← Count ways to make an amount with coins · Two-Category Counts from a Total

Count ways to make an amount with coins · 12 practice problems

2.MD.C.8

Generated variants — 12

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

Variant 1 easy answer: 3 ways

Mia has the coins shown below. To buy one toy that costs $10$ ¢, find how many different ways she can pay exactly the price of the toy.

Dimes (10¢)	Nickels (5¢)	Pennies (1¢)
1	2	7

Show solution

1 · Understandwhat's really being asked

Mia has 1 dimes, 2 nickels, and 7 pennies. Count how many different combinations of these coins pay exactly 10 cents.

Givens

Available coins: 1 dimes (10 cents each), 2 nickels (5 cents each), 7 pennies (1 cent each).
The toy costs exactly 10 cents.
She must pay the exact price.

Unknowns

The number of different ways to make exactly 10 cents from her coins.

Constraints

She cannot use more dimes, nickels, or pennies than she actually has.
Two ways are different if they use different counts of any coin.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #8 Analyze the Units

This is a 'how many ways' question over a small, bounded set of coins, so I make an organized list ordered by the number of dimes, tracking the cent value of each coin to keep the total at 10.

3 · Execute4 carry out the plan

1Set up the goal in cents

#8 Analyze the Units 2.MD.C.8

Each dime is 10 cents, each nickel 5 cents, each penny 1 cent. I need combinations that total 10 cents, using at most 1 dimes, 2 nickels, and 7 pennies.

10d + 5n + 1p = 10

Turning every coin into its cent value makes the total easy to add up and compare to 10.

2Try 1 dime

#2 Make a Systematic List 2.MD.C.8

With 1 dime (10 cents), the remaining 0 cents can be made in 1 way: 1 dime.

(10) + (0) + (0)

After fixing 1 dime, the leftover splits among nickels and pennies in a few neat ways.

3Try 0 dimes

#2 Make a Systematic List 2.MD.C.8

With 0 dimes (0 cents), the remaining 10 cents can be made in 2 ways: 1 nickel + 5 pennies; 2 nickels.

(0) + (5) + (5),\ (0) + (10) + (0)

After fixing 0 dimes, the leftover splits among nickels and pennies in a few neat ways.

4Count all the ways

#2 Make a Systematic List 2.MD.C.8

Add the cases grouped by number of dimes to get 3 ways in all.

1 + 2 = 3

The cases by number of dimes don't overlap, so the total is just their sum.

Answer: 3 ways

4 · Reviewdoes it hold up?

Each listed combination sums to exactly 10 cents and never exceeds her supply of 1 dimes, 2 nickels, and 7 pennies, so 3 ways is consistent.

Another way: Build a table with columns for dimes, nickels, and pennies, fill in every row whose values total 10 cents within the limits, and count the rows.

Standardsmin grade 2

2.MD.C.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies — Counting coin combinations that make exactly 10 cents.

💡Takeaway. Organize your guesses by how many dimes you use, and the rest of the coins fall into place!

Variant 2 easy answer: 3 ways

Mia has the coins shown below. To buy one toy that costs $15$ ¢, find how many different ways she can pay exactly the price of the toy.

Dimes (10¢)	Nickels (5¢)	Pennies (1¢)
1	2	8

Show solution

1 · Understandwhat's really being asked

Mia has 1 dimes, 2 nickels, and 8 pennies. Count how many different combinations of these coins pay exactly 15 cents.

Givens

Available coins: 1 dimes (10 cents each), 2 nickels (5 cents each), 8 pennies (1 cent each).
The toy costs exactly 15 cents.
She must pay the exact price.

Unknowns

The number of different ways to make exactly 15 cents from her coins.

Constraints

She cannot use more dimes, nickels, or pennies than she actually has.
Two ways are different if they use different counts of any coin.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #8 Analyze the Units

This is a 'how many ways' question over a small, bounded set of coins, so I make an organized list ordered by the number of dimes, tracking the cent value of each coin to keep the total at 15.

3 · Execute4 carry out the plan

1Set up the goal in cents

#8 Analyze the Units 2.MD.C.8

Each dime is 10 cents, each nickel 5 cents, each penny 1 cent. I need combinations that total 15 cents, using at most 1 dimes, 2 nickels, and 8 pennies.

10d + 5n + 1p = 15

Turning every coin into its cent value makes the total easy to add up and compare to 15.

2Try 1 dime

#2 Make a Systematic List 2.MD.C.8

With 1 dime (10 cents), the remaining 5 cents can be made in 2 ways: 1 dime + 5 pennies; 1 dime + 1 nickel.

(10) + (0) + (5),\ (10) + (5) + (0)

After fixing 1 dime, the leftover splits among nickels and pennies in a few neat ways.

