Word Problems: Coins: Example 1

The numbers of each type should add to 30.

Express this algebraically, using q, d, and n.

30 = q + d + n

Each coin has a different value, a fact we use when we determine the value of 12 quarters, for example.
What is the value of 12 quarters (in cents)?

The value of 12 quarters is 12 * 25 = 300 cents.

More generally, what is the value of "q" quarters?

Since each has the value of 25 cents, we can multiply the number by 25. We don't know the number, but have labeled it as q. Consequently, q quarters has the value q * 25.

Express the value of d dimes in a similar way.

Value of d dimes = d * 10.

Express the value of n nickels.

Value of n nickels = n * 5.

We were give the number of quarters and dimes in this collection of 30 coins.
How do we find the number of nickels?

The numbers of coins have to add up to 30.
30 = q + d + n = 12 + 5 + n

Solve for n.

Subtract 17 from both sides to get
30 – 17 = 17 + n – 17, or
n = 13.

Now determine the value of the coins. How is the total value related to the values of the quarters, dimes, and nickel?

(Total value of the collection, T) =
(sum of values of the various types), or
T = q * 25 + d * 10 + n * 5

Substitute our q = 12, d = 5, and n = 13 to find the total value.

T = 12 * 25 + 5 * 10 + 13 * 5, or
T = 300 + 50 + 65 = 415 cents.

Summarize these results.

The collection of 30 coins consists of 12 quarters, 5 dimes, and 13 nickels.
The total value of the coins is $4.15.

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