Question

Joey decides to empty his piggy bank and his money. His bank is with only nickels and dimes. Joey counted a total of that up to $3.15. How many nickels and how many dimes does Joey have in his pi bank?

Answer 1

Potentially there are a number of solutions but I am just going to give you 1
3 nickels + 30 dimes `=$3.15`

Se the explanation for the algebraic represenation

Explanation:

Known:

Note that `-=` means 'equivalent to'

`1" dime "-= 10" cents"`
`1" nickel "-=5" cents"`
`$1 -= 100" cents"`

Let the count of dimes be `d`
Let the count of nickels be `n`

Standardizing everything into cents we have

`5n+10d=315`

Splitting the 315 into 300+15

10 will not divide exactly into 15 but 5 will

Set `5n=15`

`=>n=15/5=3" "color(red)("so we choose to have 3 nickels")`
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Having 3 nickels leaves us 300 cents to deal with

Set `10d=300`

`=>d=300/10=30`

`color(red)("so we have 30 dimes to make up the difference")`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

There after you count nickels in pairs and make up the difference in dimes. Remember that we 'get rid' of the 15c by having `color(magenta)(3n=15)`. So the complete system is:

Using the count of `n` slightly differently coin count wise we have:

The 2n makes sure that we have an even number of nickels

`color(green)(color(magenta)(3)+2n+d="total coin count")`

Total count of nickels is 3+2n
Total count of dimes is d

Value wise we have:
`5(3+2n)+10d=315" cents"`