Coin Word Problem Solver
Solve classic coin word problems step by step: N coins totalling a given amount, twice as many quarters as dimes, three-coin mixtures with count relationships, and fewest coins to make an amount. Animates the coin stacks.
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About Coin Word Problem Solver
Coin word problems are one of the most common ways algebra textbooks teach you to translate a sentence like "I have 20 coins worth $1.40 in nickels and dimes" into a pair of equations that you can solve. They look harmless, but they're the gateway to systems of linear equations, substitution, and elimination. This solver covers every common pattern: count plus value, count ratios, three-coin mixtures with a relation, and the classic "fewest coins to make a value" puzzle. It writes the algebra step by step, so you can see exactly how the words become equations.
What you can solve here
- Count + value: "I have N coins of two denominations totaling V dollars. How many of each?"
- Value + ratio: "I have K times as many of coin A as coin B; total value is V. How many of each?"
- Three-coin mixture: "N coins of three denominations total V; the number of coin C is K times the number of coin B."
- Fewest coins: "What is the smallest number of coins that make exactly V?" — solved by dynamic programming so the answer is truly optimal, even for non-canonical denomination sets where greedy fails.
The two universal equations
Count equation
\(x + y + z + \dots = N\)
The number of each coin adds up to the total number of coins.
Value equation
\(d_1 x + d_2 y + d_3 z + \dots = V\)
Each coin's denomination times its count, summed, equals the total value (in minor units).
Relation (optional)
\(x = K y\) or \(x = y + M\)
Many problems add a third equation tying one count to another.
How to use this solver
- Pick a problem pattern that matches what your textbook (or boss, or kid's homework) is asking.
- Choose the currency and the specific coin denominations involved.
- Type the known totals — coin count, total value, and any ratio or offset between counts.
- Hit "Solve the coin problem". The result panel shows the count of each denomination, animated coin stacks, the equations you set up, and a step-by-step derivation.
Worked example — nickels and dimes
Problem: I have 20 coins totaling $1.40 in nickels and dimes. How many of each?
Setup: Let \(x\) be the number of dimes and \(y\) be the number of nickels. Each dime is worth 10¢ and each nickel is worth 5¢, so:
- \(x + y = 20\)
- \(10x + 5y = 140\) (cents)
Substitute \(y = 20 - x\) into the second equation: \(10x + 5(20 - x) = 140\) → \(5x + 100 = 140\) → \(x = 8\). So 8 dimes and 12 nickels. Verify: \(10(8) + 5(12) = 140\) ✓ and \(8 + 12 = 20\) ✓.
Why "fewest coins" can be tricky
For US, UK, and Euro coins, the greedy algorithm — pick the largest coin that fits, then repeat — always gives the optimal answer. But this is not true for every denomination set. The classic counter-example is denominations {1, 3, 4} making 6: greedy gives 3 coins (4+1+1) but the optimum is 2 coins (3+3). This solver uses dynamic programming, which is guaranteed to find the true minimum regardless of the coin set. Enable "Use custom denominations" and try 1, 3, 4 for value 6 to see the difference live.
Coin denominations by currency
| Currency | Standard coins |
|---|---|
| USD ($) | 1¢ penny, 5¢ nickel, 10¢ dime, 25¢ quarter, 50¢ half-dollar, $1 dollar |
| GBP (£) | 1p, 2p, 5p, 10p, 20p, 50p, £1, £2 |
| EUR (€) | 1¢, 2¢, 5¢, 10¢, 20¢, 50¢, €1, €2 |
FAQ
What is a coin word problem?
A coin word problem describes a collection of coins of two or more denominations using sentences about the total count of coins, the total monetary value, and ratios between coin counts. It translates into a small system of linear equations that you can solve for the count of each denomination.
How do I solve "I have 20 coins totaling $1.40 in nickels and dimes"?
Let \(x\) be the number of dimes and \(y\) be the number of nickels. Then \(x + y = 20\) and \(10x + 5y = 140\). Subtract 5 times the first equation from the second to get \(5x = 40\), so \(x = 8\) dimes and \(y = 12\) nickels.
What does "twice as many quarters as dimes" mean?
It means the count of quarters equals two times the count of dimes. If you have \(d\) dimes, you have \(2d\) quarters. Substitute into the total-value equation to get a single-variable equation in \(d\), then solve.
What is the fewest-coins problem?
Given a target value and a set of coin denominations, find the smallest number of coins that sum exactly to the target. For US coins the greedy strategy works, but for non-canonical denomination sets greedy can be wrong. The solver uses dynamic programming to find the true minimum every time.
Why does my problem have no whole-number solution?
The algebra gives a unique real solution, but coin counts must be whole numbers. If the math yields a fraction, the original puzzle is inconsistent (the totals are impossible with those coin denominations). Try adjusting the total value by a small amount or pick different coin types.
Does the solver support pounds and euros?
Yes. Pick GBP for British coins (1p through £2) or EUR for euro coins (1¢ through €2). The solver handles each currency natively, and you can also enable custom denominations for the fewest-coins scenario.
Can I use this for cash-register puzzles or change-making?
Yes — the fewest-coins scenario is exactly the change-making problem. Type the change amount as the target value and the solver finds the optimal way to give that change.
Related tools
You may also like the Age Word Problem Solver, the Mixture Problem Solver, or the Linear Equation Solver for the underlying algebra.
Reference this content, page, or tool as:
"Coin Word Problem Solver" at https://MiniWebtool.com/coin-word-problem-solver/ from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: 2026-05-11
You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.
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