Kenny has nickels and dimes. He has $3.80 made from 44 coins. How many dimes are there?

2 Answers
Apr 21, 2018

There are #32# dimes and #12# nickels

Explanation:

We can set up a system of equations to solve this problem

Begin by assigning variables

nickels#=n#
dimes #=d#

So #n+d=44#

Nickels are worth 5 cents #=5n#
Dimes are worth 10 cents #=10d#
#$3.80 = 380 cents#

#5n + 10d = 380#

The system becomes

#n+d = 44#
#5n+10d=380#

Rearrange the first equation to isolate a variable
d = 44 -n

Now plug the first equation value into the second equation for #d#

#5n+10(44-n) = 380#

Use the distributive property
#5n+440-10n =380#

Combine like terms
#440-5n =380#

Use additive inverse to isolate the variable term
#cancel440 -5n cancel(-440) = 380-440#
#-5n = -60#

Use multiplicative inverse to isolate the variable
#cancel((-5)n)/cancel(-5) = (-60)/-5#

#n=12#

#d=44-n#
#d=44-12#
#d=32#

Apr 21, 2018

Just for fun! A different approach using ratio. Once you get used to how this works it is very fast.

Count of dimes #=32#
Count of nickels #=44-32=12#

Explanation:

10 dimes = $1
20 nickels = $1

Total coin count =44

Target value $3.80
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Using the straight line graph approach and counting only dimes. Choose dimes as they give the highest value for 44 coins.

#color(blue)("If all 44 dimes the value is "$44/10 =$4.40)larr" 44 dimes"#

#color(red)("If all 44 nickels the value is "$44/20=$2.20) larr" 0 dimes"#

The slope (gradient) for part is the same as the slope for all of it

Let the count if dimes be #d#

Then the count of nickels is #44-d #

Tony B

#($4.40-$2.20)/44=($3.80-$2.20)/d#

#d=(44(3.8-2.2))/(4.4-2.2) =32 #