Train a and b speed math help

rain A has a speed 20 miles per hour greater than that of train B. If train A travels 160 miles in the same time train B travels 120 miles, what are the speeds of the two trains?

and


Find the number you would add to both the numerator and denominator of 6/17 so the result would be 2/3.

I have a lot more problems like this, so I just need someone to explain how to do these. Thank you!

1) 160/(B+20) = 120/B
160B = 120B+2400
B = 60 mph
A = 80 mph

2) (6+x)/(17+x) = 2/3
18+3x = 34+2x
x = 16

1st problem:
Let a = A's speed. This makes b's speed = a - 20. Let t = time both travel.
at = 160
(a - 20)t = 120
solve the system of equations:
a = 160/t
(160/t - 20)t = 120
160 - 20t = 120
-20t = -40
t = 2
a = 160/2 = 80mph
B's speed = 80 - 20 = 60mph

2nd problem:
(6 + x)/(17 + x) = 2/3
3(6 + x) = 2(17 + x)
18 + 3x = 34 + 2x
x = 16

Let speed of train B = x then speed of train A is (x + 20)
time for A for 160 miles = 160/(x + 20)
time for B for 120 miles = 120/x
160/(x + 20) = 120/x
160x = 120x + 3200
40x = 3200
x = 80 mph of B
x + 20 = 120 mph of A

if x is added to both
(6 + x)/(17 + x) = 2/3
3(6 + x) = 2(17 + x)
18 + 3x = 34 + 2x
x = 16
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Lol @ u asking here but I feel bad I'll answer one
Train a 80 mph
Train b 60 mph
Cause in 2 hours train a traveling at 80 mph will reach 160 (80 x 2 = 160miles)
train b runs 20mph less, just subtract 20 to 80 and plug in the numbers.

In time the difference in their travel is 40 miles, which is two hours (40/20), so train A travels at 160/2 = 80 mph and train B at 120/2 = 60 mph

Distance = rate * time

We've got two distances here, so we'll call the variables dA (distance of train A), rA (rate of train a), tA (time of train a), dB, rB, and tB.