GMAT Math Review - Fractions and Decimals
Fractions and Fractional Notation
Fractions and fractional notation provide a basic way of working with numbers that are not whole numbers, but instead have a portion that falls between integers. At the most basic level, a fraction can be said to represent a part-to-whole relationship. The standard fractional notation used on the GMAT consists of a numerator (the top number), a horizontal fraction bar, and a denominator (the bottom number).
In the part-to-whole relationship, the numerator tells us how many specific parts there are, while the denominator tells us how many parts are needed to make a whole. A common way to visualize this relationship is as a circle divided into segments as if it were a pie. In the pie diagram above, the fraction $3/4$ is visualized with 3 out of the 4 segments shaded.
Fractions can be either less than or greater than 1. If a fraction is less than 1, the numerator will be less than the denominator, as in $1/3$ or $7/9$. A fraction that is less than 1 is referred to as a proper fraction. When a fraction is greater than 1, the numerator will be greater than the denominator, as in $3/2$ or $15/14$. A fraction with a numerator greater than the denominator is called an improper fraction. Any improper fraction can also be expressed as a mixed fraction, which is an expression that contains both a whole number part and a fractional part. For example, the improper fraction $3/2$ can also be written as the mixed fraction $1{1/2}$.
To convert between mixed and improper fractions it can be helpful to understand that at the same time a fraction represents a part-to-whole relationship, it also represents an expression of division where the numerator is divided by the denominator. In other words $3/4$ and $3 ÷ 4$ are two ways of writing the same equation.
To convert an improper fraction to a mixed fraction, perform the division. The quotient from the division becomes the whole number of the mixed fraction while the remainder becomes the numerator. For example if you have an improper fraction $7/4$ the result of the division would be a quotient of 1 with a remainder of 3, therefore the mixed fraction equivalent would be $1{3/4}$.
To convert a mixed fraction to an improper fraction multiply the whole number part by the denominator and then add the result to the numerator which effectively reverses the process described above: $$1{3/4} = {(1 * 4) + 3}/4 = 7/4$$
Basic Operations Using Fractions
The four basic arithmetic operations, addition, subtraction, multiplication and division, can all be performed on fractions. In order to add or subtract fractions, the denominators must be equal. With equal denominators you simply perform the operation on the numerators while the denominator remains unchanged: $$1/4 + 2/4 = {1 + 2}/4 = 3/4$$ or $$3/4 - 1/4 = {3 - 1}/4 = 2/4$$
To perform multiplication or division on fractions it is not necessary for the denominators to be equal. To multiply fractions, multiply the numerators and denominators together: $$2/5 * 3/7 = {2 * 3}/{5 * 7} = 6/35$$
To divide fractions you first invert the second fraction switching the numerator and denominator - this is called taking the reciprocal of the fraction: $$3/7 ⇒ 7/3$$
After taking the reciprocal of the second fraction, division is now accomplished by multiplying the fractions $${2/5} ÷ {3/7} = {2/5} * {7/3} = {2 * 7}/{5 * 3} = 14/15$$
Fractional Equivalence
Because fractions represent a part to whole relationship, it is possible to have two fractions that appear different but are in fact equal, provided the proportional relationship between the numerator and denominator is equal: $$1/2 = 2/4 = 3/6 = 4/8$$
Cross multiplication is a technique which allows you to quickly determine whether two fractions are equivalent. To use this technique multiply the numerator of each fraction by the denominator of the other fraction. If the two results are equal then the fractions are equal:
Any fraction can be manipulated to produce an equivalent fraction by either multiplying or dividing the numerator and denominator by the same number: $$5/7 = {5 * 3}/{7 * 3} = 15/21$$ or in reverse $$15/21 = {15 ÷ 3}/{21 ÷ 3} = 5/7$$
Reducing Fractions
Reducing a fraction is the process of taking a given fraction and finding the equivalent fraction with the smallest possible terms in the numerator and denominator. Performing arithematic operations on fractions will frequently produce a result with unnecessarily large terms. Reduction is often the last step in solving a GMAT problem involving fractions. The correct GMAT answer choice will always be reduced.
