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Word problem questions(taking more than 2 mins on average)

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by \'manpreet singh » Wed Jun 20, 2012 11:39 am
Hi, i have just covered the word problems part and i found one disturbing thing that i clocked on average 2min and 45 sec for 20 such questions.i am kind of worried over my timing as i think i am consuming way too much time on such question. For me the problem is in allocating the variables quickly to the questions and setting up the equations.I am not able to do this process that quickly i think.Please let me know ways to improve it.Here are some sample question which took me a longer to evaluate, still got them wrong.

Q1)A certain fruit stand sold apples for $0.70 each and
bananas for $0.50 each. If a customer purchased both
apples and bananas from the stand for a total of
$6.30, what total number of apples and bananas did
the customer purchase?
(A) 10
(B) 11
(C) 12
(D) 13
(E) 14

Q.2)A store currently charges the same price for each
towel that it sells. If the current price of each towel
were to be increased by $1, 10 fewer of the towels
could be bought for $120, excluding sales tax. What is
the current price of each towel?
(A) $ 1
(B) $ 2
(C) $ 3
(D) $ 4
(E) $12

Q.3)A part-time employee whose hourly wage was
increased by 25 percent decided to reduce the
number of hours worked per week so that the
employee's total weekly income would remain
unchanged. By what percent should the number of
hours worked be reduced?
(A) 12.5%
(B) 20%
(C) 25%
(D) 50%
(E) 75%
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by Anurag@Gurome » Wed Jun 20, 2012 12:04 pm
'manpreet singh wrote:Q1)A certain fruit stand sold apples for $0.70 each and bananas for $0.50 each. If a customer purchased both apples and bananas from the stand for a total of $6.30, what total number of apples and bananas did the customer purchase?
Say, the customer bought m apples and n bananas.
We need to find (m + n).

So, (0.70)*m + (0.50)*n = 6.30
--> (7m + 5n) = 63

Now note that m and n are positive integers.
Also 7m and 63 both are divisible by 7. Hence, 5n = (63 - 7m) must be divisible by 7 too. It is only possible when n is a a multiple of 7.

Only possible value of n is 7.
Hence, m = (63 - 5*7)/7 = 4
Hence, (m + n) = (4 + 7) = 11

The correct answer is B.
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by Anurag@Gurome » Wed Jun 20, 2012 12:11 pm
'manpreet singh wrote:Q.2)A store currently charges the same price for each towel that it sells. If the current price of each towel were to be increased by $1, 10 fewer of the towels could be bought for $120, excluding sales tax. What is the current price of each towel?
Say, the current price of each towel is $P.
Hence, in $120, we get 120/P towels.

If the price of each towel is increased by $1, the new price will be $(P + 1).
Hence, in $120, we will get 120/(P + 1) towels.

So, 120/P - 120/(P + 1) = 10
--> 12(P + 1) - 12P = P(P + 1)
--> P(P + 1) = 12

Now, P and (P + 1) are consecutive integers.
12 can be expressed as the product of two consecutive integers in only one way, 12 = 3*4

Hence, P = 3

The correct answer is C.
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by Anurag@Gurome » Wed Jun 20, 2012 12:18 pm
'manpreet singh wrote:Q.3)A part-time employee whose hourly wage was increased by 25 percent decided to reduce the number of hours worked per week so that the employee's total weekly income would remain unchanged. By what percent should the number of hours worked be reduced?
Say, previous hourly wage was W and number of hours worked per week was H.
Hence, total weekly income was W*H.