3Try 0 dimes

#2 Make a Systematic List 2.MD.C.8

With 0 dimes (0 cents), the remaining 15 cents can be made in 1 way: 2 nickels + 5 pennies.

(0) + (10) + (5)

After fixing 0 dimes, the leftover splits among nickels and pennies in a few neat ways.

4Count all the ways

#2 Make a Systematic List 2.MD.C.8

Add the cases grouped by number of dimes to get 3 ways in all.

2 + 1 = 3

The cases by number of dimes don't overlap, so the total is just their sum.

Answer: 3 ways

4 · Reviewdoes it hold up?

Each listed combination sums to exactly 15 cents and never exceeds her supply of 1 dimes, 2 nickels, and 8 pennies, so 3 ways is consistent.

Another way: Build a table with columns for dimes, nickels, and pennies, fill in every row whose values total 15 cents within the limits, and count the rows.

Standardsmin grade 2

2.MD.C.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies — Counting coin combinations that make exactly 15 cents.

💡Takeaway. Organize your guesses by how many dimes you use, and the rest of the coins fall into place!

Variant 3 easy answer: 3 ways

Mia has the coins shown below. To buy one toy that costs $15$ ¢, find how many different ways she can pay exactly the price of the toy.

Dimes (10¢)	Nickels (5¢)	Pennies (1¢)
1	1	10

Show solution

1 · Understandwhat's really being asked

Mia has 1 dimes, 1 nickels, and 10 pennies. Count how many different combinations of these coins pay exactly 15 cents.

Givens

Available coins: 1 dimes (10 cents each), 1 nickels (5 cents each), 10 pennies (1 cent each).
The toy costs exactly 15 cents.
She must pay the exact price.

Unknowns

The number of different ways to make exactly 15 cents from her coins.

Constraints

She cannot use more dimes, nickels, or pennies than she actually has.
Two ways are different if they use different counts of any coin.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #8 Analyze the Units

This is a 'how many ways' question over a small, bounded set of coins, so I make an organized list ordered by the number of dimes, tracking the cent value of each coin to keep the total at 15.

3 · Execute4 carry out the plan

1Set up the goal in cents

#8 Analyze the Units 2.MD.C.8

Each dime is 10 cents, each nickel 5 cents, each penny 1 cent. I need combinations that total 15 cents, using at most 1 dimes, 1 nickels, and 10 pennies.

10d + 5n + 1p = 15

Turning every coin into its cent value makes the total easy to add up and compare to 15.

2Try 1 dime

#2 Make a Systematic List 2.MD.C.8

With 1 dime (10 cents), the remaining 5 cents can be made in 2 ways: 1 dime + 5 pennies; 1 dime + 1 nickel.

(10) + (0) + (5),\ (10) + (5) + (0)

After fixing 1 dime, the leftover splits among nickels and pennies in a few neat ways.

3Try 0 dimes

#2 Make a Systematic List 2.MD.C.8

With 0 dimes (0 cents), the remaining 15 cents can be made in 1 way: 1 nickel + 10 pennies.

(0) + (5) + (10)

After fixing 0 dimes, the leftover splits among nickels and pennies in a few neat ways.

4Count all the ways

#2 Make a Systematic List 2.MD.C.8

Add the cases grouped by number of dimes to get 3 ways in all.

2 + 1 = 3

The cases by number of dimes don't overlap, so the total is just their sum.

Answer: 3 ways

4 · Reviewdoes it hold up?

Each listed combination sums to exactly 15 cents and never exceeds her supply of 1 dimes, 1 nickels, and 10 pennies, so 3 ways is consistent.

Another way: Build a table with columns for dimes, nickels, and pennies, fill in every row whose values total 15 cents within the limits, and count the rows.

Standardsmin grade 2

2.MD.C.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies — Counting coin combinations that make exactly 15 cents.

💡Takeaway. Organize your guesses by how many dimes you use, and the rest of the coins fall into place!

Variant 4 easy answer: 3 ways

Mia has the coins shown below. To buy one toy that costs $20$ ¢, find how many different ways she can pay exactly the price of the toy.

Dimes (10¢)	Nickels (5¢)	Pennies (1¢)
1	3	9

Show solution

1 · Understandwhat's really being asked

Mia has 1 dimes, 3 nickels, and 9 pennies. Count how many different combinations of these coins pay exactly 20 cents.

Givens

Available coins: 1 dimes (10 cents each), 3 nickels (5 cents each), 9 pennies (1 cent each).
The toy costs exactly 20 cents.
She must pay the exact price.

Unknowns

The number of different ways to make exactly 20 cents from her coins.

Constraints

She cannot use more dimes, nickels, or pennies than she actually has.
Two ways are different if they use different counts of any coin.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #8 Analyze the Units

This is a 'how many ways' question over a small, bounded set of coins, so I make an organized list ordered by the number of dimes, tracking the cent value of each coin to keep the total at 20.

