Help - Solving Employer Payroll word problem

[mcruz65]

Employer Payroll. An employer has a daily payroll of $1225 when employing some workes at $80/day and others at $85/day. When the numbers of $80 workers is increased by 50% and the number of $85 workers is decreased by 1/5, the new daily payroll is $1540. How many were originally employed at each rate?

So far this is what I have:

80x + 85y = 1225

(80x + 50%) + (85y - 1/5) = 1540

80x + 40 = 120x, 85y - .2 = 84.8y

equation 1 = 80x + 85y = 1225
equation 2 = 120x + 84.8y = 1540

Did I translated the problem correctly? I'm stuck because I don't know what method to use to solve for x or y.

 

[mmm4444bot]

Did I translated the problem correctly? Not yet.

80x + 85y = 1225 This equation makes clear the definitions of x and y, but it's good form to define your variables with actual statements. Write them down!

Let x = the original number of $80/day workers

Let y = the original number of $85/day workers

(80x + 50%) + (85y - 1/5) = 1540 Here is where you went wrong.

Hi Cruz:

You wrote: 80x + 50%

Since 80x represents money, your expression is trying to increase money by 50%. That's wrong.

It is not the money that's increasing 50%; it is the number of workers earning $80/day that's increasing by 50%.

In other words, the variable x itself is what you need to increase by 50%.

It seems, also, that you need to learn how to calculate the percentage of a number:

--------------------------------------------

x is some number of workers

We want to increase x by 50%

50% is 50/100 which is 0.5

So, 50% of x is 0.5x

Therefore, to increase x by 50%, we need to add 0.5x to x

x + 0.5x

Of course, this can be simplified.

x(1 + 0.5)

1.5x

So, at the end of the day, we see that increasing some number by 50% requires only multiplying by the factor 1.5

--------------------------------------------

80(1.5x) + 85(?????) = 1540

Next, you need to fix the similar issue with reducing the number of workers at $85/day by one-fifth.

You wrote: 85y - 1/5

Again, the expression 85y is a dollar amount.

So, by subtacting 1/5 from 85y, you're actually subtracting 1/5th of a dollar (20 cents).

That's wrong.

You need to subtract 1/5th of the workers.

In other words, rewrite your expression so that 1/5th of y is subtracted from y. That difference is the new number of workers earning $85/day.

Once you finish fixing your second equation, you can then solve the system of two equations by the substitution method.

Solve the first equation for y, and then substitute the result for y in the second equation. Doing this yields an equation that contains only x, for which you can solve.

Of course, once you know the value of x, finding the corresponding value of y is easy.

Can you continue?

Cheers ~ Mark

MY EDITS: Specified "original" in definitions for x and y

 

[mcruz65]

Ok Mark. This is what I have so far.

x = $80/day workers
y = $85/day workers
x + y = $1225

Convert 1/5 to decimal we get 0.20
So, we want to decrease y by 0.20
y - 0.20y this can be simplied to 0.80y

x + y = 1225
1.5x + 0.80y = 1540

-0.80 (x + y) = (1225) -0.80 multiply both sides of the equation by 0.80 to eliminate y
1.5x + 0.80y = 1540

-0.80x - 0.80y = -980
1.5x + 0.80y = 1540
.7x = 560
.7x/.7 = 560/.7
x = 800

Insert x value into the first equation to get y.
800 + y =1225
800 -800 + y = 1225 - 800
y = 425

Check:
Insert the x value and the y value into both equations.

x + y = 1225
800 + 425 = 1225
So we divide 800 by 80 to get the number of workers earning $80/day = 10 workers
We divide 425 by 85 to get the number of workers earning $85/day = 5 workers

1.5x + 0.80y = 1540
1.5(800) + 0.80(425) = 1540
1200 + 340 = 1540
So we divide 1200 by 80 to get the number of increased workers earning $80/day = 15 workers
We divide 340 by 85 to get the number of workers decreased that were earning $85/day = 4 workers

Mark did I get it right?

 

[mmm4444bot]

?

x = $80/day workers

y = $85/day workers

x + y = $1225 This is silly.

Hi M:

It seems like you do not yet understand the meaning of x and y.

Let's start over, by writing down correct definitions for x and y. I will be more specific, this time, by editing my previous definitions to explicitly state the adjective "original".

?

Let x = the original number of $80/day workers

Let y = the original number of $85/day workers

In other words, x and y each represent a number of workers.

So, the sum of x and y is definitely not money!

EGs:

If x = 9, then there are originally nine workers who earn $80/day.

If x = 22, then there are originally twenty-two workers who earn $80/day.

If y = 41, then there are originally fourty-one workers who earn $85/day.

If y = 3, then there are originally 3 workers who earn $85/day.

Okay, so far?

Now let's look at some expressions that contain the symbols x and y.

If we pay x workers $80 each, then the total amount of money paid to these x workers is expressed as: 80x.

If y workers each get paid $85, then the expression for the total amount of money paid to these y workers is: 85y.

x + y = 1225 ? Do you understand now why this equation is incorrect?

80x + 85y = 1225 ? Do you understand why this equation is correct?

?

1.5x + 0.80y = 1540

You made the same mistake, on your second equation.

1.5x represents some new number of workers. Specifically, it represents the original workforce x increased by 50%.

0.8y represents some new number of workers. Specifically, it represents the original workforce y reduced by 1/5th.

Again, when totaling workers, the result is not money!

In other words, you forgot to incorporate the factors of 80 and 85 in your second equation, too.

1.5x workers each earning $80/day means that 80(1.5x) dollars is paid to this group.

0.8y workers each earning $85/day means that 85(0.8y) dollars is paid to this group.

The combined daily payroll for these two groups is $1,540.

Correct your two equations, and try again.

If you're still confused about the equations, let me know.

Cheers ~ Mark

 

[mcruz65]

Hello Mark,

So the correct equations looks like this?

80x + 85y = 1225
80(1.5x) + 85(.8y) = 1540 = 120x + 68y = 1540

80x + 85y = 1225
120x + 68y = 1540

Is this correct?

 

[mmm4444bot]

Yes. That is the correct system of two equations.

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