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1) If 15% of x is 42, what is x?

2)I need to make \$500 per week after tax in order to pay all my bills. The income tax is 20%. What is the smallest pre-tax weekly salary I can earn and still be able to pay my bills after I pay my income tax?

3)After my salary was increased by 12%, it was 62,720. What was my salary before the increase?

4)After some bad news (due to a revolutionary but ultimately doomed process for making toast in a microwave oven), stock in Seraj Food Industries fell by 40%, to \$324 a share. What was the original price of a share?

5)Three generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a 50% discount as children. The two members of the oldest generation receive a 25% discount as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs \$6.00, is paying for everyone. How many dollars must he pay?

6)The city imposed a 0.08% tax on the value of dinosaur collections. Sheila had to pay \$12 in dinosaur tax. How much is her dinosaur collection worth?

7)A city has an assessed valuation of \$8,000,000. The rate for school taxes is 60 cents per \$100 valuation. If all but 4% of the school taxes were collected, how many dollars were still owed?

8)On Tuesday, a radio store reduces all of its prices by 20% from what they were on Monday.

On Wednesday, the store again reduces all of its prices by some percentage from what they were on Tuesday. If the resulting price of a radio is half of the original Monday price, then by what percentage did the store reduce its prices from Tuesday to Wednesday?

9)Winston grows green dragon apples. This year, he shipped 80%of these apples to sell at supermarkets and kept the rest to make cider. Of all the green dragon apples he sold to the supermarkets, only 60% of them were bought. If a total of 420 green dragon apples were bought, how many green dragon apples did Winston keep to make cider?

10)At a party at the middle school over on third avenue next to the bank, 60% of the students are girls. After these students are joined by 20 more boy students, all of whom like to dance, the party is now 58% girls. How many students are now at the party?

11)Ms. Forsythe gave the same algebra test to her three classes. The first class averaged 80%, the second class averaged 85%, and the third 89%. Together, the first two classes averaged 83%, and the second and third classes together averaged 87%. What was the average for all three classes combined? Express your answer to the nearest hundredth.

Oct 14, 2018

#1
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1) If 15% of x is 42, what is x?

.15x  =  42     divide both sides by .15

x  = 42 / .15  =  280

2)I need to make \$500 per week after tax in order to pay all my bills. The income tax is 20%. What is the smallest pre-tax weekly salary I can earn and still be able to pay my bills after I pay my income tax?

Call  A the amount you need to make

So......   A  -  .20A  = 500

.80A  = 500     divide both sides by .80

A = 500 / .80   = \$625

3)After my salary was increased by 12%, it was 62,720. What was my salary before the increase?

Call S  the original salary...so..

1S + .12S  = 62720

1.12S = 62720     divide both sides by 1.12

S  = 62720 / 1.12  = \$56000

4)After some bad news (due to a revolutionary but ultimately doomed process for making toast in a microwave oven), stock in Seraj Food Industries fell by 40%, to \$324 a share. What was the original price of a share?

Call the original value of the stock, V

If the shares fell by 40%, they must be worth only 60%  of their original value

So

.60V  = 324    divide both sides by .60

324 / .60  = V  = \$540

Oct 14, 2018
edited by CPhill  Oct 15, 2018
#2
+103049
+1

5)Three generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a 50% discount as children. The two members of the oldest generation receive a 25% discount as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs \$6.00, is paying for everyone. How many dollars must he pay?

Call the price of the middle generation ticket, P

Call the price of the youngest generation ticket, .50P   [ 50% discount ]

Call the price of  the senior ticket, .75P  [ 25% discount ]

So.....we know that .75P  = 6    divide both sides by .75

P  = 6/.75  = \$8

So....the yoiungest generation pays  8 * .50   = \$4 per ticket

And the middle generation pays  \$8

So...the total amount paid   =  2 [ 4 + 8  + 6]  = 2 [ 18]   = \$36

6)The city imposed a 0.08% tax on the value of dinosaur collections. Sheila had to pay \$12 in dinosaur tax. How much is her dinosaur collection worth?

Call the value of the collection, V

And .08%  =  .0008  as a decimal

So  .08% of V  =  12

V * .0008  =  12   divide both sides  by .0008

V =  12  / .0008   = \$15000

Oct 14, 2018
#3
+103049
+1

7)A city has an assessed valuation of \$8,000,000. The rate for school taxes is 60 cents per \$100 valuation. If all but 4% of the school taxes were collected, how many dollars were still owed?

The amount that should have been collected  is

[ 8000000 / 100] * .60    = \$48000

If all but 4% have been collected, then   .04 * 48000  = \$1920  is still owed

8)On Tuesday, a radio store reduces all of its prices by 20% from what they were on Monday.

On Wednesday, the store again reduces all of its prices by some percentage from what they were on Tuesday. If the resulting price of a radio is half of the original Monday price, then by what percentage did the store reduce its prices from Tuesday to Wednesday?

