

5Find the snacks and confirm with a quick list

#2 Make a Systematic List 4.OA.A.3

The rest of the 10 items are snacks; check the bill.

10 - 3 = 7,\quad 7 \times 900 + 3 \times 1500 = 10800

7 snacks and 3 juice boxes.

Answer: 7 snacks and 3 juice boxes

4 · Reviewdoes it hold up?

The bill 10800 sits between all-snacks (9000) and all-juice (15000), so a mix was right -- and it leans towards snacks, which is what 7 to 3 shows.

Another way: Starting from all juice boxes instead costs 15000 won, which is 4200 too much; dividing that by 600 gives 7 snacks directly.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Costing the all-snacks starting guess.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Finding how far the guess falls short.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the price gap as the value of one swap.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Counting the swaps and splitting the items between the two kinds.

💡Takeaway. When the number of items is fixed, trading one for the other changes the bill by the same step every time -- so the gap counts the trades.

Variant 7 medium

At the store, Wujin bought snacks and juice boxes. Each snack costs $400$ won and each juice box costs $650$ won. He bought $20$ items in all and paid $11000$ won. How many snacks and how many juice boxes did Wujin buy?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Wujin bought 20 things altogether -- snacks at 400 won each and juice boxes at 650 won each -- and spent 11000 won. We need how many of each.

Givens

A snack costs 400 won; a juice box costs 650 won.
He bought 20 items in total.
He paid 11000 won.

Unknowns

How many snacks and how many juice boxes.

Constraints

The number of items is fixed, so buying one more juice box means buying one fewer snack.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #5 Look for a Pattern#2 Make a Systematic List

Start from the simplest possible basket -- all snacks -- and swap one snack for one juice box at a time. Each swap moves the bill by the same amount, so the shortfall says how many swaps are needed.

3 · Execute5 carry out the plan

1Start with an easy guess: all 20 are snacks

#6 Guess and Check 4.NBT.B.5

If every item were a snack, the bill would be 20 lots of 400.

20 \times 400 = 8000

That guess costs 8000 won.

2See how far off the guess is

#6 Guess and Check 4.NBT.B.4

The real bill was 11000, so the guess falls short.

11000 - 8000 = 3000

3000 won unaccounted for.

3Find the steady jump per swap (the pattern)

#5 Look for a Pattern 5.OA.A.2

Swapping one snack for one juice box keeps the count at 20 but raises the bill by the price difference.

650 - 400 = 250

Every swap adds 250 won.

4Count how many swaps reach the gap

#5 Look for a Pattern 4.OA.A.3

Divide the shortfall by the size of one swap.

3000 \div 250 = 12

12 juice boxes.

5Find the snacks and confirm with a quick list

#2 Make a Systematic List 4.OA.A.3

The rest of the 20 items are snacks; check the bill.

20 - 12 = 8,\quad 8 \times 400 + 12 \times 650 = 11000

8 snacks and 12 juice boxes.

Answer: 8 snacks and 12 juice boxes

4 · Reviewdoes it hold up?

The bill 11000 sits between all-snacks (8000) and all-juice (13000), so a mix was right -- and it leans towards snacks, which is what 8 to 12 shows.

Another way: Starting from all juice boxes instead costs 13000 won, which is 2000 too much; dividing that by 250 gives 8 snacks directly.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Costing the all-snacks starting guess.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Finding how far the guess falls short.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the price gap as the value of one swap.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Counting the swaps and splitting the items between the two kinds.

💡Takeaway. When the number of items is fixed, trading one for the other changes the bill by the same step every time -- so the gap counts the trades.

Variant 8 medium

At the store, Wujin bought snacks and juice boxes. Each snack costs $700$ won and each juice box costs $1000$ won. He bought $14$ items in all and paid $11300$ won. How many snacks and how many juice boxes did Wujin buy?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Wujin bought 14 things altogether -- snacks at 700 won each and juice boxes at 1000 won each -- and spent 11300 won. We need how many of each.

Givens

A snack costs 700 won; a juice box costs 1000 won.
He bought 14 items in total.
He paid 11300 won.

Unknowns

How many snacks and how many juice boxes.

