Problem Solving

Can anyone suggest a simplified approach to this problem? My first issue is simply interpreting exactly what the question is asking and my second problem concerns the messy algebra involved in the book's solution.

I'm not even sure how I could pick numbers to work through this one.

Problem 163, p. 175

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 has 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

Search the question, boss! You will get plenty of solutions.

This is an extremely tough question. There are a lot of parts and I think that it is very difficult to solve algebraically even after reviewing.

I would recommend that you plug and chug, especially if you are having a hard time understanding the algebra from the OG solutions.

Here's how I did it in the plug in chug.

1. They don't tell us how much Henry makes which must mean that the equation can be solved without having a specific value for his income. This means you can make an assumption about how much he makes.

I assumed $200

2. They tell us that he will get 1 + r dollars for every one dollar he saves. R is another plug and chug number, and you can verify this because they want the answer in terms of r. I chose my r to be $2. (I never choose 1 or 0 for my plug and chug number because otherwise when I plug into the answers I may get two options. Always choose either 2 or 5 for plug and chug as they are both prime and easy to work with) For every dollar he saves he gets 3.00, which means he will have triple what he saves to spend in the year he has no income.

Now we can solve for the equation.

We want the amount he has from saving = half of what he spends this year. Begin plug and chug.

Amount he has from saving 3*(s) = (1/2)*(200 - s). In other words, three time what he saves needs to equal 1/2 of $200 minus what he saves. Keep in mind that 200 minus what he saves equals what he is currently spending.

Solve for S, keeping in mind this is what he is SAVING this year.

6s=200-s

7s=200

s=200/7 (this looks ugly but isn't as bad as it looks so keep going)

Now remember that we only found how much he must SAVE but that's not what the problem asks for. It wants current year savings/current year income. If you stop here you will get it wrong!

That amount is (200/7)/200 or (200/7)*(1/200) or 1/7

Now for plug and chug.

I always start at the bottom in plug in chug probs like this because the test makers know that 99% of us will start at A when we plug and chug and mess up or get frustrated by the time we get to E.

So always start with E when plugging and chugging as lower half of the answers have a greater chance of being right when it looks like plug and chug is the fastest way to solve.

1/(2r +3)

Our r was 2

1/(2(2) +3)=1/7= the amount we got in our plug and chug. E is correct.

Note. I still generally check D through A just to make sure that I don't get duplicate right answers. Keep in mind it's important to pick fairly unique values for R to make sure that everything works. Had we chosen 1 for r C and E would have yielded unique values.

We can solve using simple algebra.

This year
----------------

Income=I

Savings =S

Money spent=(I-S)

Next year
---------------------

For each dollar(take it as 1 dollar) he saves previous year he will have (1+R) dollars available to spend.

1 dollar=(1+R)

So for S dollars which he saves,he will have S(1+R) dollars available to spent.

Now come to the next part of the question stem.

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

What fraction of Income means S/I?

Amount he spent this year=(I-S)

Amount he has available to spend next year=S(1+R)

-->S(1+R)=1/2(I-S)

S+RS=I/2-S/2

3S/2+RS=I/2

S[3/2+R]=I/2

S[3+2R]=I

S/I=1/(2R+3)

Hence E

Hope this clarify.

I always love working backwards.

Let, this year: savings = s
expenditure = e
so, total Income = s+e.

we need to find out s/(s+e).....(1)

given: e/2 = s(1+r).
so, s =e/2(1+r)

substitute s in 1 above.
u will get E.

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 has available to spend will be equal to half the amount that he spends this year?

Let r=2.
Now let's plug in values that follow the words in the problem, step by step.

For each dollar that Henry saves this year, he will have 1+r = 1+2 = 3 dollars available to spend [next year]:
If he saves $1 this year, he can spend $3 next year.

Next year the amount he has available to spend will be equal to half the amount that he spends this year:
Since he has $3 to spend next year, he has $6 to spend this year.

What fraction of his income should Henry save this year?
Since he saves $1 and spends $6, saved/total = 1/(1+6) = 1/7. This is our target.

Now we plug r=2 into the answers to see which yields our target of 1/7.