With smaller numbers you will many times be able reduce the fraction intuitively. In the above example of $15/21$, it is likely you would simply "know" that both the numerator and denominator are divisible by 3 and that the result, $5/7$ is not reducible any further. For larger numbers there are two basic approaches. The first approach is to divide the terms by small common factors repeatedly until the fraction can be reduced no further: $$60/84 = {60 ÷ 2}/{84 ÷ 2} = 30/42 = {30 ÷ 2}/{42 ÷ 2} = 15/21 = {15 ÷ 3}/{21 ÷ 3} = 5/7$$
The second, more formal, approach is to use the techniques described in the tutorial on Factors and Prime Factorization to find the greatest common factor (GCF) of the numerator and denominator and then divide the two terms by this number. Review that tutorial if you need a refresh on factoring techniques.
Finding a Common Denominator
When called upon to add or subtract fractions, or when called upon to determine if one fraction is greater or less than another fraction, you may need to convert two or more fractions such that they each share the same denominator. If, for example, the GMAT asks you to add $3/4 + 2/3$, you would need to determine two equivalent fractions that share the same denominator. A very straightforward approach to this situation is to multiply both the numerator and denominator of each fraction by the denominator of the other fraction: $$3/4 + 2/3 = {3 * 3}/{4 * 3} + {2 * 4}/{3 * 4} = 9/12 + 8/12$$
The only drawback to this approach is that it may result in a common denominator that is larger than necessary - especially when dealing with larger numbers or more than two fractions. It is ideal to be able to work with the lowest common denominator. Again, it might be helpful to review the Factors and Prime Factorization tutorial where there is a section on finding the least common multiple (LCM). Finding the LCM of two denominators will give you lowest common denominator.
Compound Fractions
A compound fraction occurs when a fraction has one or more additional fractional parts: $${1/4 + 1/3}/{5/7}$$
The GMAT loves this type of expression. To simplify the expression start by solving the addition of $1/4$ and $1/3$ in the numerator. With a common denominator of 12 we can rewrite the expression as follows: $${1/4 + 1/3}/{5/7} = {3/12 + 4/12}/{5/7} = {7/12}/{5/7}$$
We are now left with a fraction in the numerator and a fraction in the denominator. Recalling that a fraction is a representation of division, the expression can be rewritten as: $${7/12} ÷ {5/7}$$
To complete the division, take the reciprocal of the second fraction and then multiply: $${7/12} ÷ {5/7} = {7/12} * {7/5} = 49/60$$
Decimals
Decimals provide another method of working with numbers that have a fractional component. When a number with a fractional component is written in decimal notation, the portion of the number to the left of the decimal point represents the whole number and the portion to the right of the decimal point represents the fractional part of the number.
In fact, you can think of the portion to the right as a fraction where the denominator is a multiple of 10 derived from the number of digits to the right of the decimal point. $$0.1 = 1/10$$ $$0.16 = 16/100$$ $$0.163 = 163/1000$$
Basic Operations With Decimals
Performing basic arithmetic operations (addition, subtraction, multiplication, and addition) with decimals is straightforward. In each case the arithmetic operation is carried out as it would be with whole numbers, adjusting the alignment of the decimal point(s) either before or after the operation.
To setup an addition or subtraction problem, write out the two numbers with the decimal points aligned. Then when you perform the operation, position the decimal point in the answer following the same alignment:
To multiply decimal numbers, first carry out the multiplication as if they were whole numbers. Then count out the total digits to the right of the decimal point in the each of the numbers being multiplied and place the decimal point in the answer such that the number of digits to the right is equal to that sum. In the following example where 32.5 is multiplied by .65, the total number of digits to the right of the decimal is 3 (32.5 has 1 digit to the right of the decimal and .65 has 2 digits to the right of the decimal).
To perform long division on decimal numbers, the position of the decimal point in the dividend (the dividend is divided by the divisor) indicates the position of the decimal point in the answer. If the divisor has a decimal component, then before you carry out the operation shift the decimal to the right an equal number of digits in both the divisor and the dividend such that the divisor no longer has a decimal component. If the divisor has more digits to the right than the dividend, you will add zeros to make up the difference:
Converting Fractions to Decimals
Converting a number from fractional notation to decimal notation is very similar to the conversion between improper and proper fractions - perform the division implied by the fraction:
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