# Page 2 Mathematics: Computation Study Guide for the TABE

#### Decimals

##### Addition and Subtraction

The numbers to be added or subtracted are written vertically - digits are aligned according to their place value, just like when you add non-decimal numbers. All decimal points are aligned, as well.

Addition follows in much the same manner as the addition of whole numbers shown earlier, except for the decimal point which separates the whole and decimal numbers. Regrouping will be necessary if a column’s sum is over 9, just like in regular addition.

Subtraction of decimal numbers is similar to subtraction of whole numbers, too. Regrouping will also be necessary when a digit is larger than the digit it must be subtracted from.

##### Multiplication

Multiplying decimal numbers starts in the same way we multiply whole numbers, and differs only in the last few steps. Simply multiply the numbers, ignoring the decimals for now.

If you multiply 7.65 by 9.8, you would work as if you were multiplying 765 by 98. The answer would be 74970.

Count the total number of decimal places in both the multiplier and the multiplicand: $1 + 2$ decimal places. Move the decimal 3 places from the right toward the left, and get the final answer of $74.970$. The last zero(es) after the decimal can be dropped and the number will retain its value, giving you $74.97$

##### Division

Division of decimals is similar to division of whole numbers, except for the presence of the decimal point and a few added steps.

• First, move the decimal point in the divisor to the right until it becomes a whole number.
• Move the decimal point in the dividend to the right the same number of places that you moved the decimal in the divisor.
• Now, put a decimal point in the quotient area, directly above its spot in the dividend. That is where it will be in the answer.

Note: You may add zero and proceed with the division up to the required number of decimal places.

#### Fractions

##### Addition and Subtraction

Fractions with the same denominators:

Add or subtract the numerator of fractions with the same denominators as usual, then copy the common denominator.

Fractions with different denominators:

To add or subtract fractions with unlike denominators, first find the least common multiple (LCM) of the denominators. Rename or find the equivalent of the fractions with the common denominator, then proceed to addition or subtraction.

Example: $\frac{1}{7} + \frac{2}{3}$

The LCM of 7 and 3 is 21.
To find the fraction equivalent to $\frac{1}{7}$ with the denominator of 21, multiply the fraction by $\frac{3}{3}$:

Now multiply the fraction $\frac{2}{3}$ by $\frac{7}{7}$ to get the fraction of $\frac{14}{21}$.

We may now proceed to addition of fractions with common denominators.

The same method is applied when subtracting fractions with unlike denominators, except that numerators are subtracted in the last step.

##### Multiplication

Multiplying fractions is quite straightforward - multiply the numerators and multiply the denominators to get the product of the fractions.

##### Division

Division of fractions involves two steps. First get the reciprocal of the divisor. Then proceed to multiplication of fractions.

#### Integers (Positive and Negative Numbers)

##### Addition and Subtraction

To add integers with like signs: add in the usual manner, then affix the common sign to the sum.

To add integers with unlike signs: get the difference of the integers, then affix the sign of the larger integer.

To subtract integers: first change the sign of the subtrahend, then proceed to addition of integers.

##### Multiplication and Division

Multiply and divide integers with like signs as usual, then affix a positive sign (or no sign).

Multiply and divide integers with unlike signs as usual, then affix a negative sign.

#### Percents

Presenting a number in the percent form is a way of showing the value of a part compared to a whole, or the quantity for every hundred. Percent can be converted to a fraction or to a decimal and back. For instance, 50% is a way to show that there is 50 for every hundred, or 0.50, or $\frac{1}{2}$ of something.

To change 40% to fraction form, write 40 as the numerator and 100 as denominator, then reduce to the lowest equivalent fraction:

To change 40% to its decimal form, simply divide 40 by 100:

Some TABE questions will require the computation of the percentage of a given number. For example, to solve for 35% of 60, simply multiply 0.35 by 60:

Percent questions are often simplified by this useful equation:

Example: 77 is what % of 92?

#### Order of Operations

Lengthy and seemingly complicated mathematical operations can be simplified by following the PEMDAS rule which states the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Start with performing the operations inside the parentheses:

$(4+1) = 5$ and $(5^2 \cdot \frac{1}{5}) = \frac{25}{5} = 5$

The expression then becomes:

#### Operations in Algebra

In algebra, we solve unknown values with the use of other known values and relationships between these values. Unknown values are often assigned letters called variables. We don’t know their values yet, hence, we assign x, y, or any letter to represent them momentarily. The equal sign (=) is placed between expressions on the left and right to denote that the expression on the right is equal to the expression on the left.

In this equation, for instance:

$x + 3 = 7$, we mean that if x is added to 3, we get 7.

Mathematical operators, such as subtraction (-), addition (+), multiplication (x), and division () are used to define relationships between values. Instead of the equality sign (=), inequality signs such as less than or equal (), greater than or equal (), less than (<) and greater than (>) can also be used.

Think of algebra as solving puzzles and finding values based on clues given. Be familiar with buzzwords, too.
Less than, take away, and less indicate subtraction.
More than, total, all in all, combine and sum indicate addition.
Twice, of, times, and product indicate multiplication.
Ratio, out of, and per indicate division.