Math Q pls help(rational expressions and functions)

1.
Let x miles be the driving distance between the two cities.

Driving time when averaged 51 mph = (x/51) h
Driving time when averaged 60 mph = (x/60) h

(x/51) = (x/60) + 3

Multiply all terms on the both sides by 3060:
60x = 51x + 9180
9x = 9180
x = 1020

The driving distance between the two cities
= 1020 miles.

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2.
Let y miles be the distance of the first part of the trip.
Then, the distance of the second part of the trip = (y + 10) miles

Time taken for the first part of the trip = (y/60) h
Time taken for the second part of the trip = [(y + 10)/50] h
30 min = (30/60) h

[(y + 10)/50] = (y/60) + (30/60)

Multiply all terms on the both sides by 300:
6(y + 10) = 5y + 150
6y + 60 = 5y + 150
y = 90

The total distance
= [y + (y + 10)] miles
= [90 + (90 + 10)] miles
= 190 miles

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3.
Let t hours be the time taken after Dixie starts to paint the room.

Then, working time of Dixie = t hours
and working time of Trixie = (t – 1) hours

Fraction of the job finished when Dixie paints for 1 h = 1/3
Fraction of the job finished when Trixie paints for 1 h = 1/6

(1/3)t + (1/6)(t – 1) = 1

Multiply all terms on the both sides by 6:
2t + (t – 1) = 6
2t + t – 1 = 6
3t = 7
t = 2and(1/3)

Time taken to finish painting the room after Dixie starts to paint
= 2and(1/3) hours

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4.
Let t hours be the time taken to fill the vat if both pipes are open.

Fraction of the vat to be filled when only the inlet pipe is open = 1/10
Fraction of the vat to be emptied when only the outlet pipe is open = 1/20

(1/10)t – (1/20)t = 1

Multiply all terms on the both sides by 20:
2t – t = 20
t = 20

Time taken to fill the vat if both pipes are open
= 20 hours
=

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