3 · Execute4 carry out the plan

1Set up the goal in cents

#8 Analyze the Units 2.MD.C.8

Each dime is 10 cents, each nickel 5 cents, each penny 1 cent. I need combinations that total 20 cents, using at most 1 dimes, 3 nickels, and 9 pennies.

10d + 5n + 1p = 20

Turning every coin into its cent value makes the total easy to add up and compare to 20.

2Try 1 dime

#2 Make a Systematic List 2.MD.C.8

With 1 dime (10 cents), the remaining 10 cents can be made in 2 ways: 1 dime + 1 nickel + 5 pennies; 1 dime + 2 nickels.

(10) + (5) + (5),\ (10) + (10) + (0)

After fixing 1 dime, the leftover splits among nickels and pennies in a few neat ways.

3Try 0 dimes

#2 Make a Systematic List 2.MD.C.8

With 0 dimes (0 cents), the remaining 20 cents can be made in 1 way: 3 nickels + 5 pennies.

(0) + (15) + (5)

After fixing 0 dimes, the leftover splits among nickels and pennies in a few neat ways.

4Count all the ways

#2 Make a Systematic List 2.MD.C.8

Add the cases grouped by number of dimes to get 3 ways in all.

2 + 1 = 3

The cases by number of dimes don't overlap, so the total is just their sum.

Answer: 3 ways

4 · Reviewdoes it hold up?

Each listed combination sums to exactly 20 cents and never exceeds her supply of 1 dimes, 3 nickels, and 9 pennies, so 3 ways is consistent.

Another way: Build a table with columns for dimes, nickels, and pennies, fill in every row whose values total 20 cents within the limits, and count the rows.

Standardsmin grade 2

2.MD.C.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies — Counting coin combinations that make exactly 20 cents.

💡Takeaway. Organize your guesses by how many dimes you use, and the rest of the coins fall into place!

Variant 5 medium answer: 4 ways

Mia has the coins shown below. To buy one toy that costs $25$ ¢, find how many different ways she can pay exactly the price of the toy.

Dimes (10¢)	Nickels (5¢)	Pennies (1¢)
2	2	10

Show solution

1 · Understandwhat's really being asked

Mia has 2 dimes, 2 nickels, and 10 pennies. Count how many different combinations of these coins pay exactly 25 cents.

Givens

Available coins: 2 dimes (10 cents each), 2 nickels (5 cents each), 10 pennies (1 cent each).
The toy costs exactly 25 cents.
She must pay the exact price.

Unknowns

The number of different ways to make exactly 25 cents from her coins.

Constraints

She cannot use more dimes, nickels, or pennies than she actually has.
Two ways are different if they use different counts of any coin.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #8 Analyze the Units

This is a 'how many ways' question over a small, bounded set of coins, so I make an organized list ordered by the number of dimes, tracking the cent value of each coin to keep the total at 25.

3 · Execute4 carry out the plan

1Set up the goal in cents

#8 Analyze the Units 2.MD.C.8

Each dime is 10 cents, each nickel 5 cents, each penny 1 cent. I need combinations that total 25 cents, using at most 2 dimes, 2 nickels, and 10 pennies.

10d + 5n + 1p = 25

Turning every coin into its cent value makes the total easy to add up and compare to 25.

2Try 2 dimes

#2 Make a Systematic List 2.MD.C.8

With 2 dimes (20 cents), the remaining 5 cents can be made in 2 ways: 2 dimes + 5 pennies; 2 dimes + 1 nickel.

(20) + (0) + (5),\ (20) + (5) + (0)

After fixing 2 dimes, the leftover splits among nickels and pennies in a few neat ways.

3Try 1 dime

#2 Make a Systematic List 2.MD.C.8

With 1 dime (10 cents), the remaining 15 cents can be made in 2 ways: 1 dime + 1 nickel + 10 pennies; 1 dime + 2 nickels + 5 pennies.

(10) + (5) + (10),\ (10) + (10) + (5)

After fixing 1 dime, the leftover splits among nickels and pennies in a few neat ways.

4Count all the ways

#2 Make a Systematic List 2.MD.C.8

Add the cases grouped by number of dimes to get 4 ways in all.

2 + 2 = 4

The cases by number of dimes don't overlap, so the total is just their sum.

Answer: 4 ways

4 · Reviewdoes it hold up?

Each listed combination sums to exactly 25 cents and never exceeds her supply of 2 dimes, 2 nickels, and 10 pennies, so 4 ways is consistent.

Another way: Build a table with columns for dimes, nickels, and pennies, fill in every row whose values total 25 cents within the limits, and count the rows.

Standardsmin grade 2

2.MD.C.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies — Counting coin combinations that make exactly 25 cents.

