This page takes you through some important concepts on the topics of number and algebra, with activities to consolidate your knowledge along the way. In particular, it covers how to do the following (use the drop-down menu above to jump to a different section as required):
For more sample questions on these concepts, in particular if you are preparing for the Literacy and Numeracy Test for Initial Teacher Education (LANTITE), you may like to visit the LANTITE page. Alternatively, if you would like to build up some more advanced skills on the topic of algebra, you may like to visit the Algebra module.
Finally, as you work through the content keep in mind that there are often different ways of solving the same problem. While in some cases alternative methods have been explained, for many examples just one method of working is provided. If you already know a different technique for solving the problem then that’s great (provided it is also correct, of course). Don’t feel you need to change your method - the important thing is that you use a technique that makes sense to you.
Are you a Curtin student who would like to work through a module and gain a certificate for learning more about key numerical skills? Access the Maths 4 University module.
Knowing what order to perform operations in is essential in order to obtain a correct answer for any problem involving multiple operations. Similarly, it is important to know how to use the order of operations to write expressions correctly so that others (including your calculator!) will evaluate them as you intended. The rule for this is known by a few different acronyms, one of which is BIMDAS. This stands for:
Example: Michelle the maths teacher would like to purchase a licence for some maths software for her students to use for a ten week term. The licence has a
Solution: One way to figure this out is to consider that for each student, it is going to cost
Next we need to multiply the cost per student by
Plugging this into a calculator (or you could always solve it without the aid of a calculator if wished, but this almost certainly wouldn’t be a requirement), and the answer for the total cost is
Alternatively, another possible formula for the total cost is
Once you have worked through the example above, have a go at using BIMDAS to evaluate the problems in the following activity:
A decimal number is a number with two parts; the whole number part is the part to the left of the decimal point, while the part to the right is the fractional part.
Note that if the whole number part of a decimal number is
Being able to round decimal numbers is important as often it is impractical and unnecessary to state them in their entirety. Furthermore, in the case of irrational numbers like
Note however that when solving a problem involving multiple operations you should retain all accurate values as you work and not round until your final answer, in order to avoid an inaccurate solution.
When rounding a decimal number to a certain number of decimal places, or to the nearest whole number, you should follow these steps:
Find the relevant ‘round-off place’ in your number. In general, if you are rounding to
Determine the number that is one place to the right of your round-off place.
If this number is greater than or equal to
Example: Suppose you want to round
Solution: Following the steps above, we have:
Since we are rounding to two decimal places the round-off place is the number in the second decimal place, which in
The number one place to the right of this round-off place is the number in the third decimal place, which is
Since
Once you have worked through the example above, have a go at the rounding problems in the following activity:
A percentage refers to the number of something out of
You can calculate a percentage of a given quantity (including determining percentage increase and decreases), and you can also express one quantity as a percentage of another.
In certain situations, such as when finding a percentage increase or when expressing one quantity as a percentage of another, it is possible for the percentage to be more than
Percentages are commonly used to analyse and compare data, and to express things such as sales discounts, commissions and interest rates.
Since a percentage represents an amount out of
Example: Suppose that
Solution: To calculate
Hence
As an extension of this, often you may wish to calculate the result of a percentage increase or decrease. For example, the new price of an item after a
First calculate the specified percentage of the original value, as described previously, then either add (for an increase) or subtract (for a decrease) this value from the original value; or
First add (for an increase) or subtract (for a decrease) the specified percentage from
Example 1: Suppose you want to find the total cost of a
Solution: One way of calculating this would be as follows:
So the total cost of the computer is
Example 2: Suppose you’d like to buy a TV, and you find one that has a
Solution: One way of calculating this would be as follows:
So the new price of the TV will be
Alternatively, sometimes you may be required to calculate an original value before a percentage was applied, given a new value. For example, calculating the original price of an item before a
You can do this by adding (for an increase) or subtracting (for a decrease) the specified percentage to or from
Example: Suppose you know that the cost including GST of a tablet is
Solution: To calculate this you would do:
So the original price of the tablet was
Once you have worked through the examples above, have a go at the percentage problems in the following activity:
While you can generally use a calculator to evaluate percentages using the formulas above, it is also good to have an understanding of how to approximate percentages without a calculator. You can do this by performing combinations of the following as required, using rounding to aid in the calculations when needed:
To calculate
To calculate
To calculate
To calculate
To calculate
Example: Approximate
Solution: You could do this as follows:
So
Once you have worked through the example above, have a go at completing the percentage problems in the following activity without the aid of a calculator:
Another way you might need to use percentages is to express one quantity as a percentage of another. To do this, you simply need to divide the first quantity by the second and then multiply by
Example: Suppose that a student scored
Solution: To calculate what percentage
So the student scored
Finally, sometimes you may wish to calculate a percentage change; for example the percentage change on a power bill from the previous bill amount to the current bill amount. To do this you should first subtract one value from the other to find the difference, then divide this by the original amount and finally multiply the result by
Example: Suppose a student obtains a mark of
Solution: To calculate the percentage change from
So the student’s mark has increased by
Once you have worked through the examples above, have a go at the percentage problems in the following activity:
A fraction is just part of a whole number. The bottom number of the fraction (the denominator) tells you how many parts the whole is divided into, while the top number (the numerator) tells you how many parts you have.
