Problem
Count ways to make an amount with coins
Mia has 2 dimes, 2 nickels, and 10 pennies. Count how many different combinations of these coins pay exactly 20 cents.
Measurement & data
Your answer
ways
Watch out She can use at most the coins she actually has — only 10 pennies — so ways that would need 15 or 20 pennies don't count.
How to solve
Strategy Make a Systematic List — 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.
1STEP 1
Set up the goal in cents
Turn each coin into cents — dime 10, nickel 5, penny 1 — and hunt combinations totaling 20 cents within her supply.
10d + 5n + 1p = 20
Turning every coin into its cent value makes the total easy to add up and compare to 20.
2.MD.C.8Analyze The Units2STEP 2
Try 2 dimes
Two dimes already make 20 cents, so no nickels or pennies are needed — that's 1 way.
10 + 10 = 20
Two dimes hit 20 exactly, leaving nothing more to add.
2.MD.C.8Make A Systematic List3STEP 3
Try 1 dime
One dime leaves 10 cents: 2 nickels; 1 nickel + 5 pennies; or 10 pennies — all within her limits, so 3 ways.
10 + (2×5), 10 + 5 + (5×1), 10 + (10×1)
After one dime, the leftover 10 cents can be split among nickels and pennies in a few neat ways.
2.MD.C.8Make A Systematic List4STEP 4
Try 0 dimes
With no dimes, only 2 nickels + 10 pennies reach 20 cents; fewer nickels would need 15 or 20 pennies she lacks — 1 way.
(2×5) + (10×1) = 20
With only 10 pennies on hand, she must use both nickels, leaving exactly one workable combination.
2.MD.C.8Make A Systematic List5STEP 5
Count all the ways
Add the cases — 1 with 2 dimes, 3 with 1 dime, 1 with 0 dimes: 1 + 3 + 1 = 5.
1 + 3 + 1 = 5
The cases by number of dimes don't overlap, so the total is just their sum.
2.MD.C.8Make A Systematic ListThe total number of ways is the sum of the ways counted in each separate case grouped by how many dimes are used.
Why?
Sorting every payment by how many dimes it uses breaks all the ways into three groups that miss nothing and share nothing, so the group counts simply add up to the total.
Why?
A payment can use only 0, 1, or 2 dimes since Mia has just 2 dimes, and each payment has one fixed number of dimes, so every way falls into exactly one of the three groups with none left out and none counted twice.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Answer
5 ways
1 + 3 + 1 = 5
Each listed combination sums to exactly 20 cents and never exceeds her supply of 2 dimes, 2 nickels, and 10 pennies, so 5 distinct ways is consistent.
Takeaway
Organize your guesses by how many dimes you use, and the rest of the coins fall into place by twenties!
- Set up the goal in cents
- Try 2 dimes
- Try 1 dime
- Try 0 dimes
- Count all the ways
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested