Marcy has total of 100 dimes and quarters. If the total value of the coins is $14.05, how many quarters does she have?

I think I have American currency correct!

100 dimes = 1000 cents = $10
100 quarters = 100 x 25 = 2500 cents = $25

#color(red)("If these numbers are wrong just substitute and recalculate")#

This problem type looks a little confusing until you realise that you are blending 2 materials to obtain the target property. In this case the materials are coin types and the property is value

Based on this being correct for 100 coins we have:

The horizontal axis is the count of quarters. The extreme left is where there are no quarters. Consequently we have 100 dimes.

The extreme right is where all the coins are quarters so there are no dimes.
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#color(brown)("Think of this as a straight line graph.")#

The slope of part of it is the same as the slope of all of it so we can #ul("use ratio")#

#color(white)(.)#

Let the ratio be#=("100 quarters")/($25-$10) -> 100/15 -=("x quarters")/($14.05-$10.00)#

Write as #" "x/4.05=100/15#

#x=(100xx4.05)/15 =27#

The above is upside down for slope (gradient) but this does not matter for this calculation.
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So you have 27 quarters