GMAT Quantitative: Fractions and Percents

GMAT Quantitative: Fractions and Percents

 

Fractions

Dividing by 5 is the same as multiplying by 2/10. For example:

840/5 = ?

840/5 = 840*(2/10) = 84*2 = 168

Multiplying or dividing by 10’s and 2’s is generally easier than using 5’s.

 

90% of the time, fractions will be easier to perform arithmetic. Decimals are sometimes more useful when comparing numbers relative to one another, such as in a number line, but these questions are the exception. Even if given a decimal (or percent) looks easy, quickly convert to a fraction. Some common ones to memorize:

1/9 = 0.111 repeating

1/8 = 0.125

1/7 = ~0.14

1/6 = 0.166 repeating

1/5 = 0.20

1/4 = 0.25

1/3 = 0.333 repeating

1/2 = 0.5 repeating

Note: Multiples of these, such as 3/8 (0.375) are also important to remember, but can easily be derived by multiplying the original fraction (1/8 * 3 = 3/8 = 0.125 * 3 = 0.375)

Denominators are super important. A denominator of a reduced fraction with a multiple of 7 will not have a finite decimal, for example. Keep in mind what you can logically combine, and what you cannot.

This list is by no means extensive. There are many many more shortcuts. If you have some, leave them in the comment field, but generally practice and familiarity with the numbers helps a lot in doing quick arithmetic.

 

Converting Fractions, Decimals, and Percentages

Fractions, decimals and percents are different ways of expressing the same value.  Here’s how to convert them from one form to another.

Converting decimals to fractions

For any decimal, you should be able to figure what the last place in your decimal is.  For example, .125 has digits in the tenths’, hundredths’ and thousandths’ place.  Thus, .125 is essentially 125 thousandths which translates to 125/1000.

Converting decimals to percentages

Percent literally means “over 100”.  i.e. x% = x/100.

So if you want to find what percent 12.5 is, you are trying to find x in this equation

If you rearrange the equation to isolate x on one side, you’ll see that to find x, you just need to multiply your decimal by 100.  Thus, in this case, 12.5 = 1250%

Converting fractions to decimals

Converting any fraction to a decimal involves long division.  A fraction is essentially the numerator divided by the denominator.  Thus 3/5 is simply 3 divided by 5, which you can work out by long division to be 0.6

Of course the GMAT is not going to use such as fractions, so be sure to know how to do long division!

Converting fractions to percents

As mentioned earlier, x% is x/100 – meaning that it is a fraction with 100 as the denominator.  So to find out what percent a fraction is, you need to manipulate the fraction you have to have 100 in the denominator and the numerator will be the percentage.

Suppose we need to convert  75/200 into its percentage form.  You need to convert it to an equivalent fraction with 100 as the denominator and find the numerator.

Thus, you are solving. (Solve this on your own and see if you get 37.5%)

Converting percents to fractions

This one is very easy and you should know by now that x% is x/100.  Thus if you wanted to convert 320% to a fraction,  it would be 320/100 which you can simplify to be 32.

Converting percents to decimals

From earlier, we learned that to convert decimals to percents, we multiplied the decimal by 100.  To do the reverse (i.e. convert percents to decimals) we do the opposite – divide by 100.

 

Percents

Now let’s discuss some common problems students encounter with percent problems, which can come in a variety of formats. Here are some quick pointers:

  • Percents MUST be APPLIED to something

    A percent means nothing on it’s own.  For example: 16% of men, or 30% off the sales price means nothing without a description of what it’s attached to.

  • Percents are basically fractions with a denominator of 100

    Learn your common percents, and convert to fractions whenever possible. Example: 20% = 1/5, 62.5% = 5/8

  • The word “of” means multiply

    Example: 80% of men = 4/5 * (total # of men)

  • Percents higher than 100 are numbers higher than 1

    Example: 125% = 100% + 25% = 1 + 0.25 = 1.25

  • Recognize the difference between percent MORE/LESS THAN and percent OF

    Example:

    What is 25% less than 8?
    ¼* 8 = 2, so 8 – 2 = 6

    Example:
    What is 25% of 8?

    ¼*8 = 2

  • Use shortcuts

    20% less than means 80% of. So instead of taking 20%, then subtracting from the original, just take 80% and be done. Conversely, 50% more than 10 should be calculated by multiplying 10*3/2 [10*(1 + 0.5)] in one neat step, versus two tougher ones.

    See the previous example:
    What is 25% less than 8?
    ¾*8 = 6, and we’re done! On easy numbers like this, it might not seem necessary, but as numbers get larger, it will save lots of time.

  • The higher the number, the higher the resulting percent

    Applying the same percent to a higher number will yield a higher number.

    Example:
    A certain positive integer x is increased by 10%, and then decreased by 10%. Which is bigger, x or the resulting number?
    The 10% increase of x in the first round increases x by a certain amount. The 10% decrease in the 2nd round is applied to a higher number, so will yield a larger change. The original x will be bigger.

  • Percent change = Total Change/Original Value

    Example:

    Before trading began, James’ investment portfolio was worth $10,000. At the end of market close, James’ investment portfolio grew by $2,000. What was the percent change in James’ portfolio?

    Percent change = $2,000/$10,000 = .2 or 20%

  • Don’t add constants and percents

    You should never find yourself trying to figure out what 5 + 6% equals. In this case, you are probably missing what to apply the percent to.

Percent Practice Questions

Let’s take a look at two examples!

Example 1

A tour group of 25 people paid a total of $630 for entrance to a museum. If this price included a 5% sales tax, and all the tickets cost the same amount, what was the face value of each ticket price without the sales tax?

A. $22

B. $23.94

C. $24

D. $25.20

E. $30

Explanation

Without a calculator, fractions are always easier. They cancel well, and are typically neater.

5% = 1/20 since 5*20 = 100.

Now we set up the equation, setting x = ticket price before tax.

25 people * x dollars/person * 1.05 (with tax) = $630

Note we can convert to fractions, cancel and simplify. Look how easy it gets?

25*(21/20)*x = 630

5*(21/4)*x = 630

x = 630*4 / 5*21

x = $24

Choice C

Example 2

During an auction, Jerome sold 75% of the first 1,000 items he offered for sale, and 30% of his remaining items. If he sold 40% of the total number of items he offered for sale, how many items did Jerome offer for sale?

A. 750

B. 1,050

C. 1,800

D. 3,500

E. 4,500

Explanation

Again, we want to set up the equation – this will make things a lot easier. And again, switching to fractions is always best.

3/4*1000 + 3/10*R = 4/10*T

We have 2 equations, and 1 unknown. This is a good hint that there may be a hidden 2nd equation.

1000 + R = T

Now, we have 2 equations and 2 unknowns. We can solve!

750 + 3R/10 = 400 + 4R/10

350 = R/10

R = 3,500

We always look back to the original question to see exactly what we are looking for. In this case, T. Not R.

T = R + 1,000 = 3,500 + 1,000 = 4,500

Choice E

 

GMAT questions are notorious for seeming harder than they actually are. The writers recognize time is short, and will give you ostensibly time-consuming calculations. One way to mitigate this is by retaining a rockstar aptitude in manipulating fractions, which occur in a large portion of the questions.