Math Study Guide for the CBEST

Computation and Problem Solving

Operations with Whole Numbers

Math problems involving addition, subtraction, multiplication, and division of whole numbers are much easier to compute compared to fractions, decimals, and radicals. Whole numbers are the numbers from zero to infinity, with no fractions or decimals. They are sometimes called natural numbers. Counting numbers are also whole numbers, but exclude zero. To operate with whole numbers, always take note of the digits’ place value.

Examples:

When adding 50,560 and 175 vertically, align 560 and 175. The same reminder is applicable when subtracting.

When multiplying 1234 by 56 manually, always start multiplying from the ones place value to the tens and higher place values.

When dividing, always check your answer by multiplying the quotient (your answer) with the divisor (the smaller of the two numbers), and see if you get the dividend (the bigger number). If so, your division was executed correctly.

Operations with Integers

When referring to both positive and negative whole numbers, the term integer is used. The numbers 100, 435, –5, and –77 are integers, while 99 ½ and –3.25 are not.

Adding Integers

To add integers with like (the same) signs, use the usual method of adding numbers and affix the common sign to the result:

Adding positive integers: 9 + 7 = 16 Adding negative integers: -9 + –7 = –16

To add integers with unlike (different) signs, subtract the smaller integer from the bigger integer and affix the sign of the larger integer to the result:

Small positive integer + larger negative integer: 7 + (–9) = –2 Large positive integer + smaller negative integer: 9 + (–7) = 2

Subtracting Integers

Subtracting integers sometimes sounds confusing. There is one general rule, however, that applies to all cases of subtracting integers:

Rule: Change the sign of the subtrahend and proceed to addition (following the previous rules for adding integers).

Examples:

Large positive integer – smaller negative integer: 9 – 7 = 9 + (–7) = 2
Small positive integer - larger negative integer: 7 – 9 = 7 + (–9) = –2
Positive integer – negative integer: 7 – (–9) = 7 + 9 = 16
Negative integer – positive integer: –7 - (–9) = –7 + 9 = 2

Multiplying and Dividing Integers

Doing multiplication and division with integers is really similar to performing these operations with whole numbers. The only difference is that you need to determine the sign appropriate for the answer when you have finished. To do so, remember these guidelines:

Two numbers with like signs (+ and + or - and -) result in a positive answer.
Two numbers with unlike signs (+ and -) result in a negative answer.

Fractions, Decimals, and Percentages

Fractions, decimals, and percentages are other ways of representing numbers that are not whole numbers or integers.

A fraction represents part of a whole and is written with a numerator (the part) over a denominator (the whole). The fraction ⅔ refers to 2 parts of a whole, which is made up of 3 parts. This concept of fractions is similar to decimals and percentages.

The fraction ⅔, for instance, can be expressed as a decimal by dividing 2 by 3. The result is 0.67, which is the decimal equivalent of ⅔. Take note that we only used 2 decimal places or rounded to the nearest hundredths. Should the question specify 3 decimal places, the correct result should be 0.667.

To express ⅔ or 0.67 as percentage, do the following:

\[\frac{2}{3} = 0.67\] \[0.67 \cdot 100\% = 67\%\]

Operations with Fractions

To add or subtract fractions, make sure that they have a common denominator. If they don’t, then determine the least common denominator (LCD) first.

\[\frac{1}{2} + \frac{2}{3} = \frac{(3+4)}{6} = \frac{7}{6}\]

To multiply fractions, multiply numerators first, then multiply denominators next. Reduce the resulting fraction to its lowest form.

\[\frac{1}{2} \cdot \frac{2}{3} = \frac{2}{6} = \frac{1}{3}\]

To divide fractions, take the reciprocal of the divisor and proceed to multiplication.

\[\frac{1}{2} ÷ \frac{2}{3} = \frac{1}{2} \cdot \frac{3}{2} = \frac{3}{4}\]

Operations with Decimals

Mathematical operations involving decimals are simpler and more similar to operations involving whole numbers, except that there is the decimal point to consider.

When adding vertically and doing subtraction, make sure that the decimal points are aligned.
When multiplying, count the number of decimal places in the factors because this will determine the number of places in the product. To multiply 1.05 by 5.25, multiply as you would whole numbers:

\[105 \cdot 525 = 55125\]

Then count the number of places in both factors (there are four decimal places). Move four decimal places to the left in the result of 55125, and you get 5.5125, the correct answer.

When dividing decimals, it’s easier to convert them first to whole numbers before proceeding. To divide 6.54 by 3.2, for example, move the decimal place in 6.54 two places to the right to make it a whole number (654). Move the decimal in 3.2 the same number of places to the right to make it 320. Divide 654 by 320 and get the result of 2.04.

Operations with Percentages

Percentages can be added and subtracted as they are:

\[50\% + 40\% = 90\%\]

Percentages are seldom multiplied or multiplied, but if a question asks for such an operation, it will be easier to convert them into decimal or fraction forms first before proceeding to multiplication or division.

Practical Problem Solving

Many math problems are so practical that they don’t need specific formulas. For instance, a word problem may give the unit cost of Brand A notebooks as $\$0.90$ and Brand B notebooks as $\$1.25$, and ask for the total amount paid if a dozen of notebooks is bought per brand. This type of problem is ordinarily encountered in everyday life and can be solved by simply multiplying $12$ and $0.90$ and adding this to the result of $12$ multiplied by $1.25$.

Algebraically, this can be represented by:

\[x = (12 \cdot 0.90) + (12 \cdot 1.25) = 12 \cdot (0.90 + 1.25) = 25.80\]

where $x$ is the sum paid for the purchase.

Analyzing Mathematical Problems

When solving word problems, follow these simple steps:

Read the whole problem.
Reread if you don’t get it the first time, sketching as you go along.
Determine what data is given in the problem.
Find out what the problem is asking for.
Assign variables to the unknown.
Work out a formula, or recognize a formula related to the problem given.

Figure out mathematical problems by recognizing the math equivalent of English words commonly used in math, such as the following:

is, was, are, were―indicate equals or equal to
sum, total, more than, added to, plus, increased by, and combined with―indicate addition
difference of, difference between, less, less than, fewer than, fewer by, taken away from, minus, reduced by, and decreased by―indicate subtraction
of, times,* multiplied by, *product, and factor―indicate multiplication
out of, per, quotient, and ratio―indicate division