Call Monday's price, P

Tuesday's price must just be .80P   [ 20% reduction]

Call Wednesday's price reduction ( 1 - R)

So...we have that

Tuesday's price  *  Wednesday's reduction  = (1/2)P

(.80)P  *( 1 - R)  = (1/2)P    divide through by  P  and note that 1/2  = .50

.80 (1 - R)  = .50        divide both sides by .80

1 - R  =  .50/.80

1 - R  = 5/8

1 - 5/8 = R

3/8   = R = .375  = 37.5%  = the reduction from Tues. to Wed.

Proof of this...let Monday's price  = 100

Tuesdays price  = .80 * 100   = 80

Wed price  = 80 (1 - R)  = 80 (1 - .375)  = 50 =  half of  Mon. price

Oct 14, 2018
edited by CPhill  Oct 15, 2018
#4
+103049
+1

9)Winston grows green dragon apples. This year, he shipped 80%of these apples to sell at supermarkets and kept the rest to make cider. Of all the green dragon apples he sold to the supermarkets, only 60% of them were bought. If a total of 420 green dragon apples were bought, how many green dragon apples did Winston keep to make cider?

Call the number he shipped to the store, N

So

.60N  = 420   divide both sides by  ,60

420 / .60  = N  = 700 were shipped to the stores

But this number was  80%  of the total he grew = T

So

.80T  = 700   divide both sides by  .80

700/.80 = T  = 875

But..he kept 20% of these =   875 * .20  = 175

Oct 14, 2018
#6
0

9)

This is what I think of this problem:

He shipped 80% of his apples to the Supermarket, but only 60% were sold!.

0.8 x 0.6 = 0.48 were sold.

0.48x =420

x =420 /0.48 =875 apples were shipped to the Supermarket. But this represents only 80% of his produce.

875 / 0.8 =1,094 apples - his total production.

1,094 - 420 =674 apples - he kept to make cider!.

Guest Oct 14, 2018
#5
+103049
+1

10)At a party at the middle school over on third avenue next to the bank, 60% of the students are girls. After these students are joined by 20 more boy students, all of whom like to dance, the party is now 58% girls. How many students are now at the party?

Call the original  number of students, S...and the number of girls  = .60S

Total students  after 20 boys arrive = S + 20

And we have that

Number of  girls / total number of students  =  58%

.60S  / [ S + 20 ]  =   .58    multiply both sides by  S +  20

.60S  = .58 [ S + 20]

.60S  = .58S + 11.6   subtract  .58S from  both sides

.02S = 11.6   divide both sides  by .02

S  = 11/6 / .02 =    580

So...the total number of students = 580 + 20  = 600

Proof :

.60 * 580 = 348  girls originally

And  .40 * 580 =  232 boys originally

After  20  boys arrive their %  =  [ 232 + 20 ] / 600  = 42%

So...the girls % after this  = 1  - 42%  =  1 - .42   =  .58   = 58%

Oct 14, 2018
edited by CPhill  Oct 15, 2018
#7
+103049
+1

11)Ms. Forsythe gave the same algebra test to her three classes. The first class averaged 80%, the second class averaged 85%, and the third 89%. Together, the first two classes averaged 83%, and the second and third classes together averaged 87%. What was the average for all three classes combined? Express your answer to the nearest hundredth.

The average  =  total points / number of students

So....total points = average*number of students

Let the total number of students in  each class = x , y  and z

The total number of points the first class amassed  was   80x

The total number of points amased by the second class was  85y

The total number of points amassed by the third class = 89z

So...for the first two classes we have

[ 80x + 85y ] / [ x + y] = 83

80x + 85y = 83x + 83y  subtract  80x, 83y from both sides

3x = 2y

x = 2/3y

And,,,for the second two classes we have

[ 85y + 89z ]  / [ y + z ]  =  87

[  85y + 89z ]  = 87 [y + z]

85y + 89z = 87y + 87z      subtract  85y, 87z from both sides

2z = 2y

y  = z

So...for the three classes.....

Total points by all three classes /  number of class members  = the average for all three classes

[ 80x + 85y  + 89z ] /  [ x + y + z ]   =

[sub for  x and z ]

[80 (2/3)y  + 85y  + 89y]  /  [ (2/3)y  + y  + y ]

y [ 80 (2/3)  + 85  + 89 ]  / [  y [( 2/3) + 1 + 1] ]       [cancel the y's ]

[ 80 (2/3) + 85 + 89]  [  8/3 ]

[ 160/3 + 85 + 89 ] /  [8/3] =

[ 160/3  + 255/3 + 267/3]  / [8/3]     multiply  top/bottom by 3

[160 + 255 + 267 ] / 8  =

85.25  = average for all three classes

Oct 14, 2018
edited by CPhill  Oct 15, 2018