💡Takeaway. Organize your guesses by how many dimes you use, and the rest of the coins fall into place!

Variant 6 medium answer: 5 ways

Mia has the coins shown below. To buy one toy that costs $20$ ¢, find how many different ways she can pay exactly the price of the toy.

Dimes (10¢)	Nickels (5¢)	Pennies (1¢)
2	2	10

Show solution

1 · Understandwhat's really being asked

Mia has 2 dimes, 2 nickels, and 10 pennies. Count how many different combinations of these coins pay exactly 20 cents.

Givens

Available coins: 2 dimes (10 cents each), 2 nickels (5 cents each), 10 pennies (1 cent each).
The toy costs exactly 20 cents.
She must pay the exact price.

Unknowns

The number of different ways to make exactly 20 cents from her coins.

Constraints

She cannot use more dimes, nickels, or pennies than she actually has.
Two ways are different if they use different counts of any coin.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #8 Analyze the Units

This is a 'how many ways' question over a small, bounded set of coins, so I make an organized list ordered by the number of dimes, tracking the cent value of each coin to keep the total at 20.

3 · Execute5 carry out the plan

1Set up the goal in cents

#8 Analyze the Units 2.MD.C.8

Each dime is 10 cents, each nickel 5 cents, each penny 1 cent. I need combinations that total 20 cents, using at most 2 dimes, 2 nickels, and 10 pennies.

10d + 5n + 1p = 20

Turning every coin into its cent value makes the total easy to add up and compare to 20.

2Try 2 dimes

#2 Make a Systematic List 2.MD.C.8

With 2 dimes (20 cents), the remaining 0 cents can be made in 1 way: 2 dimes.

(20) + (0) + (0)

After fixing 2 dimes, the leftover splits among nickels and pennies in a few neat ways.

3Try 1 dime

#2 Make a Systematic List 2.MD.C.8

With 1 dime (10 cents), the remaining 10 cents can be made in 3 ways: 1 dime + 10 pennies; 1 dime + 1 nickel + 5 pennies; 1 dime + 2 nickels.

(10) + (0) + (10),\ (10) + (5) + (5),\ (10) + (10) + (0)

After fixing 1 dime, the leftover splits among nickels and pennies in a few neat ways.

4Try 0 dimes

#2 Make a Systematic List 2.MD.C.8

With 0 dimes (0 cents), the remaining 20 cents can be made in 1 way: 2 nickels + 10 pennies.

(0) + (10) + (10)

After fixing 0 dimes, the leftover splits among nickels and pennies in a few neat ways.

5Count all the ways

#2 Make a Systematic List 2.MD.C.8

Add the cases grouped by number of dimes to get 5 ways in all.

1 + 3 + 1 = 5

The cases by number of dimes don't overlap, so the total is just their sum.

Answer: 5 ways

4 · Reviewdoes it hold up?

Each listed combination sums to exactly 20 cents and never exceeds her supply of 2 dimes, 2 nickels, and 10 pennies, so 5 ways is consistent.

Another way: Build a table with columns for dimes, nickels, and pennies, fill in every row whose values total 20 cents within the limits, and count the rows.

Standardsmin grade 2

2.MD.C.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies — Counting coin combinations that make exactly 20 cents.

💡Takeaway. Organize your guesses by how many dimes you use, and the rest of the coins fall into place!

Variant 7 medium answer: 7 ways

Mia has the coins shown below. To buy one toy that costs $20$ ¢, find how many different ways she can pay exactly the price of the toy.

Dimes (10¢)	Nickels (5¢)	Pennies (1¢)
2	4	11

Show solution

1 · Understandwhat's really being asked

Mia has 2 dimes, 4 nickels, and 11 pennies. Count how many different combinations of these coins pay exactly 20 cents.

Givens

Available coins: 2 dimes (10 cents each), 4 nickels (5 cents each), 11 pennies (1 cent each).
The toy costs exactly 20 cents.
She must pay the exact price.

Unknowns

The number of different ways to make exactly 20 cents from her coins.

Constraints

She cannot use more dimes, nickels, or pennies than she actually has.
Two ways are different if they use different counts of any coin.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #8 Analyze the Units

This is a 'how many ways' question over a small, bounded set of coins, so I make an organized list ordered by the number of dimes, tracking the cent value of each coin to keep the total at 20.

3 · Execute5 carry out the plan

1Set up the goal in cents

#8 Analyze the Units 2.MD.C.8

Each dime is 10 cents, each nickel 5 cents, each penny 1 cent. I need combinations that total 20 cents, using at most 2 dimes, 4 nickels, and 11 pennies.

10d + 5n + 1p = 20

Turning every coin into its cent value makes the total easy to add up and compare to 20.

