Question 84801: Hello! I am reviewing my Algebra for this year and I am having difficulty solving this one problem. I am not sure if I need to use three variables or if I should do something else. Here is the problem:
Darrell has $2 less than five times as much money as Bob. Mary has as much money as Darrell and Bob together. If all three of them together have $80, how much money does Mary have?
~There is my question. It would be incredibly helpful if you helped me solve this. Thank you once again.
~ev10993@comcast.net
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! You have three unknowns: the amount of money Darrell has (call it D), the amount of money
that Bob has (call it B), and the amount of money that Mary has (call it M). Since there are
three unknowns, you can anticipate using three independent equations to solve for the unknowns.
.
Start with the fact that Darrell has $2 less than 5 times the amount of money that Bob has.
You can translate this into 5 times the amount of money that Bob has take away $2 is equal
to the amount of money that Darrell has. In equation form this can be written as:
.
D = 5*B - 2 <=== equation 1
.
Next you are told that Mary has as much money as Darrell and Bob combined. In equation
form this is:
.
M = B + D <=== equation 2
.
And finally you are told that if you add up all the money that these three have, the amount
is $80. In equation form this is:
.
M + B + D = 80 <=== equation 3
.
Now what you need to do is to examine the equations and start substituting from one to
the other. For example, from equation 1 you know what D is in terms of B. So you could
substitute the right side of equation 1 into equation 3 in place of D. You could also
substitute the right side of equation 1 for D in equation 2. That would result in equation
2 having M on the left side and only terms containing B on the right side. You could then
substitute the right side of this "new" equation 2 into the "new" equation 3 and it would
only then have only the unknown B. So you could solve for B and then work your way backward
find M and D.
.
Let's work it a little differently. Notice that equation 3 contains B + D. But from equation
2 you know that B + D is equal to M. So in equation 3 you can replace B + D with M and
equation 3 then becomes:
.
M + M = 80
.
The left side adds up to 2M so the equation is:
.
2M = 80
.
and dividing both sides by 2 results in:
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M = 40
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So now you know that Mary has $40.
.
Return to equation 2 and substitute 40 for M. When you do, the equation becomes:
.
40 = B + D
.
Solve this equation for D by subtracting B from both sides to get:
.
D = 40 - B
.
You can now substitute the right side of this for D equation 1 to get:
.
40 - B = 5B - 2
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Add B to both sides and this becomes:
.
40 = 6B - 2
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Next add 2 to both sides:
.
42 = 6B
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Finally, divide both sides by 6 to find B:
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B = 42/6 = 7
.
Now you know that Bob has $7
.
From equation 3 you know that the sum of all the money is $80, and you have found that
Mary has $40 and Bob has $7. Together Mary and Bob have $47 so Darrell must have the amount
remaining that it takes to get to $80. Therefore, Darrell must have $33.
.
There are other ways you can manipulate the three equations to come up with this answer also,
but this is one way that will work.
.
Hope this helps you. Good luck with your review.
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