Now, new hourly wage is W' = (W + 25% of W) = (W + W/4) = 5W/4
Say, new number of hours worked per week is H'

As total weekly income remained unchanged,
--> (W')*(H') = W*H
--> (5W/4)*(H') = W*H
--> H' = 4H/5 = (H - H/5) = (H - 20% of H)

The correct answer is B.
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by Anurag@Gurome » Wed Jun 20, 2012 12:22 pm
Another Approach for Q3
Say, previous hourly wage was $100 and number of hours worked per week was 100 hours
Hence, total weekly income was $10,000

Now, new hourly wage is ($100 + 25% of $100) = $(100 + 25) = $125
As total weekly income remained unchanged,
New number of hours worked per week is 10,000/125 = 80 hours

Hence, percentage reduction = (100 - 80)/100 = 20

The correct answer is B.
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by \'manpreet singh » Wed Jun 20, 2012 12:41 pm
Thanks Anurag for your fast response :D :) :)
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by gmattesttaker2 » Wed Jun 20, 2012 5:20 pm
Anurag@Gurome wrote:
'manpreet singh wrote:Q1)A certain fruit stand sold apples for $0.70 each and bananas for $0.50 each. If a customer purchased both apples and bananas from the stand for a total of $6.30, what total number of apples and bananas did the customer purchase?
Say, the customer bought m apples and n bananas.
We need to find (m + n).

So, (0.70)*m + (0.50)*n = 6.30
--> (7m + 5n) = 63

Now note that m and n are positive integers.
Also 7m and 63 both are divisible by 7. Hence, 5n = (63 - 7m) must be divisible by 7 too. It is only possible when n is a a multiple of 7.

Only possible value of n is 7.
Hence, m = (63 - 5*7)/7 = 4
Hence, (m + n) = (4 + 7) = 11

The correct answer is B.
Hello Anurag,

Thanks for the explanation. I had a question here though:

Only possible value of n is 7.

Sorry if my question is very basic but I was wondering why cannot n be say 14 because 5(14) is also a multiple of 7. When I plugged in the values, I got:

5(14)/7 = 9 - m
=> 10 = 9 - m
=> m = -1

So, 14 would be a wrong choice here which I found by calculation. However, I was looking for the reason why I should not even use 14 in the first place. Thanks a lot for your valuable time and help.

Best Regards,
Sri
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by eagleeye » Wed Jun 20, 2012 6:39 pm
gmattesttaker2 wrote:
Anurag@Gurome wrote:
'manpreet singh wrote:Q1)A certain fruit stand sold apples for $0.70 each and bananas for $0.50 each. If a customer purchased both apples and bananas from the stand for a total of $6.30, what total number of apples and bananas did the customer purchase?
Say, the customer bought m apples and n bananas.
We need to find (m + n).

So, (0.70)*m + (0.50)*n = 6.30
--> (7m + 5n) = 63

Now note that m and n are positive integers.
Also 7m and 63 both are divisible by 7. Hence, 5n = (63 - 7m) must be divisible by 7 too. It is only possible when n is a a multiple of 7.

Only possible value of n is 7.
Hence, m = (63 - 5*7)/7 = 4
Hence, (m + n) = (4 + 7) = 11

The correct answer is B.
Hello Anurag,

Thanks for the explanation. I had a question here though:

Only possible value of n is 7.

Sorry if my question is very basic but I was wondering why cannot n be say 14 because 5(14) is also a multiple of 7. When I plugged in the values, I got:

5(14)/7 = 9 - m
=> 10 = 9 - m
=> m = -1

So, 14 would be a wrong choice here which I found by calculation. However, I was looking for the reason why I should not even use 14 in the first place. Thanks a lot for your valuable time and help.

Best Regards,
Sri
Sri:

This is how you could infer that n=7 is the only option. After ascertaining that n was indeed a multiple of 7, we could see that
7m+5n = 63. Now on the LHS, 5*7 is 35, therefore, for the next multiple (7*2, it would be 5*7*2 = 70. But 70 is larger than RHS which is impossible since m is positive. So, by observing this, we can isolate n=7 as the only value.

Let me know if this helps :)
We had the equations
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by gmattesttaker2 » Wed Jun 20, 2012 7:09 pm
Hello Eagleeye,

Thank you very much for the thorough explanation. Now it clear to me.

Best Regards,
Sri
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