2Try 2 dimes

#2 Make a Systematic List 2.MD.C.8

With 2 dimes (20 cents), the remaining 0 cents can be made in 1 way: 2 dimes.

(20) + (0) + (0)

After fixing 2 dimes, the leftover splits among nickels and pennies in a few neat ways.

3Try 1 dime

#2 Make a Systematic List 2.MD.C.8

With 1 dime (10 cents), the remaining 10 cents can be made in 3 ways: 1 dime + 10 pennies; 1 dime + 1 nickel + 5 pennies; 1 dime + 2 nickels.

(10) + (0) + (10),\ (10) + (5) + (5),\ (10) + (10) + (0)

After fixing 1 dime, the leftover splits among nickels and pennies in a few neat ways.

4Try 0 dimes

#2 Make a Systematic List 2.MD.C.8

With 0 dimes (0 cents), the remaining 20 cents can be made in 3 ways: 2 nickels + 10 pennies; 3 nickels + 5 pennies; 4 nickels.

(0) + (10) + (10),\ (0) + (15) + (5),\ (0) + (20) + (0)

After fixing 0 dimes, the leftover splits among nickels and pennies in a few neat ways.

5Count all the ways

#2 Make a Systematic List 2.MD.C.8

Add the cases grouped by number of dimes to get 7 ways in all.

1 + 3 + 3 = 7

The cases by number of dimes don't overlap, so the total is just their sum.

Answer: 7 ways

4 · Reviewdoes it hold up?

Each listed combination sums to exactly 20 cents and never exceeds her supply of 2 dimes, 4 nickels, and 11 pennies, so 7 ways is consistent.

Another way: Build a table with columns for dimes, nickels, and pennies, fill in every row whose values total 20 cents within the limits, and count the rows.

Standardsmin grade 2

2.MD.C.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies — Counting coin combinations that make exactly 20 cents.

💡Takeaway. Organize your guesses by how many dimes you use, and the rest of the coins fall into place!

Variant 8 medium answer: 6 ways

Mia has the coins shown below. To buy one toy that costs $25$ ¢, find how many different ways she can pay exactly the price of the toy.

Dimes (10¢)	Nickels (5¢)	Pennies (1¢)
2	3	12

Show solution

1 · Understandwhat's really being asked

Mia has 2 dimes, 3 nickels, and 12 pennies. Count how many different combinations of these coins pay exactly 25 cents.

Givens

Available coins: 2 dimes (10 cents each), 3 nickels (5 cents each), 12 pennies (1 cent each).
The toy costs exactly 25 cents.
She must pay the exact price.

Unknowns

The number of different ways to make exactly 25 cents from her coins.

Constraints

She cannot use more dimes, nickels, or pennies than she actually has.
Two ways are different if they use different counts of any coin.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #8 Analyze the Units

This is a 'how many ways' question over a small, bounded set of coins, so I make an organized list ordered by the number of dimes, tracking the cent value of each coin to keep the total at 25.

3 · Execute5 carry out the plan

1Set up the goal in cents

#8 Analyze the Units 2.MD.C.8

Each dime is 10 cents, each nickel 5 cents, each penny 1 cent. I need combinations that total 25 cents, using at most 2 dimes, 3 nickels, and 12 pennies.

10d + 5n + 1p = 25

Turning every coin into its cent value makes the total easy to add up and compare to 25.

2Try 2 dimes

#2 Make a Systematic List 2.MD.C.8

With 2 dimes (20 cents), the remaining 5 cents can be made in 2 ways: 2 dimes + 5 pennies; 2 dimes + 1 nickel.

(20) + (0) + (5),\ (20) + (5) + (0)

After fixing 2 dimes, the leftover splits among nickels and pennies in a few neat ways.

3Try 1 dime

#2 Make a Systematic List 2.MD.C.8

With 1 dime (10 cents), the remaining 15 cents can be made in 3 ways: 1 dime + 1 nickel + 10 pennies; 1 dime + 2 nickels + 5 pennies; 1 dime + 3 nickels.

(10) + (5) + (10),\ (10) + (10) + (5),\ (10) + (15) + (0)

After fixing 1 dime, the leftover splits among nickels and pennies in a few neat ways.

4Try 0 dimes

#2 Make a Systematic List 2.MD.C.8

With 0 dimes (0 cents), the remaining 25 cents can be made in 1 way: 3 nickels + 10 pennies.

(0) + (15) + (10)

After fixing 0 dimes, the leftover splits among nickels and pennies in a few neat ways.

5Count all the ways

#2 Make a Systematic List 2.MD.C.8

Add the cases grouped by number of dimes to get 6 ways in all.

2 + 3 + 1 = 6

The cases by number of dimes don't overlap, so the total is just their sum.

Answer: 6 ways

4 · Reviewdoes it hold up?