Constraints

The number of items is fixed, so buying one more juice box means buying one fewer snack.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #5 Look for a Pattern#2 Make a Systematic List

Start from the simplest possible basket -- all snacks -- and swap one snack for one juice box at a time. Each swap moves the bill by the same amount, so the shortfall says how many swaps are needed.

3 · Execute5 carry out the plan

1Start with an easy guess: all 14 are snacks

#6 Guess and Check 4.NBT.B.5

If every item were a snack, the bill would be 14 lots of 700.

14 \times 700 = 9800

That guess costs 9800 won.

2See how far off the guess is

#6 Guess and Check 4.NBT.B.4

The real bill was 11300, so the guess falls short.

11300 - 9800 = 1500

1500 won unaccounted for.

3Find the steady jump per swap (the pattern)

#5 Look for a Pattern 5.OA.A.2

Swapping one snack for one juice box keeps the count at 14 but raises the bill by the price difference.

1000 - 700 = 300

Every swap adds 300 won.

4Count how many swaps reach the gap

#5 Look for a Pattern 4.OA.A.3

Divide the shortfall by the size of one swap.

1500 \div 300 = 5

5 juice boxes.

5Find the snacks and confirm with a quick list

#2 Make a Systematic List 4.OA.A.3

The rest of the 14 items are snacks; check the bill.

14 - 5 = 9,\quad 9 \times 700 + 5 \times 1000 = 11300

9 snacks and 5 juice boxes.

Answer: 9 snacks and 5 juice boxes

4 · Reviewdoes it hold up?

The bill 11300 sits between all-snacks (9800) and all-juice (14000), so a mix was right -- and it leans towards snacks, which is what 9 to 5 shows.

Another way: Starting from all juice boxes instead costs 14000 won, which is 2700 too much; dividing that by 300 gives 9 snacks directly.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Costing the all-snacks starting guess.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Finding how far the guess falls short.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the price gap as the value of one swap.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Counting the swaps and splitting the items between the two kinds.

💡Takeaway. When the number of items is fixed, trading one for the other changes the bill by the same step every time -- so the gap counts the trades.

Variant 9 hard

At the store, Wujin bought snacks and juice boxes. Each snack costs $800$ won and each juice box costs $1200$ won. He bought $12$ items in all and paid $11600$ won. How many snacks and how many juice boxes did Wujin buy?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Wujin bought 12 things altogether -- snacks at 800 won each and juice boxes at 1200 won each -- and spent 11600 won. We need how many of each.

Givens

A snack costs 800 won; a juice box costs 1200 won.
He bought 12 items in total.
He paid 11600 won.

Unknowns

How many snacks and how many juice boxes.

Constraints

The number of items is fixed, so buying one more juice box means buying one fewer snack.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #5 Look for a Pattern#2 Make a Systematic List

Start from the simplest possible basket -- all snacks -- and swap one snack for one juice box at a time. Each swap moves the bill by the same amount, so the shortfall says how many swaps are needed.

3 · Execute5 carry out the plan

1Start with an easy guess: all 12 are snacks

#6 Guess and Check 4.NBT.B.5

If every item were a snack, the bill would be 12 lots of 800.

12 \times 800 = 9600

That guess costs 9600 won.

2See how far off the guess is

#6 Guess and Check 4.NBT.B.4

The real bill was 11600, so the guess falls short.

11600 - 9600 = 2000

2000 won unaccounted for.

3Find the steady jump per swap (the pattern)

#5 Look for a Pattern 5.OA.A.2

Swapping one snack for one juice box keeps the count at 12 but raises the bill by the price difference.

1200 - 800 = 400

Every swap adds 400 won.

4Count how many swaps reach the gap

#5 Look for a Pattern 4.OA.A.3

Divide the shortfall by the size of one swap.

2000 \div 400 = 5

5 juice boxes.

5Find the snacks and confirm with a quick list

#2 Make a Systematic List 4.OA.A.3

The rest of the 12 items are snacks; check the bill.

12 - 5 = 7,\quad 7 \times 800 + 5 \times 1200 = 11600

7 snacks and 5 juice boxes.

Answer: 7 snacks and 5 juice boxes

4 · Reviewdoes it hold up?

The bill 11600 sits between all-snacks (9600) and all-juice (14400), so a mix was right -- and it leans towards snacks, which is what 7 to 5 shows.

