word problem- should be easy, but . . .

[Connie]

Q: At a Christmas party, there are five times and many men as women. If 12 more women arrive, there will be only twice as many men as women. How many men are at the party?

A: The book says 40, but how do I set this problem up to figure it out.

=======================
Q: Carlos can walk 8 miles in the same time it takes him to jog 12 miles. His jogging rate is 5 miles per hour faster than his walking rate. At what rate does he walk? How long does it take him to walk 8 miles?

A: The book says 10 mph: 48 min. Again, how do I set this up to solve?

Thanks for any help.

[gmbd97]

For QUESTION 1:

WOMEN=x (let)
So, MEN= 5x
12 WOMAN joined, so total Woman now=x+12
According to the QUESTION, 5x=2(x+12)
5x=2x+24
5x-2x=24
3x= 24
x=8
so MEN were=5*8=40


For QUESTION 2:

Let , the TIME be = t hour( same for both walking and jogging,
as mentioned)

so Walking rate= 8/t (distance, d= r (rate)* t (time))
jogging rate=12/t
so from the QUESTION, 12/t-8/t=5
4/t =5
t =4/5=4/5*60=48 min
WALKING rate= 8/(4/5)
=8*5/4
=10 miles per hour
HOPE, it helped!

[tommywallach]

Hey Connie,

GMB's explanations are correct, but I want to add a bit (particularly to his second explanation).

Briefly:

#1: This is a ratio question. GMB is making use of what is sometimes called "the unknown multiplier". Basically, when you have something in a ratio of 5:1 (like men to women here), you can set them equal to 5x and x. Now you only have one variable, so you only need one equation. GMB set that equation up correctly (5x = 2(x + 12)), so let me know if it makes sense.

#2: This is a rates question. It's hard to make a chart look good in this forum, but you always want to make a chart on your paper:

Rate * Time = Distance
Walk: x t 8
Jog: x + 5 t 12

How use R * T = D, to write 2 equations:

xt = 8
xt + 5t = 12

Substitute 8 for xt in the second equation, then solve:

8 + 5t = 12
5t = 4
t = 4/5

And 4/5 of an hour is 48 minutes.

Hope that helps!

-t