Only E works:
1/(2r+3) = 1/(2*2 + 3) = 1/7.

The correct answer is E.

GMATGuruNY 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 has available to spend will be equal to half the amount that he spends this year?

Since there is a variable in the answer choices, and the question is asking only for a fraction (not for an exact value), we can plug in our own numbers for all the unknowns in this problem.

Let r=2.

Now let's plug in values that follow the words in the problem, step by step.

for each dollar that he saves this year, he will have 1 + r = 1 + 2 = 3 dollars available to spend (next year): So if he saves this year $1, he can spend next year $3.

next year the amount he has available to spend will be equal to half the amount that he spends this year: If he spends next year $3 (from above), he spends this year $6 (3 = 1/2 * 6).

What fraction of his income should Henry save this year? If this year he saves $1 and spends $6 (both from above), he must save 1/(1+6) = 1/7. This is our target fraction.

Now we plug r=2 into the answers and look for our target fraction.

Only E works:

1/(2r+1) = 1/(2 * 2 + 3) = 1/7.

The correct answer is E.

late thanks but this explanation was great. starting at one and moving fwd is a great way to solve. just not as easy to see it all the time!!

GMATGuruNY 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 has available to spend will be equal to half the amount that he spends this year?

Let r=2.
Now let's plug in values that follow the words in the problem, step by step.

For each dollar that Henry saves this year, he will have 1+r = 1+2 = 3 dollars available to spend [next year]:
If he saves $1 this year, he can spend $3 next year.

Next year the amount he has available to spend will be equal to half the amount that he spends this year:
Since he has $3 to spend next year, he has $6 to spend this year.

What fraction of his income should Henry save this year?
Since he saves $1 and spends $6, saved/total = 1/(1+6) = 1/7. This is our target.

Now we plug r=2 into the answers to see which yields our target of 1/7.

Only E works:
1/(2r+1) = 1/(2*2 + 3) = 1/7.

The correct answer is E.

Hi Mitch,

Lets use the same plugging mechanism but assume that Henry saves 3$ this year. This means that he will get 4$, the next year. So he must be spending 8$ this year so the ratio becomes 3/(8+3) = 3/11

None of the answers match this ratio when we put r = 3. Please help me find out my mistake

Thanking you in anticipation of your help

Regards,
Vishal

vishal.pathak wrote:

GMATGuruNY 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 has available to spend will be equal to half the amount that he spends this year?

Let r=2.
Now let's plug in values that follow the words in the problem, step by step.

For each dollar that Henry saves this year, he will have 1+r = 1+2 = 3 dollars available to spend [next year]:
If he saves $1 this year, he can spend $3 next year.

Next year the amount he has available to spend will be equal to half the amount that he spends this year:
Since he has $3 to spend next year, he has $6 to spend this year.

What fraction of his income should Henry save this year?
Since he saves $1 and spends $6, saved/total = 1/(1+6) = 1/7. This is our target.

Now we plug r=2 into the answers to see which yields our target of 1/7.

Only E works:
1/(2r+3) = 1/(2*2 + 3) = 1/7.

The correct answer is E.

Hi Mitch,

Lets use the same plugging mechanism but assume that Henry saves 3$ this year. This means that he will get 4$, the next year. So he must be spending 8$ this year so the ratio becomes 3/(8+3) = 3/11

None of the answers match this ratio when we put r = 3. Please help me find out my mistake

Thanking you in anticipation of your help

Regards,
Vishal

r does not represent the value that Henry saves this year; r represents the RATIO of the amount SAVED THIS YEAR to the amount SPENT NEXT YEAR.
Let r=3.

For each dollar that Henry saves this year, he will have 1+r= 1+3 = 4 dollars available to spend [next year]:
If he saves $1 this year, he can spend $4 next year.

Next year the amount he has available to spend will be equal to half the amount that he spends this year:
Since he has $4 to spend next year, he has $8 to spend this year.

What fraction of his income should Henry save this year?
Since he saves $1 and spends $8, saved/total = 1/(1+8) = 1/9. This is our target.

Now we plug r=3 into the answers to see which yields our target of 1/9.

Only answer choice E works:
1/(2r+3) = 1/(2*3 + 3) = 1/9.