Each listed combination sums to exactly 25 cents and never exceeds her supply of 2 dimes, 3 nickels, and 12 pennies, so 6 ways is consistent.

Another way: Build a table with columns for dimes, nickels, and pennies, fill in every row whose values total 25 cents within the limits, and count the rows.

Standardsmin grade 2

2.MD.C.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies — Counting coin combinations that make exactly 25 cents.

💡Takeaway. Organize your guesses by how many dimes you use, and the rest of the coins fall into place!

Variant 9 hard answer: 7 ways

Mia has the coins shown below. To buy one toy that costs $30$ ¢, find how many different ways she can pay exactly the price of the toy.

Dimes (10¢)	Nickels (5¢)	Pennies (1¢)
2	4	14

Show solution

1 · Understandwhat's really being asked

Mia has 2 dimes, 4 nickels, and 14 pennies. Count how many different combinations of these coins pay exactly 30 cents.

Givens

Available coins: 2 dimes (10 cents each), 4 nickels (5 cents each), 14 pennies (1 cent each).
The toy costs exactly 30 cents.
She must pay the exact price.

Unknowns

The number of different ways to make exactly 30 cents from her coins.

Constraints

She cannot use more dimes, nickels, or pennies than she actually has.
Two ways are different if they use different counts of any coin.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #8 Analyze the Units

This is a 'how many ways' question over a small, bounded set of coins, so I make an organized list ordered by the number of dimes, tracking the cent value of each coin to keep the total at 30.

3 · Execute5 carry out the plan

1Set up the goal in cents

#8 Analyze the Units 2.MD.C.8

Each dime is 10 cents, each nickel 5 cents, each penny 1 cent. I need combinations that total 30 cents, using at most 2 dimes, 4 nickels, and 14 pennies.

10d + 5n + 1p = 30

Turning every coin into its cent value makes the total easy to add up and compare to 30.

2Try 2 dimes

#2 Make a Systematic List 2.MD.C.8

With 2 dimes (20 cents), the remaining 10 cents can be made in 3 ways: 2 dimes + 10 pennies; 2 dimes + 1 nickel + 5 pennies; 2 dimes + 2 nickels.

(20) + (0) + (10),\ (20) + (5) + (5),\ (20) + (10) + (0)

After fixing 2 dimes, the leftover splits among nickels and pennies in a few neat ways.

3Try 1 dime

#2 Make a Systematic List 2.MD.C.8

With 1 dime (10 cents), the remaining 20 cents can be made in 3 ways: 1 dime + 2 nickels + 10 pennies; 1 dime + 3 nickels + 5 pennies; 1 dime + 4 nickels.

(10) + (10) + (10),\ (10) + (15) + (5),\ (10) + (20) + (0)

After fixing 1 dime, the leftover splits among nickels and pennies in a few neat ways.

4Try 0 dimes

#2 Make a Systematic List 2.MD.C.8

With 0 dimes (0 cents), the remaining 30 cents can be made in 1 way: 4 nickels + 10 pennies.

(0) + (20) + (10)

After fixing 0 dimes, the leftover splits among nickels and pennies in a few neat ways.

5Count all the ways

#2 Make a Systematic List 2.MD.C.8

Add the cases grouped by number of dimes to get 7 ways in all.

3 + 3 + 1 = 7

The cases by number of dimes don't overlap, so the total is just their sum.

Answer: 7 ways

4 · Reviewdoes it hold up?

Each listed combination sums to exactly 30 cents and never exceeds her supply of 2 dimes, 4 nickels, and 14 pennies, so 7 ways is consistent.

Another way: Build a table with columns for dimes, nickels, and pennies, fill in every row whose values total 30 cents within the limits, and count the rows.

Standardsmin grade 2

2.MD.C.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies — Counting coin combinations that make exactly 30 cents.

💡Takeaway. Organize your guesses by how many dimes you use, and the rest of the coins fall into place!

Variant 10 hard answer: 5 ways

Mia has the coins shown below. To buy one toy that costs $30$ ¢, find how many different ways she can pay exactly the price of the toy.

Dimes (10¢)	Nickels (5¢)	Pennies (1¢)
3	2	10

Show solution

1 · Understandwhat's really being asked

Mia has 3 dimes, 2 nickels, and 10 pennies. Count how many different combinations of these coins pay exactly 30 cents.

Givens

Available coins: 3 dimes (10 cents each), 2 nickels (5 cents each), 10 pennies (1 cent each).
The toy costs exactly 30 cents.
She must pay the exact price.

Unknowns

The number of different ways to make exactly 30 cents from her coins.

Constraints

She cannot use more dimes, nickels, or pennies than she actually has.
Two ways are different if they use different counts of any coin.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #8 Analyze the Units

This is a 'how many ways' question over a small, bounded set of coins, so I make an organized list ordered by the number of dimes, tracking the cent value of each coin to keep the total at 30.

