Problem Solving

1.Did it take Pei more than 2 hours to walk a distance of 10 miles along a certain trail? ( 1 mile = 1.6 km)
a. Pei walked this distance at an average rate of less than 6.4 km/hr
b. On average, it took Pei more than 9 minutes per kilometer to walk this distance

2.Lines n and p lie in the xy-plane. Is the slope of the line n less than the slope of the line p?
a. Lines n and p intersect at the points (5,1)
b. The Y intercept of line n is greater than the y-intercept of line p

3. Of the 800 sweaters at a certain store, 150 are red. How many of the red sweaters at the store are made of pure wool?
a. 320 of the sweaters at the store are neither red nor made of pure wool
b. 100 of the red sweaters at the store are not made of pure wool

4. On her way home from work, Janet drives through several toll booths. Is there a pair of these toll booths that are less than 10 miles apart?
a. The 1st tool booth and the last tool booth are 25 miles apart.
b. Janet drives through 4 tool booths on her way home?

5. An integer greater than 1 that is not prime is called composite. If the two-digit integer n is greater than 20, is n composite?
a. The tens digit of n is a factor of the unit digit of n.
b. The tens digit of n is 2

Tagne wrote:1.Did it take Pei more than 2 hours to walk a distance of 10 miles along a certain trail? ( 1 mile = 1.6 km)
a. Pei walked this distance at an average rate of less than 6.4 km/hr
b. On average, it took Pei more than 9 minutes per kilometer to walk this distance

2.Lines n and p lie in the xy-plane. Is the slope of the line n less than the slope of the line p?
a. Lines n and p intersect at the points (5,1)
b. The Y intercept of line n is greater than the y-intercept of line p

3. Of the 800 sweaters at a certain store, 150 are red. How many of the red sweaters at the store are made of pure wool?
a. 320 of the sweaters at the store are neither red nor made of pure wool
b. 100 of the red sweaters at the store are not made of pure wool

4. On her way home from work, Janet drives through several toll booths. Is there a pair of these toll booths that are less than 10 miles apart?
a. The 1st tool booth and the last tool booth are 25 miles apart.
b. Janet drives through 4 tool booths on her way home?

5. An integer greater than 1 that is not prime is called composite. If the two-digit integer n is greater than 20, is n composite?
a. The tens digit of n is a factor of the unit digit of n.
b. The tens digit of n is 2

Answers ::

1. Pei runs for 10 miles = 10*1.6 = 16 Km

Lets got to option (a) first ... average speed of Pei is 6.4km/hr
With this speed he can cover 6.4*2 = 12.8 kms in 2 hours
Hence , answer is YES, he did take more than 2 hours to walk a distance of 10 miles.

Since (a) gives us the answer we can cut short our answers choice to AD.

Option (b) now , average speed of Pei is 9 minutes / km
It took 9 * 16 mins = 144 mins which is > 120mins ( 2 hrs ) to cover the whole distance .
Hence , answer is YES, he did take more than 2 hours to walk a distance of 10 miles

We can mark the answer as " D "now.

2. n and p are two lines on xy plane.

Lets see option (a) first.. n and p intersect at point (5,1 ) , this information doesn't tell us about slope of the lines ,
Hence this option is not giving us the answer , we can limit our choices to BCE.

Option (b) now , if Y intercept of line n is greater than line p , means the slope of n is greater than line p.
It gives us the answer NO , slope of the line n is NOT less than the slope of the line p .

We can go for the answer as "B".

3. Total sweaters in store = 800
Red sweaters = 150

Option (a) . 320 are neither red and nor wool. It doesn't give us enough information to get any answer.

Option (b). 100 red sweaters are not of pure wool , means 50 ( 150 - 100 ) are of pure wool.
It gives us the answer. We can mark the answer as "B".


4.

Option (a) . distance in 1st toll and last one is 25 miles.
We cannot conclude the answer with this much information.

Option (b). there are 4 toll booths on the way to her home.
This also doesn't give us the answer.

But if we add the information from both the options , we can get the answer as the distance is 25 miles and booths are 4 , there has to be a pair of booth less than 10 miles apart.

We can mark the answer as "C".

5. A two digit number n greater than 20 is there.

Option (a) tens digit of n is factor of units digit. since tens digit is not 1 ( as n is greater than 20 ) and every n ( with both the digits same i.e. 22,33,44 etc ) can be divisible by 11.

We can say this number is not at all prime and greater than 20 and hence composite. We got the answer.

Option (b) Tens digit is 2 , we have 23, 29 etc as prime and 24,26 as composites, we cannot conclude to the answer with this information.

We can mark the answer as "A".

Q3. stat 1 says not red not wool = 320 not sufficient
stat 2 says 100 are red and not wool. total red = 150 (given)
hence red and wool = 150 - 100 = 50
Stat 2 alone is sufficient

Q4. stat 1: difference between 1st and last toll booth tell us nothing about dispersion of toll booths in betweeen. not sufficient
stat 2: there are 4 toll booths but no data on total distance. hence NOt sufficient

together 1 and 2 - sufficient as no matter how widely or closely dispersed are 4 toll booths within 25 miles there has to be 2 booths within 10 miles of each other hence C

Q5. to find if n >20 is a composite number?
stat 1: tens digit factor of units digit so numbers could be:
39 : not prime
26 : not prime
hence sufficient

stat 2: tens digit is 2 hence numbers could be:
21: not prime
23 - prime
hence not sufficient

A alone is sufficient

2. n and p are two lines on xy plane.

