Algebra II application problems

[jonboy]

Hi everyone I need help with these problem. I'm pretty lost but I'll show you my effort though it probably won't help.

2) Before the final exam, a student has test scores 72, 80, 65, 78 and 60. If the final exam counts as one-third of the final grade, what score must the student receive in order to have a final average of 76?

I know that: \(\displaystyle final\,average\,=\,\frac{cumulative\,scores}{number\,of\,tests}\)

I don't see how to incorporate the final exam as one-third of the grade though.

4) A couple does not wish to spend more than $70 for dinner at a restaurant. If a sales tax of 6% is added to the bill and they plan to tip 15% after the tax has been added, what is the most they can spend for the meal ?

Let \(\displaystyle x\) be the price of the meal.

So I know that: \(\displaystyle \L \;x\,+\,.6x\,+\,.15(x\,+\,.6x)\,=\,70\)

Add like terms: \(\displaystyle \;1.6x\,+\,.15(x\,+\,.6x)\,=\,70\)

Distribute: \(\displaystyle \L \;1.6x\,+\,.15x\,+\,.09x\,=\,70\)

Multiply by 100: \(\displaystyle \;160x\,+\,15x\,+\,9x\,=\,7000\)

...\(\displaystyle \L \;184x\,=\,7000\;\Rightarrow\;x\,\approx\;38.04\)

So the cost of the dinner should be about $38.04 right?

5) The cost of installing insulation in a particular two-bedroom home is $1080. Present monthly heating costs average $60, but the insulation is expected to reduce heating cost by 10%. How many months will it take to recover the cost of insulation?

Let \(\displaystyle x\) be the amount of months.

So the amount of insulation reduction needs to be equal to 1080.

The amount of insulation reduction is \(\displaystyle \;.10(60x)\)

Now I have: \(\displaystyle .10(60x)\,=\,1080\;\Rightarrow\;x\,=\,180\)

My book differs; gets an answer of 26/7. What have I done wrong?

6) A workman's basic hourly wage is $10, but he receives one and a half times his hourly rate for any hours worked in excess of 40 per week. If his paycheck for the week is $595, how many hours of overtime did he work?

Since he made 595, I know he started out working 40 hrs making 400.

Now he make one and half his hourly pay, which is 15 per hour.

With the remaining $195, he worked: \(\displaystyle \;\frac{195}{15}\,=\,12.8\,\) hours.

That's our overtime hours. Plugging everything back in works. Is this setup correct? Thanks you everyone for any help!

 

[skeeter]

#2 ... let x = final exam grade

\(\displaystyle \L \frac{2}{3}\left(\frac{72+80+65+78+60}{5}\right) + \frac{1}{3}x = 76\)

solve for x.

#4 ... let x = price of the meal

(1.15)(1.06)x < 70

solve the inequality.

#5 ... present cost for heat is $60. A 10% reduction is a savings of $6 per month.

number of months = 1080/6

#6 ... uhh, 195/15 = 13.

 

[jonboy]

Thank you skeeter! You're amazing. I understand most of it; I don't understand how you got the (1.15)(1.06)x on 4 but I'll look at it again later.

 

[Mrspi]

4) A couple does not wish to spend more than $70 for dinner at a restaurant. If a sales tax of 6% is added to the bill and they plan to tip 15% after the tax has been added, what is the most they can spend for the meal ?

Hey Jonboy....if the tax is 6%, then the tax on a meal costing x dollars is 0.06x.....NOT .6x.....

Try that and see if you get a better result.

 

[Denis]

I don't understand how you got the (1.15)(1.06)x on 4 but I'll look at it again later.

x + .06x + .15(x + .06x) = 70
1.06x + .15(1.06x) = 70
1.06x(1 + .15) = 70
1.06x(1.15) = 70
1.06x = 70 / 1.15
x = 70 / [1.15(1.06)] ... Yokay?

 

[jonboy]

Thanks Denis for that detailed solution (hope I didn't make you type too much ) and Mrspi. I wasn't thinking cleverly.