Need help with speed - fastest way to do these problems?

[mayonnai5e]

I had several problems in the last few tests that were of the same nature. I can solve them all easily using the tried and true method of algebra and equations, but I could not solve them fast enough and eventually guessed and moved on. I need help in figuring out the quickest way to get the solution. Thanks in advance:

In a nationwide poll, N people were interviewed. If
1/4 of them answered “yes” to question 1, and of
those, 1/3 answered “yes” to question 2, which of the
following expressions represents the number of
people interviewed who did not answer “yes” to both
questions?
(A) 7/N
(B) 6N/7
(C) 5N/12
(D) 7N/12
(E) 11N/12

A hiker walked for two days. On the second day the
hiker walked 2 hours longer and at an average speed
1 mile per hour faster than he walked on the first day.
If during the two days he walked a total of 64 miles
and spent a total of 18 hours walking, what was his
average speed on the first day?
(A) 2 mph
(B) 3 mph
(C) 4 mph
(D) 5 mph
(E) 6 mph

One night a certain motel rented 3/4
of its rooms, including 2/3 of its air-conditioned
rooms. If 3/5 of its rooms were air-conditioned,
what percent of the rooms that were not rented
were air-conditioned?
(A) 20%
(B) 33 1/3%
(C) 35%
(D) 40%
(E) 80%



One-fifth of the students at a nursery school are 4 years old or older. If 16 students have not yet reached their third birthday, and a total of 40 students are not between 3 years old and 4 years old, how many children are in the nursery school?
(A) 120
(B) 96
(C) 70
(D) 60
(E) 24


HMMM most of them look like candidates for plugging in the answer. Is that the trick?

[ri2007]

Since you already know how to solve the questions and are only looking for a short cut will reply to question 1 & 3 (not sure of a short cut for Q 2)

Q1) You see that the fractions given have a denominator of 3 & 4. So Assume there are 3*4 = 12 people (you also see most of the answer choice have 12 as a denominator)

1/4 ans yes to question 1, so that is 14/4 = 3
1/3 of the 1/4 ans yes to q 2 = 3/3 = 1
So people who ans yes to both question is 1 of 12
People who did not ans yes to both questions is 12/12 - 1/12 = 11/12
Since this is the ration the actual no will be 11/12 N (where N is the total no of people.)

Q3 Solve in the same way as Q1

LCM of all denominators is 4* 3 * 5 = 60

Total rooms rented = 3/4* 60 = 45
So total rooms not rented = 60 - 45 = 15
Total Airconditioned rooms = 3/5*60 = 36
Total Airconditioned rooms not rented =1/3*36 = 12
So % of rooms that were not rented that were airconditioned is 12 of 15 = 80%

Here is my take on Q4

The class has 3 catageories of studients

1) < 3 = 16
2) 3 to 4 (but less than 4)
3) => 4 = 1/5 of the class

Total students not in 3 to 4 group is 40
So total in group =>4 is 40-16 =24
=1/5 of total

So total is 24*5 = 120

Pls confirm if I got the ans correct or have made some mistakes in either of the answers.

Hope this helps!!!

[wizardofwashington]

Qeustion 3:

Let S1 and S2 are the speeds on Day 1 and Day 2 respectively.

Setting the equation as follows:
Day 1 Distance + Day 2 Distance = Total Distance
S1xT1 + S2xT2 = 64 ===> Equ. 1

We know that S2 = S1+1, from the problem, which states that he walked one mph faster on Day 2 than on Day 1.

Also, we know that a total of 18 hrs can be split into 8hrs on Day 1 and 10hrs on Day 2. This breakdown of hours is based on the problem statement that he walked two hours more on Day 2 than on Day 1. Any other way of breaking down is not possible (such as 7 and 11 since the difference is not 2hrs if you do that)

Now, lets start plugging the values in the above Equn 1, we get :
s1x8 + (s1+1)x10 = 64
Solving for S1,
8S1+10S1+10 = 64
18S1 = 54
S1 = 3
Hence the answer is B. Is this correct??

[wizardofwashington]

Sorry about the earlier post.. That solution is actually applicable to Question 2 and NOT for Q3, which ri2007 has wonderfully solved. Hope it helps.

[wizardofwashington]

Here is my take on Q4:
The class has 3 catageories of studients (as defined by ri2007)

1) < 3 = 16
2) 3 to 4 (but less than 4)
3) => 4 = 1/5 of the class

Assume total # of students = X = 16+40+(1/5X)

X -1/5X = 56
4/5X = 56
X = 70

I think this is straight forward and doesn't have that twist as perceived by r12007 in his solution. Please correct me if I'm wrong, which may very well be the case here..hehehe

[mayonnai5e]

Since you already know how to solve the questions and are only looking for a short cut will reply to question 1 & 3 (not sure of a short cut for Q 2)

Q1) You see that the fractions given have a denominator of 3 & 4. So Assume there are 3*4 = 12 people (you also see most of the answer choice have 12 as a denominator)

1/4 ans yes to question 1, so that is 14/4 = 3
1/3 of the 1/4 ans yes to q 2 = 3/3 = 1
So people who ans yes to both question is 1 of 12
People who did not ans yes to both questions is 12/12 - 1/12 = 11/12
Since this is the ration the actual no will be 11/12 N (where N is the total no of people.)

