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Smart Numbers Problem

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Smart Numbers Problem

by jroach31 » Tue Feb 04, 2014 3:50 pm
This question really confused me. I know the strategy for it is to pick "smart numbers" to stand in for the unknown incomes, but is there some way to do it if you kept them as unknowns? It seems like you should be able to figure out the fraction of their combined monthly income if you know what fraction of their monthly income each of them have to donate. I guess they key here is that the problem tells you that Carrie earns twice as much? If you didn't know that, would it be possible to solve this problem?

Lisa spends 3/8 of her monthly paycheck on rent and 5/12 on food. Her roommate,
Carrie, who earns twice as much as Lisa, spends 1/4 of her monthly paycheck on rent
and 1/2 on food. If the two women decide to donate the remainder of their money
to charity each month, what fraction of their combined monthly income will they
donate?
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by Brent@GMATPrepNow » Tue Feb 04, 2014 4:15 pm
jroach31 wrote: Lisa spends 3/8 of her monthly paycheck on rent and 5/12 on food. Her roommate,
Carrie, who earns twice as much as Lisa, spends 1/4 of her monthly paycheck on rent
and 1/2 on food. If the two women decide to donate the remainder of their money
to charity each month, what fraction of their combined monthly income will they
donate?
Let's start with the "smart numbers" approach.
We want some salaries that work well with the fractions 5/12, 3/8, 1/4 and 1/2.
The LEAST COMMON MULTIPLE of 12, 8, 4 and 2 is 24, so let's say that Lisa earns $24/month.
Since Carrie earns twice as much as Lisa, Carrie earns $48/month.

LISA
Paycheck: $24
Rent = 3/8 of $24 = $9
Food = 5/12 of $24 = $10
So, charity = $24 - $9 - $10
= $5

CARRIE
Paycheck: $48
Rent = 1/4 of $48 = $12
Food = 1/2 of $48 = $24
So, charity = $48 - $12 - $24
= $12

Combined paychecks = $24 + $48 = $72
Combined charity = $5 + $12 = $17

So, fraction of combined income that goes to charity = 17/72

Cheers,
Brent
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by jroach31 » Tue Feb 04, 2014 4:19 pm
Thanks. Yeah, my question was more focused on what is the alternative way to doing this if you didn't pick numbers?
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by Brent@GMATPrepNow » Tue Feb 04, 2014 4:28 pm
jroach31 wrote: Lisa spends 3/8 of her monthly paycheck on rent and 5/12 on food. Her roommate,
Carrie, who earns twice as much as Lisa, spends 1/4 of her monthly paycheck on rent
and 1/2 on food. If the two women decide to donate the remainder of their money
to charity each month, what fraction of their combined monthly income will they
donate?
Here's the algebraic approach.
Let x = Lisa's monthly paycheck
So 2x = Carrie's monthly paycheck

LISA
Lisa spends 3/8 of her monthly paycheck on rent and 5/12 on food
Fraction for charity = 1 - 3/8 - 5/12 = 24/24 - 9/24 - 10/24 = 5/24
Amount given to charity = 5/24 of x = 5x/24

CARRIE
Carrie spends 1/4 of her monthly paycheck on rent and 1/2 on food
Fraction for charity = 1 - 1/4 - 1/2 = 4/4 - 1/4 - 2/4 = 1/4
Amount given to charity = 1/4 of 2x = 2x/4 = x/2

Combined paychecks = x + 2x = 3x
Combined charity = 5x/24 + x/2 = 5x/24 + 12x/24 = 17x/24

So, fraction of combined income that goes to charity = (17x/24)/3x = 17/72

Cheers,
Brent
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by jroach31 » Tue Feb 04, 2014 5:09 pm
Awesome. Thank you so much!
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by jroach31 » Tue Feb 04, 2014 5:11 pm
One additional question...if you weren't given the ratio of Lisa's income to Carrie's income, you wouldn't be able to solve this, right?
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by Brent@GMATPrepNow » Tue Feb 04, 2014 5:15 pm
jroach31 wrote:One additional question...if you weren't given the ratio of Lisa's income to Carrie's income, you wouldn't be able to solve this, right?
Exactly.

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by [email protected] » Wed Feb 05, 2014 1:15 am
Hi jroach31,

Brent's explanation that focused on TESTing Values is a great approach to learn and master. It's a tactic that is easy to use and can be used on many GMAT Quant questions.

As to your last question, if you weren't given the ratio of the two incomes, there IS a way to solve this problem using algebra, but the solution would not be as pretty:

Lisa spends 3/8 on rent and 5/12 on food, so she spends 9/24 + 10/24 = 19/24 of her paycheck
She would end up giving 5X/24 to charity

Carrie spends 1/4 on rent and 1/2 on food, so she spends 1/4 + 2/4 = 3/4 of her paycheck
She would end up giving 1Y/4 to charity

The total that the two women would have given to charity = 5X/24 + Y/4

If we go one more step and create a common denominator, we'd have:

5X/24 + 6Y/24

(5X + 6Y)/24

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by Jeff@TargetTestPrep » Mon Dec 11, 2017 11:37 am
jroach31 wrote:
Lisa spends 3/8 of her monthly paycheck on rent and 5/12 on food. Her roommate, Carrie, who earns twice as much as Lisa, spends 1/4 of her monthly paycheck on rent and 1/2 on food. If the two women decide to donate the remainder of their moneyto charity each month, what fraction of their combined monthly income will they
donate?
We can let p = Lisa's monthly paycheck. Thus, Carrie's monthly paycheck = 2p. Now we can determine each person's rent expense, food expense, and remaining money in terms of p.

Lisa:

Rent = (3/8)p

Food = (5/12)p

Remainder = p - (3/8)p - (5/12)p = (24/24)p - (9/24)p - (10/24)p = (5/24)p

Carrie:

Rent = (1/4)(2p) = (1/2)p

Food = (1/2)(2p) = p

Remainder = 2p - (1/2)p - p = (1/2)p

From the above, we see that the combined money they donate to charity is the sum of their remaining money:

(5/24)p + (1/2)p = (5/24)p + (12/24)p = (17/24)p

Also, the combined money they earn in a month is p + 2p = 3p.

Thus, the fraction of their monthly income donated to charity is [(17/24)p]/(3p) = (17/24)/3 = 17/72.

Jeffrey Miller
Head of GMAT Instruction
[email protected]

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