An understanding of fractions is required for many every day concepts, from cooking, to telling the time, reading maps, calculating discounts and more. Furthermore, knowledge of how to work with fractions is a vital skill required for many other mathematical concepts, from ratio and proportion to algebra, geometry and statistics.
Some of the things you may commonly be required to do with fractions are to simplify them, convert them to equivalent fractions, add or subtract them and multiply or divide with them. While many of these operations can be performed using the fractions feature on a calculator, it is also good to have an understanding of how to evaluate them manually - and this section details how. First though, some important things to note about fractions are as follows:
If the numerator of a fraction is smaller than the denominator, the fraction is less than
If the numerator of a fraction is bigger than the denominator, the fraction is greater than
If the denominator of a fraction is
If the numerator and denominator are the same, the fraction is equal to
Unless otherwise stated, fractions should always be given in their simplest form - that is, using the smallest numbers possible. You can convert a fraction to simplest form by following these steps:
Determine the highest common factor of the numerator and denominator (i.e. the largest number that divides into both a whole number of times). Note that you may need to list out all the factors of both the numerator and denominator in order to find this, as unfortunately there is no ‘magic’ way of doing it. Alternatively, rather than trying to find the highest common factor you can use any common factor of the numerator and denominator (i.e. any number that divides into both a whole number of times). This just means you will need to repeat these steps until there are no more common factors. Note that when doing this it is generally easiest to look for ‘easy’ factors such as
Divide both the numerator and denominator by this highest common factor or a factor to obtain a new numerator and denominator. If the latter is used, repeat as necessary until there are no more factors.
Example: If
Solution: Following the steps above, we have:
The factors of
Since the largest number on both these lists is
Dividing the numerator of
So
The fractions
Divide the denominator of your incomplete fraction by the denominator of your complete fraction.
Multiply the numerator of your complete fraction by the value obtained previously.
Note that you can always do the division in the first step the other way around if you prefer, in which case you will need to divide in the second step instead of multiplying (if you are ever in doubt about whether to multiply or divide in the second step, take note of the fractions and whether it would make sense for the new denominator, or numerator, to be bigger or smaller than for the original fraction).
Alternatively, you may want to work out the unknown value by multiplying the known value in the incomplete fraction (e.g. the denominator) by the ‘opposite’ value in the complete fraction (e.g. the numerator), and then dividing by the remaining value (e.g. the denominator of the complete fraction).
Example: If a student scores
Solution: Following the steps above, we have:
The denominator of the incomplete fraction is
So
The numerator of the complete fraction is
So
Alternatively, you could multiply
Once you have worked through the examples above, have a go at the fraction problems in the following activity:
When it comes to adding or subtracting fractions, there are two different types of problems you need to consider. The first is when the fractions have like denominators (i.e. the same denominators). When this is the case, adding or subtracting them is straightforward as you just add or subtract the numerators of the fractions, and keep the denominator of your answer the same as the original fractions.
Example: Nick eats
Solution: Adding the three fractions simply requires adding the numerators, as follows:
So
Adding or subtracting fractions with unlike, or different, denominators requires more work, as you need to convert the fractions to equivalent fractions with like denominators first before you can add or subtract them. You can do this by following these steps:
Determine the lowest common denominator of all fractions in the problem (i.e. the smallest number that all the denominators can divide into a whole number of times). If you can’t find this you can always just use the product of the denominators, which is always a common denominator but may not be the lowest.
Convert each fraction in the problem to an equivalent fraction with denominator as specified.
Once your fractions all have the same denominator, you can add or subtract them as usual.
Simplify your result if necessary.
