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.