GRE Exam: A Simple Guide To Part-to-Whole Relationships - Fractions

A part-to-whole relationship can be expressed as a fraction, decimal, percent, or ratio.

It may seem simplistic, but envisioning a pie in these questions often helps conceptualize the situation.

It’s equally important that you remember that the part must always refer to a whole.

People often have a tendency to say things like, “I have one-fourth,” or “Jack has 20%.” But these statements beg the question: one-fourth of what? 20 percent of what? When expressing these relationships algebraically, remember that the part must always be a piece of some quantity.


This guide discusses fractions.

A fraction is a way to represent a piece of a whole.

Fractions look like the following:

When thinking of fractions, imagine one pie. The denominator tells you how many slices the pie is broken into, and the numerator tells you how many of those slices you have.

So the fraction means you have 1 slice of a pie that has been cut into 4 slices.

Now let’s look at what happens when the numerator and denominator of a fraction change.

Let’s compare to .

In this case, you are still dealing with one pie, but now the pie is broken up into 8 slices instead of 4.

One slice out of 8 is less than 1 slice out of 4. From this, you can derive a general rule:
As the denominator of a positive fraction increases, the value of the fraction decreases.

Now let’s compare to .

Again, each fraction represents one pie, and in both cases, the number of slices is the same.

Two slices from a 4-slice pie is more than 1 slice from a 4-slice pie.

Thus is greater than . From this, you can derive a general rule:
As the numerator of a positive fraction increases, the value of the fraction increases.

Quantitative Comparison Strategy: Fractions
When the quantities in a Quantitative Comparison question are fractions, focus on the relationships between numerator and denominator. If the denominators are the same, the fraction with the larger numerator will be greater. If the numerators are the same, the fraction with the smaller denominator will be greater.

SOLUTION: Since the numerators of the fractions are the same, the fraction with the smaller denominator will be greater. 100 < 1002. Thus Quantity A is greater.


SOLUTION: The fractions in the two quantities have the same numerator. Thus the fraction with the smaller denominator will be greater. Since the numerator in is smaller than the numerator in is a smaller fraction, which means the denominator in Quantity A is smaller. Since the denominator in Quantity A is smaller, the value of the fraction in Quantity A is greater. The correct answer is A.

What if the fractions are negative? The opposite relationships apply:
As the denominator of a negative fraction increases, the value of the fraction increases.
As the numerator of a negative fraction increases, the value of the fraction decreases.

To understand why this is the case, compare and .

You know that is a larger piece of a pie than is .

But since , the negative version of will be more negative than the negative version of . If is more negative than , then must be smaller.

See the number line below for illustration:

 

Improper Fractions
Any fraction in which the numerator is larger than the denominator is an improper fraction.

Look at the fraction . In this case, the denominator is 4, which means the pie you are working with is cut up into 4 slices.

The numerator tells you that you have 7 slices of this pie.

If you have 7 slices of a 4-slice pie, then you have more than one pie. In fact, you have one 4-slice pie and 3 additional slices of that pie .

You can write this as a mixed number: .

A mixed number is any number that is written as an integer and a fraction.

To convert from a mixed number to an improper fraction, do the following:
1.   Multiply the integer by the denominator of the fraction: in this case, 1 × 4 = 4
2.   Take that product and add it to the numerator of the fraction: in this case, 4 + 3 = 7
3.   Take the result of Step 2 and put it above the original denominator: .

Practice: Improper Fractions
Write each of the following as an improper fraction. 




Adding and Subtracting Fractions
When adding or subtracting fractions, your goal is to manipulate the fractions to have the same denominator.

If you are asked to solve for , the answer would be since you are adding 1 slice of a 4-slice pie to 2 slices of a 4-slice pie.

You will end up with 3 slices of a 4-slice pie, which means you will have .

However, addition and subtraction of fractions will not always be so straightforward.

Sometimes the fractions you are adding or subtracting will have different denominators, for example, .

When adding or subtracting fractions with different denominators, you must first find a common denominator.

To do so, find the smallest number that is a multiple of both denominators.

In this case, the smallest number that is a multiple of 3 and 4 is 12. Now you need to manipulate both fractions to have a denominator of 12. To do so, you will multiply.

Note that when you multiply the denominator of the fraction by a value, you must multiply the numerator of the fraction by the same value.

By doing so, you are essentially multiplying the fraction by 1, which means that the value of the fraction will not change.

Comparing Fractions with Different Numerators and Denominators
Using a common denominator is also helpful when comparing fractions whose numerators and denominators differ.