Another way: Starting from all juice boxes instead costs 14400 won, which is 2800 too much; dividing that by 400 gives 7 snacks directly.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Costing the all-snacks starting guess.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Finding how far the guess falls short.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the price gap as the value of one swap.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Counting the swaps and splitting the items between the two kinds.

💡Takeaway. When the number of items is fixed, trading one for the other changes the bill by the same step every time -- so the gap counts the trades.

Variant 10 hard

At the store, Wujin bought snacks and juice boxes. Each snack costs $600$ won and each juice box costs $850$ won. He bought $16$ items in all and paid $11600$ won. How many snacks and how many juice boxes did Wujin buy?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Wujin bought 16 things altogether -- snacks at 600 won each and juice boxes at 850 won each -- and spent 11600 won. We need how many of each.

Givens

A snack costs 600 won; a juice box costs 850 won.
He bought 16 items in total.
He paid 11600 won.

Unknowns

How many snacks and how many juice boxes.

Constraints

The number of items is fixed, so buying one more juice box means buying one fewer snack.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #5 Look for a Pattern#2 Make a Systematic List

Start from the simplest possible basket -- all snacks -- and swap one snack for one juice box at a time. Each swap moves the bill by the same amount, so the shortfall says how many swaps are needed.

3 · Execute5 carry out the plan

1Start with an easy guess: all 16 are snacks

#6 Guess and Check 4.NBT.B.5

If every item were a snack, the bill would be 16 lots of 600.

16 \times 600 = 9600

That guess costs 9600 won.

2See how far off the guess is

#6 Guess and Check 4.NBT.B.4

The real bill was 11600, so the guess falls short.

11600 - 9600 = 2000

2000 won unaccounted for.

3Find the steady jump per swap (the pattern)

#5 Look for a Pattern 5.OA.A.2

Swapping one snack for one juice box keeps the count at 16 but raises the bill by the price difference.

850 - 600 = 250

Every swap adds 250 won.

4Count how many swaps reach the gap

#5 Look for a Pattern 4.OA.A.3

Divide the shortfall by the size of one swap.

2000 \div 250 = 8

8 juice boxes.

5Find the snacks and confirm with a quick list

#2 Make a Systematic List 4.OA.A.3

The rest of the 16 items are snacks; check the bill.

16 - 8 = 8,\quad 8 \times 600 + 8 \times 850 = 11600

8 snacks and 8 juice boxes.

Answer: 8 snacks and 8 juice boxes

4 · Reviewdoes it hold up?

The bill 11600 sits between all-snacks (9600) and all-juice (13600), so a mix was right -- and it leans towards snacks, which is what 8 to 8 shows.

Another way: Starting from all juice boxes instead costs 13600 won, which is 2000 too much; dividing that by 250 gives 8 snacks directly.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Costing the all-snacks starting guess.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Finding how far the guess falls short.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the price gap as the value of one swap.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Counting the swaps and splitting the items between the two kinds.

💡Takeaway. When the number of items is fixed, trading one for the other changes the bill by the same step every time -- so the gap counts the trades.

Variant 11 hard

At the store, Wujin bought snacks and juice boxes. Each snack costs $1000$ won and each juice box costs $1400$ won. He bought $11$ items in all and paid $13400$ won. How many snacks and how many juice boxes did Wujin buy?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Wujin bought 11 things altogether -- snacks at 1000 won each and juice boxes at 1400 won each -- and spent 13400 won. We need how many of each.

Givens

A snack costs 1000 won; a juice box costs 1400 won.
He bought 11 items in total.
He paid 13400 won.

Unknowns

How many snacks and how many juice boxes.

Constraints

The number of items is fixed, so buying one more juice box means buying one fewer snack.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #5 Look for a Pattern#2 Make a Systematic List

Start from the simplest possible basket -- all snacks -- and swap one snack for one juice box at a time. Each swap moves the bill by the same amount, so the shortfall says how many swaps are needed.

3 · Execute5 carry out the plan

1Start with an easy guess: all 11 are snacks

#6 Guess and Check 4.NBT.B.5

If every item were a snack, the bill would be 11 lots of 1000.

11 \times 1000 = 11000

That guess costs 11000 won.

