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algebra word problem

Find out how much Grace earned if she earned $27 more than Robert and Alexis combined.  Alexis worked 3 hours more than Robert and 7 hours less than Grace.

Robert earned $7.50/hr, Alexis earned $6/hr,  Grace earned $9/hr

 

How much did Grace earn?

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2 Answers

Make a lot of different equations and piece together to solve for one variable at a time.

g = Grace hours         $9.00g = what Grace earns per g hours

r = Robert hours        $7.50r = what Robert earns per r hours

a = Alexis hours         $6.00a = what Alexis earns per a hours

Grace earned if she earned $27 more than Robert and Alexis combined

     because earned, we need to use the equations above that have the money involved per hour

    $9.00g = ($7.50r + $6.00a) + $27.00

Alexis worked 3 hours more than Robert

   because this is just hours, no money in equation

   a = r + 3      OR     r = a - 3

Alexis worked 7 hours less than Grace.

   a = g - 7     OR     g = a + 7

Put the equations together to help you get to only have a's in one equation, and solve for a

   $9.00g = ($7.50r + $6.00a) + $27.00

   $9.00(a + 7) = ($7.50(a - 3) + $6.00a) + $27.00

   $9.00a + $63.00 = $7.50a - $22.50 + $6.00a + $27.00

   -$4.50a = -$58.50

   a = 13

Plug a back into previous equations to solve for g and r

  g = a + 7 = (13) + 7 = 20

  r = a - 3 = (13) - 3 = 10

It asked how much Grace earned, not how many hours, so take your answer for g and plug into her hourly equation

   $9.00g

   $9.00(20)

   $180.00

 Hi,

I would approach word problems starting with the 'least' or lowest' and add on others in relation to it.

What is asked ?

Grace's earning

What is given ?

relationship between her earning and those of (Robert and Alexis)

relationship between Robert and Alexis earning

rates of earning of all 3

=> there are two things here : no of hours and rate ; earning depends upon both

Starting from lowest,

Let X be the number of hours Robert worked (his rate is known $7.50 per hour)

Robert earned $7.5X

Then Alexis worked X + 3 hours (his rate is known also $6 per hour)

Alexis earned (X+3)*6 =$(6X + 18)

Therefore Alexis and Robert together earned : 7.5X + 6X + 18 = 13.5X + 18

Now Grace earned 27$ more than both Robert and Alexis together, so add 27$ to their total

So Grace earned 13.5X + 18 + 27 = $(13.5X + 45)

If we can find out what X is, we will know Grace's earning. (the solution we need)

So number of hours Grace worked is total earnings / rate per hour = (13.5X +45) / 9

But we know that Alexis worked 7 hours less than Grace so

So number of hours worked by Alexis is {(13.5X + 45) / 9 } - 7

But Alexis also worked 3 hours more than Robert. So we can equate them

X + 3 = {(13.5X + 45) / 9} - 7

Solve the equation for X as follows :

X + 3 = (13.5X/9) + (45/9) - 7

X + 3 = (13.5X/9) + 5 - 7

X = (13.5X/9) + 5 - 7 -3 (subtracting 3 from both sides)

X = (13.5X/9) - 5

9X = (13.5X) - (5 * 9) (multiplying both sides by 9 to ease solving)

9X - 13.5X = -45  (subtracting 13.5X from both sides)

-4.5X = - 45

 X = 45/4.5  (multiplying both sides by -1)

X = 10 hours

Therefore Grace earned (remember it was $(13.5X + 45))

(13.5*10) + 45 = 135 + 45 = 180$

verify that 180$ is 27 more than Alexis and Robert earnings put together!

Hope this helps!

 

 

 

 

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