Lets see option (a) first.. n and p intersect at point (5,1 ) , this information doesn't tell us about slope of the lines ,
Hence this option is not giving us the answer , we can limit our choices to BCE.

Option (b) now , if Y intercept of line n is greater than line p , means the slope of n is greater than line p.
It gives us the answer NO , slope of the line n is NOT less than the slope of the line p .

We can go for the answer as "B".

In y= mx+c how is c and m related ... you wrote c is directly proportional to m....
For this problem I got C as solution.

5m1+c1 =1 and 5m2+c2 =1 .. given c1>c2 so m2<m1 to balance this equation to 1... Kindly post your comment on this

HSPA wrote:2. n and p are two lines on xy plane.

Lets see option (a) first.. n and p intersect at point (5,1 ) , this information doesn't tell us about slope of the lines ,
Hence this option is not giving us the answer , we can limit our choices to BCE.

Option (b) now , if Y intercept of line n is greater than line p , means the slope of n is greater than line p.
It gives us the answer NO , slope of the line n is NOT less than the slope of the line p .

We can go for the answer as "B".

In y= mx+c how is c and m related ... you wrote c is directly proportional to m....
For this problem I got C as solution.

5m1+c1 =1 and 5m2+c2 =1 .. given c1>c2 so m2<m1 to balance this equation to 1... Kindly post your comment on this

Thanks HSPA for the response.

I analyzed it as m=slope= tanA ( where A is the angle with +ve X-axis ) and as tangent of any angle increases with the angle as such 0 to 90 , i took it as c will also increase gradually. ( But I was wrong as i didn't consider all the possibilities )

Your approach seems correct , but still we get the answer if we follow your way.
5m1+c1 =1 and 5m2+c2 =1

If c1 > c2 , then m1 < m2 to balance the equation.

Hence, it gives us the answer YES, slope of the line n is less than the slope of the line p .

We can mask the answer as "B". ( Not "C" )

Please comment.

1) Did it take Pei more than 2 hours to walk a distance of 10 miles along a certain trail? ( 1 mile = 1.6 km)
a. Pei walked this distance at an average rate of less than 6.4 km/hr
b. On average, it took Pei more than 9 minutes per kilometer to walk this distance

RT=d;

a) less than 6.4 . TAKE max 6.3 KM/HR. RT=D ; T= 16kms/6.3 Definitely more than 2 hours. Sufficient

b) 9 min per km = so for 16 kms it wil take =16*9 =144 min for 16 kms----> definitely grater than 2 hrs. Sufficient.

D

2.Lines n and p lie in the xy-plane. Is the slope of the line n less than the slope of the line p?
a. Lines n and p intersect at the points (5,1)
b. The Y intercept of line n is greater than the y-intercept of line p

a) Let us take the equations of two lines as

n-----> y1=m1x1+c1 ----(1)

m ----> y2= m2x2+c2 ----(2)

They both intersect at the point (5,1) So this point will satisfy both the Eqns. Putting x=5 & y = 1 in both eqns:

n---> 1= 5m1+c1
m---> 1 = 5m2+c2

Cannot determine slope of which Eqn is Higher. NS

b) Given c1>c2. Again this information alone cannot tell whether m1>m2.

Combining both : Sufficient

C

Hi Manish

Your approach seems correct , but still we get the answer if we follow your way.
5m1+c1 =1 and 5m2+c2 =1

If c1 > c2 , then m1 < m2 to balance the equation.

Hence, it gives us the answer YES, slope of the line n is less than the slope of the line p .

We can mask the answer as "B". ( Not "C" )

Please comment.

5m1+c1 =1 and 5m2+c2 =1 -----> these equations are made by using 1 . So inspite of using the eqn for calculation u still say B.Strange. Anyway let me know if i did not catch ur point.

Hi HSPA,

Option (b) now , if Y intercept of line n is greater than line p , means the slope of n is greater than line p.
It gives us the answer NO , slope of the line n is NOT less than the slope of the line p .
We can go for the answer as "B".

Ur statement in BOLD above is true for only positive values of y intercept.

Consider -3 & -5 . In this -3>-5 but the line will have slope less than the line which has Y intercept as -5. Please draw & check.

Hope it clears the point. Please post ur comments.

the bold statement in my post is a copy from previous post... Even I havent understood what manish was trying to convey

ankur.agrawal wrote:Hi Manish

Your approach seems correct , but still we get the answer if we follow your way.
5m1+c1 =1 and 5m2+c2 =1

If c1 > c2 , then m1 < m2 to balance the equation.

Hence, it gives us the answer YES, slope of the line n is less than the slope of the line p .

We can mask the answer as "B". ( Not "C" )

Please comment.

5m1+c1 =1 and 5m2+c2 =1 -----> these equations are made by using 1 . So inspite of using the eqn for calculation u still say B.Strange. Anyway let me know if i did not catch ur point.

Yes Ankur , a very silly mistake. My brain was running over B-letter and I had clear answer as C.

Thanks for mentioning it specifically.That is what happen when you run at super speed when it is not required.

Apologies HSPA and Ankur. I will take care.

Guys, in the tollbooth question it mentions there are "several" tollbooths...are we supposed to assume this is greater than 2 or not assume a number?