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How many coins of each kind are there (full question below)?

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3 Answers

n = nickels
d = dimes
q = quarters
 
1)  n + d + q = 39
 
n+d = q - 3
 
Translate to:  2)   n + d - q = -3
 
.05n + .1d + .25q = 6.7   amount of each coin times the number of coins must add to 6.7
Multiply both sides by 20...
 
3)  1n + 2d + 5q = 134
 
 
Combine 1) & 2)
n + d + q = 39
n + d  - q = -3
 
Add together
 
2n + 2d = 36
 
Divide by 2:
 
4)  n + d = 18
 
 
Combine 1) and 3)
 
n + d + q = 39         Multiply by -5 =>  -5n -5d -5q = -195
n + 2d + 5q = 134                              n + 2d + 5q = 134
 
Add together->                                 5)   -4n - 3d  = -61
 
 
Combine 4) and 5)
 
   n + d = 18   Multiply by 4 =>  4n + 4d = 72
-4n - 3d = 61                          -4n - 3d = -61
 
Add:                                               d = 11
 
11 dimes...
 
Plug that into 4)
 
n + 11 = 18
 
n = 18-11 = 7 nickels.
 
Plug that into 1.
 
n + d + q = 39
 
7 + 11 + q = 39
18 + q = 39
 
q = 39-18 = 21 quarters
 
21 quarters, 7 nickels, and 11 dimes.
 
21(.25) + 7(.05) + 11(.1) = 5.25 + .35 + 1.1 = 5.6 + 1.1 = 6.7   Check.
 
 

Comments

You are most welcome Katelyn.
Here's another way to solve.
 
n+d+q=39 and n+d=q-3
go to the first equation and replace n+d with q-3
q-3+q=39
2q-3=39
2q=42
q=21 quarters
we have $6.70 so take away 21 quarters or $5.25 to get $1.45 left in nickels and dimes
39 coins minus 21 quarters gives us 18 coins in nickels and dimes
n+d=18 and 0.05n+0.10d=$1.45
n=18-d
0.05(18-d)+0.10d=$1.45
0.90-0.05d+0.10d=$1.45
0.90+0.05d=$1.45
0.05d=$1.45-$0.90
0.05d=$0.55
5d=55
d=55/5=11 dimes and18-11=7 nickels
check:21+11+7=39
0.25(21)+0.10(11)+0.05(7)=$5.25+$1.10+$0.35=$6.70
Let x be number of nickels, y--number of dimes, and z--number of quarters. Then
 
5x+10y+25z=670
x+y+z=39
x+y+3=z
 
From the second and the third equation we obtain:
 
x+y=18
 
From the first and third we get:
 
30x+35y=595 or 6x+7y=119
 
Now we can easily determine x and y.
 
6x+6y=108
6x+7y=119
 
Thus y=119-108=11
x=18-y=7
z=x+y+3=21
 
So we have 7 nickels, 11 dimes, and 21 quarter.