Example: David has a big pile of assignments to mark, which he spreads over a number of days. He marks
Solution: Following the steps above, we have:
The three denominators in the problem are
The fractions can be converted to equivalent fractions with denominators of
The sum now becomes
This fraction cannot be simplified, so the fraction of assignments marked is
Sometimes you may be required to find a fraction ‘of’ some number, such as
To do this when the number in question is also a fraction, you can multiply the two fractions together by simply multiplying the two numerators and the two denominators to make a new fraction (simplifying as required). On the other hand, if the number to multiply by is an integer then you can simply multiply the integer by the numerator of the fraction and divide by the denominator, in whichever order is easiest to calculate and makes sense to you.
Example: Suppose you are considering taking a job that is part-time, and you would be working
Solution: This requires you to calculate
You can do this in a number of ways (note it doesn’t matter which way around you write the numbers being multiplied, so do it in a way that makes sense to you). For example:
Method 1:
Method 2:
Method 3:
So your part-time salary would be
Furthermore, if you can multiply with fractions then you are already halfway there (no pun intended!) when it comes to dividing with fractions, as dividing by a fraction with a certain numerator and denominator is the same as multiplying by the same fraction with the two values switched around (i.e. by the reciprocal). This works because multiplication and division are opposite operations.
Example: Suppose you have a piece of string that is
Solution: This requires you to divide
You can do this in any of the ways detailed above, for example:
So you would end up with
Once you have worked through the examples above, have a go at the fraction problems in the following activity:
Like fractions and percentages, a ratio is a way of comparing two or more numbers. For example, if a class contains
Ratios such as this compare values that are in the same units, and therefore the units are excluded from the ratio as we are comparing the numbers only. A rate, on the other hand, is a special type of ratio that compares two quantities with different units of measurement. When describing a rate, the word ‘per’ is used between the two measurements; for example the price of a particular food stuff may be
Just as fractions are usually expressed in simplest form, ratios are usually expressed in lowest terms. For example, if there are
Similarly, rates are typically expressed as unit rates, where the second term in the rate is one. For example, the cost of the same food stuff as a unit rate is
When two ratios or rates are equivalent, we say that they are in proportion. For example, the ratios
Finding unknown values to ensure ratios or rates are in proportion is a common requirement, and this can be done in much the same way as finding unknown values in equivalent fractions (as detailed in the Fractions section). The key thing to remember is that in order to be in proportion, the numbers in the ratios or rates need to be multiplied or divided by the same value.
Example: Consider that an excursion is planned for a year two class of
Solution: To calculate this, first note that there is one complete ratio (
To find the missing value so that the ratios are in proportion, one way is to write the ratios on separate lines:
This way it is easy to see that you need to divide the two numbers that are above one another to find out the relationship between the ratios, and then apply this relationship to the other side of the ratios.
So in this case
Alternatively, you could multiply
Once you have worked through the example above, have a go at the ratio, rate and proportion problems in the following activity:
Sometimes you may be required to calculate a simple average given a total and a number of people or things. To do this, you simply need to divide the total by the number of people or things.
Example: Suppose that in a school reading challenge a class of
Solution: To calculate the average books per student when
So the average number of books read per student is
Once you have worked through the example above, have a go at the average problems in the following activity:
Having an understanding of some basic algebra is handy in order to solve problems involving unknown values. This section details how to write simple equations and how to substitute values in order to solve them. If you would like to learn more about algebra, including techniques for rearranging in order to solve equations, please refer to the Algebra module.
Writing a simple algebraic equation involves using letters or symbols to represent unknown values (called variables) and putting these together in an equation, complete with an equals sign and the relevant mathematical operations, to represent the relationship between them.
Example: Suppose you are a teacher planning a school incursion. You investigate prices and find an incursion you like which costs
Solution: In this case our unknown values are the number of students and the total cost. We could use the letter
Generally, the purpose of writing algebraic equations is to solve them to find out unknown values. While this can be done for more than one variable at a time, note that this is more complex and requires the equivalent number of equations which is not covered here. Instead, we will focus on using one algebraic equation to find one unknown value, and in particular at how to do this by substituting values in for any other variables so that the only remaining variable is by itself on one side of the equals sign.
Example: Consider the equation from the previous example. If there are
Solution: You can solve to find the total cost by substituting (i.e. replacing) the
So the equation becomes
Alternatively, if you wanted to find out how many students could attend the incursion given a set budget, this would involve rearranging the equation so that the
Once you have worked through the examples above, have a go at the algebra problems in the following activity: