Problem Solving

This year Henry will save a certain amount of his income, and he will spend the rest. Next year Henry will have no income, but for each dollar that he saves this year, he will have 1 + r dollars available to spend. In terms of r, what fraction of his income should Henry save this year so that next year the amount he was available to spend will be equal to half the amount that he spends this year?

A. 1/r+2
B. 1/2r+2
C. 1/3r+2
D. 1/r+3
E. 1/2r+3


OA is E. Plz explain

Hi agemroy,

In the first year Income = Expenses1 + Savings (1)
In the second year Expenses2 = Savings * ( 1 + r ) (2)
and these two equations are also linked by Expenses2 = Expenses1 / 2 (3)

The problem asks what fraction of his income should Henry save, or in other words Savings / Income.

So from (2) we get
Expenses1 = 2 * Expenses2
and replacing this in (1) we get
Income = 2 * Expenses2 + Savings
and using (3) we get
Income = 2 * Savings*(1+r) + Savings
Income = (3+2r) * Savings

then
Savings / Income = 1 / (3+2r)


I'm sure there's a shorter way to do it, but I wanted to be clear.

Cheers,
Augusto

agemroy wrote:This year Henry will save a certain amount of his income, and he will spend the rest. Next year Henry will have no income, but for each dollar that he saves this year, he will have 1 + r dollars available to spend. In terms of r, what fraction of his income should Henry save this year so that next year the amount he was available to spend will be equal to half the amount that he spends this year?

A. 1/r+2
B. 1/2r+2
C. 1/3r+2
D. 1/r+3
E. 1/2r+3


OA is E. Plz explain

First thing I want to say is for every algebra word problem question with answer choices in variables it is always better to plug in numbers.

Here also we can plug in numbers.

Assume,

Income = $100
Savings = $25

Fraction = 25/100 = 1/4 (the answer that is expressed in terms of r)

Lets work with r+1,

The question says that for every dollar saved last year he will have (r+1) to spend in the next.

Therefore, By taking our assumptions he will have 25(r+1) to spend which is equal to half the amount he spends this year. (100-25)=> 75

25r+25=75/2

this results in r = 1/2

Now we insert the value of r=1/2 in the answer choices whichever yields 1/4 as the answer is the correct option.

Hence option E.

Let me know if you have doubts.

Where did you get this question from ?

Ok ... here's a detailed breakdown ... hope it helps.

I think the key with these word translation problems is being able to make sense of the information that they have provided you. Organising AND comprehending the information that they have provided you in the question is half the battle. Especially under the time constraints of the exam.

Lets take a closer look at the question:

"This year Henry will save a certain amount of his income and he will spend the rest"
This is telling us 3 pieces of information:
1) amount spent (lets call this x) this year
2) amount saved (lets call the y) this year
3) total income (lets call this i) this year
With word translation problems we should assign variables to all the unknowns.
We can also deduce that the total income is comprised up of the amount he has spent and the amount he has saved. This can be expressed using a simple equation:
i = x + y

"Next year Henry will have no income, but for each dollar he saves this year, he will have 1+r dollars available to spend."
What is this telling us ?
4) No income in 2nd year.
5) For each dollar saved this year (note: we labelled this with a variable y above - see point 2), he will have 1+r dollars available to spend next year. We can see here that we now have 2 spend values: this years spend, and next years spend. We can label these x1 (this year's spend) and x2 (next year's spend).
6) We can setup another equation using this information: x2 = y(1+r)
x2 = next years spend, y = this year's savings

"In terms of r, what fraction of his income should Henry save this year so that next year the amount he has available to spend will be equal to half the amount that he spends this year".
7) This is now the actual question be asked: "what fraction of his income should be saved this year" => y/i
8 ) "next year the amount he has available to spend will be equal to half the amount that he spends this year" => x2 = x1 / 2

So lets summarise all the information we have extracted from the question:

i = income this year
x1 = amount spend this year
y = amount saved this year
x2 = amount available to spend next year

Equations derived:
(1) i = x1 + y
(2) x2 = y(1+r) = y + yr
(3) x2 = x1 / 2

Lets start with (3):
x2 = x1 / 2
x1 = 2(x2)

Now we can substitute this for x1 into (1):
i = x1 + y
i = 2(x2) + y

Now we can use equation (2) and substitute into (1) for the x2 variable:
i = 2(x2) + y
i = 2(y+yr) + y
i = 2y + 2yr + y
i = 3y + 2yr
i = y(3+2r)

So in answering the question: y/i = 1/(3+2r)

Answer E.

