Word Problems?

Question

Brenda and her husband Randy bicycled cross-country together. One Morning Brenda rode 30 miles. By traveling only 5 miles per hour faster and putting in one more hour, Randy rode 60 miles. What are the speed of each cyclist?

Answer 1

Say Brenda's speed was u miles/hour.

Then Randy Felix's speed was u+5

If Brenda travelled for t hours, Randy travelled for t+1 hours.

speed = distance / time

For Brenda

u = 30/t ====> 30 = ut

For Randy

u+5 = 60 / (t+1) ===> (u+5)(t+1) = 60

In this substitute for t (=u/30) to get u..... ie
(u+5)(30+u) = 60u
u^2 - 25u + 150 = 0
(u-15)(u-10) = 0

This is a quadratic equation with positive root u = 10 mph, u = 15mph

Therefore Breda's speed was 10 mph
Randy's speed was 10 + 5 = 15 mph OR

Therefore Breda's speed was 15 mph
Randy's speed was 5 + 15 = 20 mph

Answer 2

Keep in mind that (rate) × (time) = (Distance). Use this format to set up your equations.

Brenda:
r·t = 30. [Both her rate and time are unknown.]

Randy:
(r + 5) · (t + 1) = 60. [His rate is 5mph faster than Brenda's "r," so his rate is (r + 5). He rode one hour longer, so his time is (t + 1). Remember that since the rate is in miles per hour, the time should be in hours... it's (t + 1) rather than (t + 60).]

From Brenda, t = 30 / r. Substitute into Randy.
(r + 5) · (t + 1) = 60
(r + 5) · (30 / r + 1) = 60
(r + 5) · (30 / r + r / r) = 60 [Get common denominators.]
(r + 5) · (30 + r ) / r = 60 [Combining like terms.]
(r + 5) · (30 + r ) = 60r [Multiplying both sides by "r" to get rid of the fraction.]
r² + 35r + 150 = 60r
r² - 25r + 150 = 0
r² - 25r + 150 = 0
(r - 15) · (r - 10) = 0
r - 15 = 0 or r - 10 = 0
r = 15 or r = 10.

Checking both answers:
If "r" (Brenda's speed) = 10, then her "t" for 30 miles is 3 hours.
Randy's speed would be 15mph, his time 4 hours, and this does amount to a distance of 60 miles.

If "r" (Brenda's speed) = 15, then her "t" for 30 miles is 2 hours.
Randy's speed would be 20mph, his time 3 hours, and this also amounts to 60 miles.

Therefore, both answers are valid.
Either Brenda went 10 and Randy went 15mph, or
she went 15 and he 20mph.

Answer 3

let the speeds of Brenda and Randy be 'x' and 'x+5' mph respectively.so the equations will be x*t=30 and
(x+5)(t+1)=60 =>xt+x+5t+5=60=>x+5t=25 sub xt=30 and the expressing 't' in terms of 'x' and framing the equation
30/x+5x=25 =>5x^2-25x+30=0 dividing throughout by 5
x^2-5x+6=0 factoring x=3 or 2. the speeds of the cyclists are
10 mph and 15 mph or 15 mph and 20 mph

Answer 4

Answer given by "blind chaemeleon" is absolutely correct. Thus one at 2 sets of speeds the distance of 30 and 60 miles can be travelled meeting the other conditions.
The speed copuld either have been 10 and 15 mph or 15 and 20 mph.

Answer 5

2 answers are possible:

She went 10mph and he 15mph: 10mph x 3hr = 30; 15mph x 4 = 60

She went 15mph and he 20mph: 15mph x 2hr = 30; 20mph x 3 = 60