3 · Execute5 carry out the plan

1Set up the goal in cents

#8 Analyze the Units 2.MD.C.8

Each dime is 10 cents, each nickel 5 cents, each penny 1 cent. I need combinations that total 30 cents, using at most 3 dimes, 2 nickels, and 10 pennies.

10d + 5n + 1p = 30

Turning every coin into its cent value makes the total easy to add up and compare to 30.

2Try 3 dimes

#2 Make a Systematic List 2.MD.C.8

With 3 dimes (30 cents), the remaining 0 cents can be made in 1 way: 3 dimes.

(30) + (0) + (0)

After fixing 3 dimes, the leftover splits among nickels and pennies in a few neat ways.

3Try 2 dimes

#2 Make a Systematic List 2.MD.C.8

With 2 dimes (20 cents), the remaining 10 cents can be made in 3 ways: 2 dimes + 10 pennies; 2 dimes + 1 nickel + 5 pennies; 2 dimes + 2 nickels.

(20) + (0) + (10),\ (20) + (5) + (5),\ (20) + (10) + (0)

After fixing 2 dimes, the leftover splits among nickels and pennies in a few neat ways.

4Try 1 dime

#2 Make a Systematic List 2.MD.C.8

With 1 dime (10 cents), the remaining 20 cents can be made in 1 way: 1 dime + 2 nickels + 10 pennies.

(10) + (10) + (10)

After fixing 1 dime, the leftover splits among nickels and pennies in a few neat ways.

5Count all the ways

#2 Make a Systematic List 2.MD.C.8

Add the cases grouped by number of dimes to get 5 ways in all.

1 + 3 + 1 = 5

The cases by number of dimes don't overlap, so the total is just their sum.

Answer: 5 ways

4 · Reviewdoes it hold up?

Each listed combination sums to exactly 30 cents and never exceeds her supply of 3 dimes, 2 nickels, and 10 pennies, so 5 ways is consistent.

Another way: Build a table with columns for dimes, nickels, and pennies, fill in every row whose values total 30 cents within the limits, and count the rows.

Standardsmin grade 2

2.MD.C.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies — Counting coin combinations that make exactly 30 cents.

💡Takeaway. Organize your guesses by how many dimes you use, and the rest of the coins fall into place!

Variant 11 hard answer: 6 ways

Mia has the coins shown below. To buy one toy that costs $40$ ¢, find how many different ways she can pay exactly the price of the toy.

Dimes (10¢)	Nickels (5¢)	Pennies (1¢)
4	3	12

Show solution

1 · Understandwhat's really being asked

Mia has 4 dimes, 3 nickels, and 12 pennies. Count how many different combinations of these coins pay exactly 40 cents.

Givens

Available coins: 4 dimes (10 cents each), 3 nickels (5 cents each), 12 pennies (1 cent each).
The toy costs exactly 40 cents.
She must pay the exact price.

Unknowns

The number of different ways to make exactly 40 cents from her coins.

Constraints

She cannot use more dimes, nickels, or pennies than she actually has.
Two ways are different if they use different counts of any coin.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #8 Analyze the Units

This is a 'how many ways' question over a small, bounded set of coins, so I make an organized list ordered by the number of dimes, tracking the cent value of each coin to keep the total at 40.

3 · Execute5 carry out the plan

1Set up the goal in cents

#8 Analyze the Units 2.MD.C.8

Each dime is 10 cents, each nickel 5 cents, each penny 1 cent. I need combinations that total 40 cents, using at most 4 dimes, 3 nickels, and 12 pennies.

10d + 5n + 1p = 40

Turning every coin into its cent value makes the total easy to add up and compare to 40.

2Try 4 dimes

#2 Make a Systematic List 2.MD.C.8

With 4 dimes (40 cents), the remaining 0 cents can be made in 1 way: 4 dimes.

(40) + (0) + (0)

After fixing 4 dimes, the leftover splits among nickels and pennies in a few neat ways.

3Try 3 dimes

#2 Make a Systematic List 2.MD.C.8

With 3 dimes (30 cents), the remaining 10 cents can be made in 3 ways: 3 dimes + 10 pennies; 3 dimes + 1 nickel + 5 pennies; 3 dimes + 2 nickels.

(30) + (0) + (10),\ (30) + (5) + (5),\ (30) + (10) + (0)

After fixing 3 dimes, the leftover splits among nickels and pennies in a few neat ways.

4Try 2 dimes

#2 Make a Systematic List 2.MD.C.8

With 2 dimes (20 cents), the remaining 20 cents can be made in 2 ways: 2 dimes + 2 nickels + 10 pennies; 2 dimes + 3 nickels + 5 pennies.