P.S. With practice this whole thinking process should become a lot quicker.

II wrote:

Where did you get this question from ?

Ok ... here's a detailed breakdown ... hope it helps.
.
.
.
Now we can use equation (2) and substitute into (1) for the x2 variable:
i = 2(x2) + y
i = 2(y+yr) + y
i = 2y + 2yr + y
i = 3y + 2yr
i = y(3+2r)

So in answering the question: y/i = 1/(3+2r)

Answer E.

P.S. With practice this whole thinking process should become a lot quicker.

Thanks for your explanation, II ... How do you have so much patience to type so much ? ? man !!!

Awesome ...

What is the best way to learn to think abstract like you did here ? ?

Thanks
Amit

amitdgr wrote:
Thanks for your explanation, II ... How do you have so much patience to type so much ? ? man !!!

Awesome ...

What is the best way to learn to think abstract like you did here ? ?

Thanks
Amit

Thanks.
Word translations have been my weak point ... so I am keen to improve in this area.
I think the key here is to be able to break down the question into smaller more manageable pieces. My approach involves reading the FULL question quickly first, and then breaking it down ... in other words SIMPLIFY.
Identify the unknowns ... and find out how they are linked with the known values ... this will enable you to create equations which you can use to solve the question.

Does anyone else have any views on this ... other approaches ?

Thanks.

I think we can make it shorter :

We know that I=S+E1 I=income , S =saving ,E1=Expense first year

E2=S(1+r) E2 = Expense year 2

and we need that E2=(1/2)E1

so S(1+r)=(1/2)(I-S) --> S(1+r)+(1/2)S = (1/2) I ---> S(1+r+(1/2))=(1/2)I ---> (2+2r+1)=I/S--> 2r+3= I/S

So S/I = 1/ (2r+3 )

Answer E

i am sowree but i found II's method of solving tooooooooo long

any other approach ???

bhumika.k.shah wrote:i am sowree but i found II's method of solving tooooooooo long

any other approach ???

Sure maam

Let me say Henry's present year income is 100 dollars and he saves x dollars out of it, which will get him x (1 + r) dollars to spend the next year. This sum is needed to be equal to half of (100 - x) dollars. Let's do math now:

x (1 + r) = ½ (100 - x) => 2 x (1 + r) = 100 - x

or x/100 = 1/(2 r + 3)

E

1st year Income = Exp1 + Saving (1)
2nd year Exp2 = Saving * ( 1 + r ) (2)
Exp2 = Exp1 / 2 =>Exp1= 2 *Exp2 (3)

sub eq 3 in eq 1

Income = 2 * Exp2 + Saving (4)

sb eq 2 in eq 4

Income = 2 * Saving*(1+r) + Saving
Income = (3+2r) * Saving

then
Saving / Income = 1 / (3+2r)

E

In this type of problems (long word problems with variables in the choices) I face 2 challenges:
1) Speed: It usually takes me more than 60 secs just to read and understand what is going in it. Is the reading speed lower than the normal for a guy who aims Q50/51 or maybe for anyone? Do i need more practice so that I can improve the speed?
2) choosing between algebra and vic(using numbers) method: how much time should one spend to decide which way one should go- algebra or vic? What indicators should we look for each approach i.e. algerbra or vic? I usually know that I have taken an inefficient approach only when I am well past half way around 3mins.

Any advice much appreciated. Thanks

Thanks for your replies Brent. However, I am not so much troubled by the solutions per se. Often, I am able to solve these questions fairly accurately by both methods. My questions are more strategic. Here they are again. Any advice?

In this type of problems (long word problems with variables in the choices) I face 2 challenges:
1) Speed: It usually takes me more than 60 secs just to read and understand what is going in it. Is the reading speed lower than the normal for a guy who aims Q50/51 or maybe for anyone? Do i need more practice so that I can improve the speed?
2) choosing between algebra and vic(using numbers) method: how much time should one spend to decide which way one should go- algebra or vic? What indicators should we look for each approach i.e. algerbra or vic? I usually know that I have taken an inefficient approach only when I am well past half way around 3mins.

Any advice much appreciated. Thanks