John has a large collection of conis with him. The coins are split between of one - dollar coins, quarters, and dimes ( one quarter is equivalent to one - fourth of a dollar and one dime is equivalent to one -tenth to one tenth of a dollar ). If the ratio of the number of one dollar coins to quarters to dimes is 3: 8: 10 and they are exactly $210, how many quarters does John have ?

Options

A. 35
B. 80
C. 105
D. 280

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Define the Variables
Let the number of one-dollar coins be \(3x\), the number of quarters be \(8x\), and the number of dimes be \(10x\), based on the given ratio of 3:8:10.

### Step 2: Calculate the Value of Each Type of Coin
- The total value of the one-dollar coins is:
\[ \text{Value from one-dollar coins} = 3x \times 1 = 3x \text{ dollars} \]
- The total value of the quarters is:
\[ \text{Value from quarters} = 8x \times \frac{1}{4} = 2x \text{ dollars} \]
- The total value of the dimes is:
\[ \text{Value from dimes} = 10x \times \frac{1}{10} = x \text{ dollars} \]

### Step 3: Set Up the Equation
Now, we can set up the equation for the total value of all the coins:
\[ 3x + 2x + x = 210 \]

### Step 4: Simplify the Equation
Combine like terms:
\[ 6x = 210 \]

### Step 5: Solve for \(x\)
To find \(x\), divide both sides by 6:
\[ x = \frac{210}{6} = 35 \]

### Step 6: Calculate the Number of Quarters
Now that we have \(x\), we can find the number of quarters:
\[ \text{Number of quarters} = 8x = 8 \times 35 = 280 \]

### Conclusion
John has **280 quarters**.

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