(20) + (10) + (10),\ (20) + (15) + (5)

After fixing 2 dimes, the leftover splits among nickels and pennies in a few neat ways.

5Count all the ways

#2 Make a Systematic List 2.MD.C.8

Add the cases grouped by number of dimes to get 6 ways in all.

1 + 3 + 2 = 6

The cases by number of dimes don't overlap, so the total is just their sum.

Answer: 6 ways

4 · Reviewdoes it hold up?

Each listed combination sums to exactly 40 cents and never exceeds her supply of 4 dimes, 3 nickels, and 12 pennies, so 6 ways is consistent.

Another way: Build a table with columns for dimes, nickels, and pennies, fill in every row whose values total 40 cents within the limits, and count the rows.

Standardsmin grade 2

2.MD.C.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies — Counting coin combinations that make exactly 40 cents.

💡Takeaway. Organize your guesses by how many dimes you use, and the rest of the coins fall into place!

Variant 12 hard answer: 10 ways

Mia has the coins shown below. To buy one toy that costs $35$ ¢, find how many different ways she can pay exactly the price of the toy.

Dimes (10¢)	Nickels (5¢)	Pennies (1¢)
3	4	15

Show solution

1 · Understandwhat's really being asked

Mia has 3 dimes, 4 nickels, and 15 pennies. Count how many different combinations of these coins pay exactly 35 cents.

Givens

Available coins: 3 dimes (10 cents each), 4 nickels (5 cents each), 15 pennies (1 cent each).
The toy costs exactly 35 cents.
She must pay the exact price.

Unknowns

The number of different ways to make exactly 35 cents from her coins.

Constraints

She cannot use more dimes, nickels, or pennies than she actually has.
Two ways are different if they use different counts of any coin.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #8 Analyze the Units

This is a 'how many ways' question over a small, bounded set of coins, so I make an organized list ordered by the number of dimes, tracking the cent value of each coin to keep the total at 35.

3 · Execute6 carry out the plan

1Set up the goal in cents

#8 Analyze the Units 2.MD.C.8

Each dime is 10 cents, each nickel 5 cents, each penny 1 cent. I need combinations that total 35 cents, using at most 3 dimes, 4 nickels, and 15 pennies.

10d + 5n + 1p = 35

Turning every coin into its cent value makes the total easy to add up and compare to 35.

2Try 3 dimes

#2 Make a Systematic List 2.MD.C.8

With 3 dimes (30 cents), the remaining 5 cents can be made in 2 ways: 3 dimes + 5 pennies; 3 dimes + 1 nickel.

(30) + (0) + (5),\ (30) + (5) + (0)

After fixing 3 dimes, the leftover splits among nickels and pennies in a few neat ways.

3Try 2 dimes

#2 Make a Systematic List 2.MD.C.8

With 2 dimes (20 cents), the remaining 15 cents can be made in 4 ways: 2 dimes + 15 pennies; 2 dimes + 1 nickel + 10 pennies; 2 dimes + 2 nickels + 5 pennies; 2 dimes + 3 nickels.

(20) + (0) + (15),\ (20) + (5) + (10),\ (20) + (10) + (5),\ (20) + (15) + (0)

After fixing 2 dimes, the leftover splits among nickels and pennies in a few neat ways.

4Try 1 dime

#2 Make a Systematic List 2.MD.C.8

With 1 dime (10 cents), the remaining 25 cents can be made in 3 ways: 1 dime + 2 nickels + 15 pennies; 1 dime + 3 nickels + 10 pennies; 1 dime + 4 nickels + 5 pennies.

(10) + (10) + (15),\ (10) + (15) + (10),\ (10) + (20) + (5)

After fixing 1 dime, the leftover splits among nickels and pennies in a few neat ways.

5Try 0 dimes

#2 Make a Systematic List 2.MD.C.8

With 0 dimes (0 cents), the remaining 35 cents can be made in 1 way: 4 nickels + 15 pennies.

(0) + (20) + (15)

After fixing 0 dimes, the leftover splits among nickels and pennies in a few neat ways.

6Count all the ways

#2 Make a Systematic List 2.MD.C.8

Add the cases grouped by number of dimes to get 10 ways in all.

2 + 4 + 3 + 1 = 10

The cases by number of dimes don't overlap, so the total is just their sum.

Answer: 10 ways

4 · Reviewdoes it hold up?

Each listed combination sums to exactly 35 cents and never exceeds her supply of 3 dimes, 4 nickels, and 15 pennies, so 10 ways is consistent.

Another way: Build a table with columns for dimes, nickels, and pennies, fill in every row whose values total 35 cents within the limits, and count the rows.

Standardsmin grade 2

2.MD.C.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies — Counting coin combinations that make exactly 35 cents.

💡Takeaway. Organize your guesses by how many dimes you use, and the rest of the coins fall into place!