Math Problem Statement

Paula’s purse contains twice as many dimes as quarters and three fewer nickels than dimes. The total of these coins is $3.15. How many dimes are there?

Solution

Let's denote the number of quarters as qq, the number of dimes as dd, and the number of nickels as nn.

According to the problem, we have the following relationships:

  1. Twice as many dimes as quarters:

    d=2qd = 2q
  2. Three fewer nickels than dimes:

    n=d3n = d - 3
  3. Total value of the coins is $3.15:
    The value equation can be expressed in cents as:

    25q+10d+5n=31525q + 10d + 5n = 315

Now, let's substitute the expressions for dd and nn in terms of qq into the value equation.

First, substitute d=2qd = 2q into n=d3n = d - 3:

n=2q3n = 2q - 3

Now substitute d=2qd = 2q and n=2q3n = 2q - 3 into the value equation:

25q+10(2q)+5(2q3)=31525q + 10(2q) + 5(2q - 3) = 315

Simplify and solve for qq:

25q+20q+10q15=31525q + 20q + 10q - 15 = 315 55q15=31555q - 15 = 315 55q=33055q = 330 q=6q = 6

Now that we know q=6q = 6, we can find dd and nn:

d=2q=2(6)=12d = 2q = 2(6) = 12 n=2q3=2(6)3=123=9n = 2q - 3 = 2(6) - 3 = 12 - 3 = 9

Thus, the number of dimes is 12.

Would you like more details on this solution or have any other questions?


Here are 5 related questions for further practice:

  1. If Paula had 4 more dimes, what would be the new total value of the coins?
  2. What is the total number of coins Paula has in her purse?
  3. If Paula added 2 quarters, how much would the total value of her coins increase?
  4. How many more dimes than nickels does Paula have?
  5. If Paula had an equal number of nickels and quarters, what would be the new total value?

Tip: Always convert the total value into cents when working with coin problems to avoid confusion with decimals.

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Math Problem Analysis

Mathematical Concepts

Equations
System of Equations
Word Problems

Formulas

Linear equations
Value equations in cents

Theorems

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Suitable Grade Level

Grades 7-9