See if you know the answer to this brain teaser!?

Question

Anna visited her aunt in a town which is 235 miles from where Anna lives. On her way to her aunt's town she had to slow down for 70 miles where one of the lanes was closed due to a traffic accident. If she had to slow down by 20 mph, and the trip took her 5 hours, find the speed at which she drove on the one lane portion. (Hint: after establishing your equation as a rational equation, convert it into a quadratic equation).

Answer 1

She went 35 mph on the one-lane portion of road. For 70 miles, that means she spent 2 hours at that speed.

She went three hours (5-2=3) at 55 mph for a toal of 165 miles.
165 miles + 70 miles = 235 miles.

Answer 2

Let r1 be the speed when slowed down, and r2 be the speed the rest of the time.

Let t1 be the time taken to travel the 70-mile one lane portion, and t2 be the time taken on the rest of the trip.

t1,t2 are in hours, and r1,r2 are in mph.

Then r1*t1 + r2*t2 = 235, but r1*t1=70, so r2*t2 = 235 - 70 = 165.

We know that r1=r2-20, and t1+t2=5, so now we have

(r2-20)*(5-t2) = 70, r2*t2 = 165.

Expand the left side of the first equation:

5*r2 - 100 - r2*t2 + 20*t2 = 70

but r2*t2 = 165, so now

5*r2 - 100 - 165 + 20*t2 = 70

or

5*r2 + 20*t2 = 335.

Divide by 5:

r2 + 4*t2 = 67

Use r2*t2 = 165 to get the rational equation mentioned at the end of the problem. Multiply by whatever is needed to get rid of any fractions, and you will have the quadratic equation.

Answer 3

Speed = Distance / time.

S1 = speed 1.
S2 = speed 2.
D1 = distance 1.
D2 = distance 2.
t1 = time 1.
t2 = time 2.

Let distance one be the distance traveled at the faster speed and distance two be the distance traveled at the slower speed.

D1 = 235 - 70 = 165 miles.
D2 = 70 miles.

D1 / S1 = t1 and D2 / S2 = t2

D1/S1 + D2/S2 = 5 hr.
But S2 = S1 - 20
D1/S1 + D2/(S1 - 20) = 5 hr.
D1(S1 - 20) + D2(S1) / S1(S1 - 20) = 5 hr.
But D1 = 165 miles and D2 = 70 miles.
165(S1 - 20) + 70(S1) / S1(S1 - 20) = 5 hr.
165S1 - 3300 + 70S1 / S1(S1 - 20) = 5 hr.
235S1 - 3300 / (S1² - 20S1) = 5 hr.
235S1 - 3300 = 5S1² - 100S1 (hr.)
0 = 5S1² - 100S1 - 235S1 + 3300 (hr.)
0 = 5S1² - 335S1 + 3300 (hr.)
0 = S1² - 67S1 + 660 (hr.)

Now use quadratic equation: (-b±√[b² - 4.a.c])/ 2a
S1 = 55 or 12 mph.
S2 = S1 - 20
→ S2 = 35 or -8 mph.

Checking values it will be seen that S1 is 55mph and S2 is 35mph.

Check answer:
t1 = D2/S2 = 70/35 = 2 hr.
t2 = D2/S2 = 165/55 = 3 hr.
t1 + t2 = 5 hr.

Speed in slow lane is 35 mph.

Answer 4

70 miles with 20 mph = 3,5 hours
the remaining 165 miles hence in 1,5 hours, hence on average 165/1,5 = 110 mph

Answer 5

35 mph

Answer 6

(235-70)/v = t1

70/(v-20) = t2

t1 + t2 = 5
165/v + 70/(v-20) = 5
(165(v-20) + 70v)/(v*(v-20)) = 5
(305v - 3300)/(v^2 - 20v) = 5

(v^2 - 20v) = (305v - 3300)/5
v^2 - 20v - 61v + 660 = 0
v^2 - 81v + 940 = 0

v = 71.81
v = 9.19

Speed at which she drove the one lane portion = v-20

Because it must be positive, the speed she drove the one lane portion was 51.81mph

Answer 7

25/35 mph

Answer 8

(235-70)/x+70/(x-20)=5

Answer 9

~55.717mph