Q3 Solve in the same way as Q1

LCM of all denominators is 4* 3 * 5 = 60

Total rooms rented = 3/4* 60 = 45
So total rooms not rented = 60 - 45 = 15
Total Airconditioned rooms = 3/5*60 = 36
Total Airconditioned rooms not rented =1/3*36 = 12
So % of rooms that were not rented that were airconditioned is 12 of 15 = 80%

Here is my take on Q4

The class has 3 catageories of studients

1) < 3 = 16
2) 3 to 4 (but less than 4)
3) => 4 = 1/5 of the class

Total students not in 3 to 4 group is 40
So total in group =>4 is 40-16 =24
=1/5 of total

So total is 24*5 = 120

Pls confirm if I got the ans correct or have made some mistakes in either of the answers.

Hope this helps!!!

Q1 and Q4 are correct, but I do not remember Q3 off the top of my head (i'm at work). The solutions you provided are really simple and easy - picking numbers. I have notes on that topic from the PR and Kaplan books, but haven't used that strat at all during my practice. I'll be sure to find some problems like these and do timed practice using this method. Thanks!

[mayonnai5e]

Here is my take on Q4:
The class has 3 catageories of studients (as defined by ri2007)

1) < 3 = 16
2) 3 to 4 (but less than 4)
3) => 4 = 1/5 of the class

Assume total # of students = X = 16+40+(1/5X)

X -1/5X = 56
4/5X = 56
X = 70

I think this is straight forward and doesn't have that twist as perceived by r12007 in his solution. Please correct me if I'm wrong, which may very well be the case here..hehehe

The OA is 120 not 70.

[mayonnai5e]

Qeustion 3:

Let S1 and S2 are the speeds on Day 1 and Day 2 respectively.

Setting the equation as follows:
Day 1 Distance + Day 2 Distance = Total Distance
S1xT1 + S2xT2 = 64 ===> Equ. 1

We know that S2 = S1+1, from the problem, which states that he walked one mph faster on Day 2 than on Day 1.

Also, we know that a total of 18 hrs can be split into 8hrs on Day 1 and 10hrs on Day 2. This breakdown of hours is based on the problem statement that he walked two hours more on Day 2 than on Day 1. Any other way of breaking down is not possible (such as 7 and 11 since the difference is not 2hrs if you do that)

Now, lets start plugging the values in the above Equn 1, we get :
s1x8 + (s1+1)x10 = 64
Solving for S1,
8S1+10S1+10 = 64
18S1 = 54
S1 = 3
Hence the answer is B. Is this correct??

I'm not sure why, but I got stuck on this one even though it seems easy in hindsight even if I used all equations and varibles.

I actually solved this one fairly quickly using a table:

T1 + T2 = 18
T2 = T1 + 2

so T1 = 8 and T2 = 10

R2 = R1 + 1

Start with A:
R1 T1 D1 R2 T2 D2 Total Distance
2 8 16 3 10 30 46
3 8 24 4 10 40 64 (winner!)

*Note: I should have started with C instead of A on this one.

[wizardofwashington]

Here is my take on Q4:
The class has 3 catageories of studients (as defined by ri2007)

1) < 3 = 16
2) 3 to 4 (but less than 4)
3) => 4 = 1/5 of the class

Assume total # of students = X = 16+40+(1/5X)

X -1/5X = 56
4/5X = 56
X = 70

I think this is straight forward and doesn't have that twist as perceived by r12007 in his solution. Please correct me if I'm wrong, which may very well be the case here..hehehe

The OA is 120 not 70.

My bad.. I was not reading the problem correctly.. I totally missed out on the word NOT in this part of the question: total of 40 students are not between 3 years old and 4 years old.. and hence the wrong approach and the terrible answer.. Kudos to ri2007 for working out the right way..Keep up the good work guys.. Its been very helpful so far..

[tanyajoseph]

With Q4:

Prob quotes 1/5 of the class are 4 yrs old or older.
So how do u actually derive 40-16 = 1/5???

Can you please ellaborate the explanation...missing smethin crazy!

Thank you

[Bharat]

Lets try to solve this problem one step at a time
1. One-fifth of the students at a nursery school are 4 years old or older.
means: 1/5th of the class is 4+ yrs old

2. If 16 students have not yet reached their third birthday,
means: 16 students are less than 3 yrs old.

3. a total of 40 students are not between 3 years old and 4 years old,
means: total number of student 4+ yrs old + students below 3 yrs old. = 40

how many children are in the nursery school?

hence total number of 4+ yrs students = 40 - 16 = 24 (which is 1/5th of the class)
Thus total students = 24*5 = 120

Let me know if there is any doubt.

[PRODG]

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I hope this will help..

[PRODG]

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