Calculating the Value of Coin Collections Using Ratios

If Jeff has a collection of dimes and quarters, and the ratio of dimes to quarters is 4:5, the task is to find the total value of his coins when he has 20 dimes. Here’s how we can approach this problem step-by-step:

Understanding the Ratio and Given Information

The ratio of dimes to quarters is given as 4:5. This implies that for every 4 dimes, there are 5 quarters. We are also given that the number of dimes Jeff has is 20.

Determining the Number of Quarters

Let’s denote the number of dimes as ( d ) and the number of quarters as ( q ). According to the ratio, we have:

[frac{4}{5} frac{20}{q}]

By solving this proportion, we can find the value of ( q ):

[4q 100 implies q frac{100}{4} 25]

Thus, the number of quarters Jeff has is 25.

Calculating the Value of Dimes and Quarters

The value of one dime is 10 cents and one quarter is 25 cents. Therefore, the total value of the dimes Jeff has is:

[20 times 0.10 2.00 , text{dollars}]

The total value of the quarters Jeff has is:

[25 times 0.25 6.25 , text{dollars}]

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Total Value of All Coins Jeff Has

Adding the value of dimes and quarters, we get the total value of all the coins Jeff has:

[2.00 6.25 8.25 , text{dollars}]

This is the final answer. Let’s explore this problem through another approach to ensure our solution is consistent:

Alternative Approach: Ratio as a Factor

We can use the fact that the ratio 4:5 means that for every 4 dimes, there are 5 quarters. Since Jeff has 20 dimes, we can determine that:

[frac{20}{4} 5]

The factor 5 indicates that for every group of 4 dimes, there are 5 quarters. Therefore, for 20 dimes (which is 5 sets of 4 dimes), there would be 5 sets of 5 quarters, totaling 25 quarters.

The value calculation remains the same:

[20 times 0.10 2.00 , text{dollars}] [25 times 0.25 6.25 , text{dollars}] [2.00 6.25 8.25 , text{dollars}]

This confirms our initial solution.