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avigutman
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Joined: 17 Jul 2019
Last visit: 30 Sep 2025
Posts: 1,285
Given Kudos: 66
Location: Canada
Schools: NYU Stern (WA)
GMAT 1: 780 Q51 V45
GMAT 2: 780 Q50 V47
GMAT 3: 770 Q50 V45
Expert reply
27 Sep 2021, 15:48
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kskumar
kskumar let me know if this helps:
This year Henry will save a certain amount of his income, and he will spend the rest.
Set up a 3-way ratio for this year (I know a ratio is a good idea because I glanced at the answer choices)...
Save : Spend : Total Income
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.
Add another column for the ratio, representing next year's spending money
Save : Spend : Total Income : Next Year's Spend
1 : ---------------- : ----------------------- : (1+r)
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?
(1+r) is half of what? Well, (1+r) is half of (2+2r), so:
Save : Spend : Total Income : Next Year's Spend
1 : (2+2r) : -------------------------- : (1+r)
And Total Income is the sum of the first two columns, so:
Save : Spend : Total Income : Next Year's Spend
1 : (2+2r) : (3+2r) : (1+r)
In terms of r, what fraction of his income should Henry save this year
Well, looking at our ratio, that's 1/(3+2r)
18 May 2023, 08:07
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Let's denote Henry's income this year as I.
Henry will save a fraction of his income, denoted as xI, where 0 ≤ x ≤ 1. He will spend the remaining fraction, which is (1 - x)I.
Next year, Henry will have no income. However, for each dollar that he saves this year, he will have 1 + r dollars available to spend.
The amount Henry saves this year is xI. Therefore, next year he will have (1 + r)(xI) available to spend.
To find the fraction of his income that Henry should save this year, we need to determine the value of x that satisfies the given condition:
Next year's amount available to spend = (1/2) * This year's amount spent
(1 + r)(xI) = (1/2)[(1 - x)I]
Simplifying the equation, we get:
(1 + r)(x) = (1/2)(1 - x)
Expanding the equation:
x + rx = (1/2) - (1/2)x
Rearranging the terms:
(3/2)x + rx = 1/2
Multiplying both sides by 2:
3x + 2rx = 1
Factoring out x:
x(3 + 2r) = 1
Dividing both sides by (3 + 2r):
x = 1 / (3 + 2r)
Therefore, in terms of r, the fraction of his income that Henry should save this year so that next year the amount he has available to spend will be equal to half the amount he spends this year is 1 / (3 + 2r).
Henry will save a fraction of his income, denoted as xI, where 0 ≤ x ≤ 1. He will spend the remaining fraction, which is (1 - x)I.
Next year, Henry will have no income. However, for each dollar that he saves this year, he will have 1 + r dollars available to spend.
The amount Henry saves this year is xI. Therefore, next year he will have (1 + r)(xI) available to spend.
To find the fraction of his income that Henry should save this year, we need to determine the value of x that satisfies the given condition:
Next year's amount available to spend = (1/2) * This year's amount spent
(1 + r)(xI) = (1/2)[(1 - x)I]
Simplifying the equation, we get:
(1 + r)(x) = (1/2)(1 - x)
Expanding the equation:
x + rx = (1/2) - (1/2)x
Rearranging the terms:
(3/2)x + rx = 1/2
Multiplying both sides by 2:
3x + 2rx = 1
Factoring out x:
x(3 + 2r) = 1
Dividing both sides by (3 + 2r):
x = 1 / (3 + 2r)
Therefore, in terms of r, the fraction of his income that Henry should save this year so that next year the amount he has available to spend will be equal to half the amount he spends this year is 1 / (3 + 2r).
25 May 2023, 19:12
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I - Income
Sa - Saved
Sp - Spend
I = Sa + Sp
If $1 is saved he has 1+r to spend.
Therefore,
Sa $ saved : Sa+Sa*r available to spend
To find Sa/I = Sa/Sa+Sp =? -------eq1
When,
Sa + Sar = Sp/2
solving
2Sa + 2Sar = Sp
Replacing Sp in eq1
=Sa/ Sa+2Sa + 2Sar
= 1/ 1+2+2r. = 1/3+2r
Sa - Saved
Sp - Spend
I = Sa + Sp
If $1 is saved he has 1+r to spend.
Therefore,
Sa $ saved : Sa+Sa*r available to spend
To find Sa/I = Sa/Sa+Sp =? -------eq1
When,
Sa + Sar = Sp/2
solving
2Sa + 2Sar = Sp
Replacing Sp in eq1
=Sa/ Sa+2Sa + 2Sar
= 1/ 1+2+2r. = 1/3+2r
