Math Study Guide for the CBEST

Numerical and Graphic Relationships

Comparing Data

Comparing change in data is often presented as percentage change, as in the comparison between a new value and an old value.

The improvement of a student after a teacher’s intervention, for example, can be measured by allowing the student to take a pre-intervention test and a post-intervention test. If the score in the pre-test is 65 and in the post-test is 88, the percentage change is solved using the following formula:

\[\text{Percentage Change} = \frac{\text{(New value – Old value)}}{\text{Old value}} \cdot 100\%\] \[\text{Percentage Change} = \frac{(88 – 65)}{65} \cdot 100\% = 35.38\%\]

Number Order

When comparing numbers, we determine their relative position on the number line―the numbers on the right are larger than numbers on the left. The number 10, for example, is to the right of 5; hence, 10 is greater than 5 ($10 > 5$). Comparing fractions, though, will not be as easy, but there are two common methods used for making such a comparison.

Convert the fractions to decimals and then compare.

Example: Compare ½ and ⅔ by converting them to decimals.

\[\frac{1}{2} = 0.5\] \[\frac{2}{3} = 0.67\] \[0.5 < 0.67\]

Therefore: $\frac{1}{2}$ < $\frac{2}{3}$

Hint: When you are comparing two or more decimals, write the numbers in column form with the decimal points aligned vertically. This way you will be able to compare the values of the numbers in each place accurately.

Convert to equivalent fractions with the same denominators.

Example: Compare ½ and ⅔ by converting them to equivalent fractions with common denominators.

$\frac{1}{2}$ = $\frac{3}{6}$

$\frac{2}{3}$ = $\frac{4}{6}$

\[3 < 4\]

Therefore: $\frac{3}{6}$ or $\frac{1}{2}$ < $\frac{4}{6}$ or $\frac{2}{3}$

Equalities and Inequalities in Data

The concept of equality or equations expands to inequalities. A term or expression can be equal (=), less than (<), or greater than (>) another term or expression.

As in equations, statements of inequality can be used to solve unknown values in mathematical problems. Solving inequalities is quite similar to solving equalities except for some rules distinct to inequalities. Here is a very important rule:

When a negative sign is introduced to the statement of inequality either by multiplying or dividing, the symbol of inequality reverses its direction.

Example:

\[-5x < 25\]

Multiplying both sides by $-\frac{1}{5}$ and reversing the inequality, we get:

\[x > -5\]

Equivalencies

Many math problems can be solved easily if you are familiar with the many forms in which a mathematical expression can be written. The fraction ¼, for example, is the same as 0.25 and 25%. It is also equivalent to the fractions 25/100, 7/28, and countless other equivalent fractions. Knowing how to “translate” numbers to their many forms shortens the time needed to solve some problems.

Example: 25% of 500 is ___?

Knowing that 25% is the same as ¼, you may actually just mentally divide 500 by 4 to get 125 (divide 500 by 2 = 250, then further divide 250 by 2 = 125). That’s a lot easier and quicker than multiplying 500 by 25%.

Rounding, Connectives, and Quantifiers

The key to solving math problems is practice. With constant practice, you develop techniques that become handy when faced with test questions and even real-life problems. Here are some tips to remember:

Rounding of numbers is useful when estimating or simplifying computation by using numbers with fewer digits (or rounded) while staying close to the value of the rounded numbers. Observe the rules for rounding numbers―round up from 5 to 9, and round down from 0 to 4.
Be attentive to the language and mathematical statements used. Connectives, such as and, or and if-then, and quantifiers, such as none, some, and all, imply important information.

Example: Is the following statement true or false?

If x is an integer and $x > 1$, then $\frac{1}{x-1}$ is a rational number.

The “if” statement says that: x is an integer and $x > 1$.

The “then” statement concludes that:

$\frac{1}{(x-1)}$ is a rational number.

This fraction will become undefined if the denominator is zero, which is what happens if x equals 1. If x is always greater than 1, the fraction remains a rational number. A rational number can always be expressed as a fraction with integers in the numerator and denominator. So the answer is true.

Missing Entries

Some math problems may ask you to fill in missing numbers in a given table of data. Here are steps to help you solve this type of problem:

Read the title or name of the table.
Identify the headings or labels.
Determine what the numbers in the rows and columns represent.
Figure out trends and relationships from one figure to another.

Example: Fill in the incomplete data in the fields marked A, B, C, and D.

Carlo and Barney’s Savings for 1 Month

Name	Week 1	Week 2	Week 3	Week 4	Total
Carlo	$5	A (?)	$12	$4	$23
Barney	$3	$10	$5	C (?)	$23
Total	$8	$12	B (?)	D (?)	$46

The correct answer is: A = 2; B = 17; C = 5; D = 9.

Using Data in Problem-Solving

Data provided in tables and graphs will help you answer questions. In the table given above, for instance, a question may ask:

“In what week was the boys’ combined savings the highest? the lowest?”

With the table filled correctly, it will now be easy to find that the highest combined savings was during Week 3, while the lowest was during Week 1.

A question may not be given directly about the table, but the trend or relationship in the table can be used to answer the question. A question may go like this:

“If Carlo saved three times than what he did on Week 1, but $3 dollars less than half of what he saved on Week 3, what would be the total combined savings after Week 4?”

The correct answer would be 47.