patricio has a collection of dimes and quarters. overall he has 51 coins and their total value is $10.35.how many quarters and how many dimes does he have?

35 quarters and 16 dimes.

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Why?

Let d represent the number of dimes Patricio has.

So, that means 51 - d is equal to the number of quarters he has.

Now, since we know the total value of all his coins is $10.35, we can put this into an equation.

The value of all his dimes is 0.1d, and the value of all his quarters is 0.25(51 - d). It’s all equal to $10.35.
—> 0.1d + 0.25(51 - d) = 10.35

Let’s solve. First, distribute 0.25 to the two terms in the bracket.
—> 0.1d + 12.75 - 0.25d = 10.35

Now, put all terms with d on the left and other numbers on the right.
—> 0.1d - 0.25d = 10.35 - 12.75

Combine like terms.
—> -0.15d = -2.4

Multiply each side by -1 to get rid of their negative values.
—> 0.15d = 2.4

Now, divide each side by 0.15 to isolate d.
—> d = 16

So, the number of dimes Patricio has is equal to 16. And we said the number of quarters he has is equal to 51 - d, or 51 - 16. So, let’s solve for the number of quarters.
—> 51 - 16 = 35

Therefore, he has 35 quarters and 16 dimes.

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Check:

35 + 16 = 51.
The number of coins he has is correct.

35(0.25) + 16(0.1) = 10.35
8.75 + 1.6 = 10.35
10.35 = 10.35
So, the amount of each coin he has is correct as well.