Word Problems Strategy Guide (3) - 6th ed. Chap. 3 Prob. Set

[RiccoR945]

Word Problems Strategy Guide (3) - 6th ed. Chap. 3 Prob. Set

In the Word Problems Strategy Guide, question number 7 of the Chapter 3 problem set states the following:

Mary and Nancy can each perform a certain task in m and n hours, respectively. Is m < n?

(1) Twice the time it would take both Mary and Nancy to perform the task together, each working at their respective constant rates, is greater than m.
(2) Twice the time it would take both Mary and Nancy to perform the task together, each working at their respective constant rates, is less than n.


I'm having some trouble with the logic on this one. I understand that the rate is (1/m)+(1/n) which equals (m+n)/mn. I also understand that since the problem states "...to perform the task together..." that we would make the total work 1. But what I'm not understanding is why the explanation makes the total time t rather than m+n.

Why is the question restated to be [(m+n)/mn] t=1 rather than [(m+n)/mn)](m+n)=1? (I do understand how the algebra to have t=mn/(m+n))

Could you help me through the logic on this one please?

[RonPurewal]

well, you can knock that possibility out with pure common sense.

m + n is bigger than either m or n individually.
if you're using time together = m + n, you're implying that the two women, AS A TEAM, actually do less work per unit time than either woman does individually!

that makes no sense, of course.

if this is still too abstract, think about it with actual numbers.
if i can paint a house by myself in 5 hours, and you can paint the same house by yourself in 4 hours, then it should be pretty obvious that it will not take us 9 hours to paint the house together! (it's going to take less than 4 hours, of course.)

[RonPurewal]

in any case, i'm awfully lazy, and i don't like doing tons and tons of algebra. here's how i would approach this problem:

• BASE FACT:
if you have two people working together at identical rates, then, together, they work twice as fast.
ok?
ok.
so... if you had 2 people working at mary's rate, they'd finish the job in m/2 hours. if you had 2 people working at nancy's rate, they'd finish in n/2 hours.

STATEMENT 1:
this means they take MORE than m/2 hours together.
if they both worked at mary's speed, they'd take exactly m/2 hours. thus, this statement means that nancy is slower than mary. sufficient.

STATEMENT 2:
this means they take LESS than n/2 hours together.
if they both worked at nancy's speed, they'd take exactly n/2 hours. thus, this statement means that mary is faster than nancy. sufficient.