Huge Roadblock: Solving Problems with Too Many Decimals

[JoeyV]

Alright, I'm having a problem that is exacerbated by my current tests. I repeatedly run into problems where I am fully, 100% capable of solving the problem 'algebraically,' but am unable to solve it in practice because the arithmetic gets daunting. In these cases, I look at the explanations offered by my Test Prep software, and it is completely unhelpful since it almost completely appears to rely on either me using a calculator, or being a calculator.

An example of a problem that I know how to solve logically, but cannot work out the arithmetics of in a 2 minute time-frame. If someone could please tell me how I can effectively solve this problem I would really appreciate it:

Question: (Barron's GMAT Test Prep software, CAT #1)
Joan started work 2 years ago. Her starting salary was 1/2 of Mike's salary at that time. Each year since then Joan has received a raise of 5 percent in her salary and Mike has received a raise of 10% in his salary. What percentage (to the nearest percent) of Mike's current salary is Joan's current salary?

• 45
• 46
• 48
• 50
• 220

I was able to solve the algebra behind the problem quite easily, and my approximation was close to 45%, so I guessed 45 (A). The exact solution to the problem, however, was 45.5%, which rounds up to 46%, making the correct answer B.

When I clicked on the explanation, I was offered this (skipping the verbal explanation):

JS = Joan's starting salary. JC = Joan's current salary.
MS = Mike's starting salary. MC = Mike's current salary.

MC = (1.10)(1.10)MS = (1.21)MS
JC = (1.05)(1.05)JS = (1.1025)JS

So,
JC = (1.05^2)JS = (1.1025)JS = .5(1.1025)MS = .55125[1.1/(1.1^2)]MC = (.55125/1.21)MC

Since we want the answer to the nearest percent, we must divide to three decimal places.

So .55125/1.21=.455, which is 46% to the nearest percent.

My problem:

I understand all the algebra behind this problem. It is difficult (though doable) for me to calculate that 1.05 squared is 1.1025, however, it is near impossible for me to calculate that .55125 / 1.21 = .455 . In fact, that latter problem would take me an eternity to solve by hand.

How can I solve this problem without really tedious calculations? It seems like I am doomed to fail these problems every single time because my brain is not wired like a calculator. If someone could show me an alternate way to solve this problem, or a proper way to solve the math by hand (.55125/1.21), I would really appreciate it.

-JV

[Patrick_GMATFix]

Honestly Joey, I don't know whether this problem is worth your time, and I would be careful about relying on a resource that repeatedly tests long decimal division/multiplication. The GMAT does this only sparingly; the exam is a reasoning exam, so most of the time when long decimal division is part of the problem, you will be asked to approximate and the answers will be far apart enough that reasonable approximation is possible.

Furthermore, I've never seen an official GMAT question (in any OG or in GMATPrep) in which getting the correct answer depends on knowing whether .5 rounds up or down.

Consider practicing your test-taking from GMATPrep (it contains over 1200 questions) and using the Solutions Engine to manage your study.

If you really don't want to test from GMATPrep, I would still advise that you think about moving away from the CATs you're using.

Good Luck,
-Patrick

[albatross86]

I agree with Patrick, the mathematics involved seem a little too dodgy for the GMAT.

However, for the sake of discussion, here is how I solved it.

Let Joan's salary be J and Mike's be 2J (2 years ago)
1 year ago it would be 1.05J and 1.1*2*J
Now the salaries are (1.05)^2 times J and (1.1)^2 times 2J

Percentage : 100/2* (1.05/1.1)^2
= 1/2 * (105/11)^2 .......Dividing 105/11 isn't too tough and yields 9.5555.. let's express this conveniently
= 1/2 * (956/100)^2
=1/2 * (956)^2 * 10^-4

How you square 956 is kinda up to you. There are some tricks out there for squaring numbers, but I'm not sure if that's necessary for the GMAT, really.

I prefer calling it (1000 - 44)^2 = 1000^2 + 44^2 - 2*44000 = 913936

Percentage: 91.39 / 2 approx 46%

So really you only had to perform division by 11 and 2, and square 44, with a bit of addition and subtraction.

I'm betting you'd be able to manipulate any GMAT problem solving sum into something workable, so don't worry!

[logik1911]

A very easy way to do this problem in a short period of time is to reason it out

FYI I gave up on using Barron's because most of their material looks nothing like what I see in the OG and i'm often confused by their explanations.

but let's take a crack at this one:

just plug in. have Mike's salary equal 100 and joan's equal 50 and increase them in both 5 and 10% increments over a 2 yr period.

you'll end up with 55/121 which reduces to 5/11 which is 46%


so joan's will go from 50 to 52.5 to 55.095 (50*.05 = 2.5, and 52.5 *.05 = 55.095)

Mike's will go from 100 to 110 to 121 (increasing it 10% gives you 110, and increasing that 10% gives you 121)

55/121 is divisible by 11 and 5/11 = 46%


Hope that helps