2See how far off the guess is

#6 Guess and Check 4.NBT.B.4

The real bill was 13400, so the guess falls short.

13400 - 11000 = 2400

2400 won unaccounted for.

3Find the steady jump per swap (the pattern)

#5 Look for a Pattern 5.OA.A.2

Swapping one snack for one juice box keeps the count at 11 but raises the bill by the price difference.

1400 - 1000 = 400

Every swap adds 400 won.

4Count how many swaps reach the gap

#5 Look for a Pattern 4.OA.A.3

Divide the shortfall by the size of one swap.

2400 \div 400 = 6

6 juice boxes.

5Find the snacks and confirm with a quick list

#2 Make a Systematic List 4.OA.A.3

The rest of the 11 items are snacks; check the bill.

11 - 6 = 5,\quad 5 \times 1000 + 6 \times 1400 = 13400

5 snacks and 6 juice boxes.

Answer: 5 snacks and 6 juice boxes

4 · Reviewdoes it hold up?

The bill 13400 sits between all-snacks (11000) and all-juice (15400), so a mix was right -- and it leans towards snacks, which is what 5 to 6 shows.

Another way: Starting from all juice boxes instead costs 15400 won, which is 2000 too much; dividing that by 400 gives 5 snacks directly.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Costing the all-snacks starting guess.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Finding how far the guess falls short.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the price gap as the value of one swap.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Counting the swaps and splitting the items between the two kinds.

💡Takeaway. When the number of items is fixed, trading one for the other changes the bill by the same step every time -- so the gap counts the trades.

Variant 12 hard

At the store, Wujin bought snacks and juice boxes. Each snack costs $550$ won and each juice box costs $750$ won. He bought $24$ items in all and paid $16200$ won. How many snacks and how many juice boxes did Wujin buy?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Wujin bought 24 things altogether -- snacks at 550 won each and juice boxes at 750 won each -- and spent 16200 won. We need how many of each.

Givens

A snack costs 550 won; a juice box costs 750 won.
He bought 24 items in total.
He paid 16200 won.

Unknowns

How many snacks and how many juice boxes.

Constraints

The number of items is fixed, so buying one more juice box means buying one fewer snack.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #5 Look for a Pattern#2 Make a Systematic List

Start from the simplest possible basket -- all snacks -- and swap one snack for one juice box at a time. Each swap moves the bill by the same amount, so the shortfall says how many swaps are needed.

3 · Execute5 carry out the plan

1Start with an easy guess: all 24 are snacks

#6 Guess and Check 4.NBT.B.5

If every item were a snack, the bill would be 24 lots of 550.

24 \times 550 = 13200

That guess costs 13200 won.

2See how far off the guess is

#6 Guess and Check 4.NBT.B.4

The real bill was 16200, so the guess falls short.

16200 - 13200 = 3000

3000 won unaccounted for.

3Find the steady jump per swap (the pattern)

#5 Look for a Pattern 5.OA.A.2

Swapping one snack for one juice box keeps the count at 24 but raises the bill by the price difference.

750 - 550 = 200

Every swap adds 200 won.

4Count how many swaps reach the gap

#5 Look for a Pattern 4.OA.A.3

Divide the shortfall by the size of one swap.

3000 \div 200 = 15

15 juice boxes.

5Find the snacks and confirm with a quick list

#2 Make a Systematic List 4.OA.A.3

The rest of the 24 items are snacks; check the bill.

24 - 15 = 9,\quad 9 \times 550 + 15 \times 750 = 16200

9 snacks and 15 juice boxes.

Answer: 9 snacks and 15 juice boxes

4 · Reviewdoes it hold up?

The bill 16200 sits between all-snacks (13200) and all-juice (18000), so a mix was right -- and it leans towards snacks, which is what 9 to 15 shows.

Another way: Starting from all juice boxes instead costs 18000 won, which is 1800 too much; dividing that by 200 gives 9 snacks directly.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Costing the all-snacks starting guess.
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Finding how far the guess falls short.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the price gap as the value of one swap.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Counting the swaps and splitting the items between the two kinds.

💡Takeaway. When the number of items is fixed, trading one for the other changes the bill by the same step every time -- so the gap counts the trades.

Fixed sum: one side falls, other rises · 12 practice problems

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