Question

1. A shoe repairman is working with his assistant, who takes 5 times long to repair of shoes. Together they can fix 20 pairs of shoes in an eight-hour day. How ling does it take the repairman to fix one pair of shoes by himself?

A: 8/24 hour

B: 2/3 hour

C: 1 hour

D: 4/25 hour

2. The total cost of an item including sales tax is directly proportional to its price. If the cost of a 24$ item is 25.44$, what is the total cost of a 75$ item?

A: 795

B: 76.44

C: 79.50

D: 106

3. Divide and state the quotient in simplest form. X^2-4-45/x+6 ÷ x^2-3x-40/x+6

A:9/8

B:x^2-4x-45/x^2-3x-40

C: x-9/x-8

D: x-8/x-9

4. Simplify: 7-(7/3x)/(5/x)-(5/3x^2)

A: 35x

B: 14x/5

C: 21x^2-7x/15x-5

D: 7x/5

can someone show me step by step?

Answer 1

1. Let x = no. of shoes repaired by assistant in an 8 hour day

Then 5x = no. of shoes repaired by repairman in an 8 hour day

Together they fix 20 shoes ie ,

x + 5x = 20

6x = 20 ; x = 20/6 = 10/3

So repairman himself takes 5*10/3 = 50/3 shoes repaired in 8 hour

So to repair one pair of shoes he take 8/(50/3) = 24/50 = 12/25 hour

Which is approximately half an hour. None of the answer is correct in choices.

2. Let c = total cost of item and p = price of an item

Since c is proportional to p we can write , c = k*p ; where k is the proportionality constant

Given c = 25.44 and p = 24

Which gives k = 25.44/24 = 1.06

For p = 75 , total cost c = 1.06*75 = 79.5 ,, which is option D.

3. The question has a small mistake. There should be x in ^{​​​​​​}x^{​​​​​​ 2} - 4 -45

Giving x become( x ² - 4x - 45 ) / ( x+6) / ( x ² - 3x - 40) / (x+6)

x ² - 4x - 45 = x ² - ( 9x - 5x) - (9*5) = (x + 5) ( x - 9)

x ² - 3x - 40 = x^{​​​​​​ 2} - ( 8x - 5x) - (8*5) = (x + 5) (x - 8)

On dividing we get ( x - 9 )/( x - 8) which is option C.

4. The equation can be written as

7(1 - 1/3x) / 5/x(1 - 1/3x)

On eliminating 1 - 1/3x from numerator and denominator

We get 7/(5/x) which is 7x/5 option D is correct.