SOLUTION: this is the question: “How can I have $23.50 in change consisting of dimes, quarters, halves, and dollars with one third as many halves as dimes, twice as many quarters as dimes

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Question 1158500: this is the question: “How can I have $23.50 in change consisting of dimes, quarters, halves, and dollars with one third as many halves as dimes, twice as many quarters as dimes and 7 more dollars than halves”
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39623) About Me  (Show Source):
You can put this solution on YOUR website!
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$23.50 in change consisting of dimes, quarters, halves, and dollars with one third as many halves as dimes, twice as many quarters as dimes and 7 more dollars than halves
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d, q, h, D
dimes, quarters, half-dollars, DOLLARS

system%2810d%2B25q%2B50h%2B100D=2350%2C3h=d%2Cq%2Fd=2%2CD-h=7%29

system%282d%2B5q%2B10h%2B20D=470%2Cd=3h%2Cq=2d%2CD=h%2B7%29

Substitute for D:
2d%2B5q%2B10h%2B20h%2B140=470
2d%2B5q%2B30h=470
2d%2B5q%2B30h=330
substitute for q:
2d%2B5%2A2d%2B30h=330
12d%2B30h=330
highlight_green%282d%2B5h=55%29

Sub for d in that two-variable equation:
6h%2B5h=55
11h=55
highlight%28h=5%29
and from this find the numbers of the other coins.

Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


No! DON'T use 4 variables and write and solve a system of 4 equations...!

Take the time to analyze the given information and try to write a single equation.

The first piece of information is "one-third as many halves as dimes". We could let our variable x be the number of dimes; but then the number of halves would be (1/3)x. Fractions always make calculations more complicated; so let's start with our variable x as the number of halves. So

let x = # of half dollars
then 3x = # of dimes ["one third as many halves as dimes"]
then 6x = # of quarters ["twice as many quarters as dimes"]
then x+7 = # of dollars ["7 more dollars than halves"]

Now we have a single equation in one variable to solve -- far less work than solving a system of 4 equations and 4 unknowns.

Writing the equation in cents (again to avoid the difficulty of working with decimals):
50%28x%29%2B10%283x%29%2B25%286x%29%2B100%28x%2B7%29+=+2350
50x%2B30x%2B150x%2B100x%2B700+=+2350
330x+=+1650
x+=+1650%2F330+=+5

ANSWER:
dollars: x+7 = 12
halves: x = 5
quarters: 6x = 30
dimes: 3x = 15

CHECK: 12%28100%29%2B5%2850%29%2B30%2825%29%2B15%2810%29+=+1200%2B250%2B750%2B150+=+2350