Look at the following Quantitative Comparison question.

SOLUTION: Express both fractions with a denominator of 72. Quantity A: . Quantity B: .

Now the comparison is to .

The fractions have the same denominator, but the numerator of the second fraction is greater. Thus Quantity B is greater.

 

Multiplying Fractions
The product of two positive fractions will always be smaller than either of the factors, for example, .
Why is this the case?

When you multiply fractions, you are essentially taking a piece of a piece.

In other words, multiplying by means that you are looking for of . Or it can mean that you are taking of .

In either case, the result is a piece of both original fractions, meaning that the result will be smaller than either fraction.

Look at the following Quantitative Comparison question.

SOLUTION: Since x and y are between 0 and 1, their product must be a fraction. Thus the value in Quantity A must be a fraction. Since x and y are between zero and 1, their respective reciprocals must be greater than 1. Thus the factors of the product in Quantity B are each greater than 1. Since both factors are greater than 1, the product must be greater than 1. Thus Quantity B is greater.

Generally, when multiplying fractions, you should multiply all the numerators and all the denominators.

For example:

However, sometimes you will be able to reduce the fractions before multiplying:


SOLUTION: Before multiplying, cancel out the common factors:

The answer is 1.

Dividing Fractions
When dividing by a fraction, multiply the numerator by the reciprocal of the fraction in the denominator.

To find the reciprocal of a fraction, simply switch the terms in the numerator and denominator.


SOLUTION: Multiply by the reciprocal of :


Fractions in Word Problems
Many word problems will test your understanding of fractions in real-world contexts. In many of these questions, you will be given a value for the part and will be asked to solve for the whole or vice versa.

In these questions, it is essential to remember the following relationship: part = fraction × whole.

As you are given information, track how that information fits into that formula.

Look at the following example:
Of the 120 boys in a school, are taking chemistry. If of the students in the school are boys, the number of boys taking chemistry is what fraction of all the students at the school?

SOLUTION: You are being asked to solve for the number of boys taking chemistry as a fraction of all students at the school, so the numerator should be “number of boys taking chemistry” and the denominator should be “the number of students in the school.”

Now use the information from the question to solve for these values.

SOLUTION: In the first sentence, you are told that of all the boys are taking chemistry.

Since there are 120 students in the school, the number of boys taking chemistry . Now figure out the number of students in the school. The whole is the number of students, the part is the number of boys, and is the fraction. Thus , where n is the number of students in the school.

Multiply both sides of the equation by to solve for n:


SOLUTION: So the fraction you arrive at is , which reduces to .

Plugging in Numbers with Fractions
In many fraction questions, you will be given unspecified amounts and asked to solve for a relationship between these amounts. In these situations, the best strategy is to choose values that satisfy the relationships in the question.

When plugging in numbers for fraction questions with no specified amounts, keep the following tips in mind:
 

- When choosing a value, choose a value for the whole.

For example, if you are told that of the employees in a company are administrators and that of the administrators are men, then choose a value for the number of employees (the whole), and not the administrators or men.
- Choose a value for the whole that will be divisible by the denominators of all the fractions in the question.

For example, if you are told that of the employees in a company are administrators and that of the administrators are men, then the value you choose for the number of employees should be divisible by 3 (the denominator of the first fraction) and 4 (the denominator of the second fraction).

Good numbers here would be multiples of 12. Why is this strategy important? Because to the extent that it is possible, you want to work with integers. If the number that you choose is not divisible by 3 or 4, then the value for one of the parts will end up being a fraction, and the math will almost certainly be messier.

Let’s look at an example:
Two-thirds of the cars in a lot are sedans, and the rest are trucks. If of the sedans are new and of the trucks are new, what fraction of the cars in the lot are used?

SOLUTION: Since the question does not provide any amounts, you should choose numbers to determine values for the total number of cars and the number of used cars.

Based on the first tip provided earlier, you should choose a value for the whole, which in this case is the total number of cars. Based on the second tip, the value you choose should be divisible by 3, 8, and 10.

The most obvious value here is 3 × 8 × 10 = 240.

If there are 240 cars, then are sedans, and the other 80 are trucks.

Now determine how many sedans and how many trucks are used.

If of the sedans are new, then the remaining are used. of 160 = 16, so there are 16 used sedans.

If of the trucks are new, then are used. of 80 = 50, so 50 of the trucks are used. In total, there are 50 + 16 = 66 used cars.

The final fraction of used cars/total cars .

Divide the numerator and denominator by 6 and arrive at .