Acollection of dimes and nickels is worth $4.70 there are 60 coins in all how many of each are there?

Step-by-step explanation:

If all were nickels, the value would be 60×$0.05 = $3.00. Their $4.70 value is $1.70 more than that. For each nickel replaced by a dime, the value goes up by $0.05, so there must be 1.70/0.05 = 34 replacements. That is, ...

... there are 34 dimes and 60-34 = 26 nickels.

Using an equation

In mixture problems, it often works well to let the variable represent the quantity of the largest contributor. Here, we can let "d" represent the number of dimes, the coin with the greatest value. Then 60-d will be the number of nickels, and the total value of the coins (in dollars) is ...

... (60 -d)×0.05 + d×0.10 = 4.70

... 0.05d + 0.05×60 = 4.70 . . .simplify

... 0.05d = 4.70 -3.00 . . . . . . . subtract 3.00

... d = 1.70/0.05 = 34 . . . . . . . . divide by the coefficient of d

There are 34 dimes and 26 nickels.

The sequence of math steps should look a lot like the sequence of steps described in words, above.

As a mixture problem

The average coin value is 4.70/60 = 0.07833... = 7 5/6¢. Then the proportion that are dimes is given by ...

... (7 5/6 - 5)/(10 - 5) = (2 5/6)/5 = 17/30 . . . . . 10 and 5 are the cents values of the coins in the mix

Multiplying by the 60 coins, we find (17/30)×60 = 34 are dimes. The remaining 26 are nickels.

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The formula used above is ...

... proportion of highest contributor = ...

... ((mix value) - (least contributor)) / ((greatest contributor) - (least contributor))

This is the generic solution for any mixture problem.