Topic outline
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In this course you will learn about:
- Ratio and rate.
- Direct and indirect proportion.
- Simple interest.
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To compare quantities of the same type we use ratios.
Example
A recipe requires 1 cup of water for every 2 cups of flour.
This means that the ratio of water to flour (in that order) is \(1:2\) or we can write the ratio as a fraction as \(\frac{1}{2}\) or \(1\) to \(2\).
Only quantities of the same kind can be compared using ratios. For example, litres with litres, miles with miles, grams with grams, cups with cups, and so on.
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Working with ratio and rate
Some ratios can be simplified to an equivalent form. For example, \(5:10\) can be simplified to \(1:2\).
\(\frac{5}{10}=\frac{1}{2}\) by dividing the numerator and denominator by \(5\) we can simplify the fraction and rewrite the fraction as its equivalent \(1:2\).
Rate
To compare different types of quantities we use rate. An example is the speed of a car, which is measured in kilometres (km) per hour.
For example, if a car uses 2 litres of petrol to travel 40 km then we can find the amount of petrol needed for 20 km by dividing both the litres and kilometres by 2. We find that the rate is \(1:20\) so the car uses 1 l of petrol to travel 20 km.
Proportion
When ratios are equal we say they are in proportion. We use proportion to solve for unknown values in a ratio equation.
Example
To make an orange juice drink you must mix juice concentrate with water in the ratio \(1:3\). You need to make enough for an entire class and you have \(500\) ml of orange juice concentrate, how much water should you use?
You can set up the proportion equation to calculate the amount of water required:
In the next video you will learn how to apply ratio and rate to answer questions.
Try the next activity to make sure you understand how to answer questions on ratio and rate. Some of the questions in the activity include examples before the question.
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Direct proportion
If one quantity increases as another quantity increases or decreases as the other decreases then they are directly proportional. For example, as the number of items you buy increases the amount you pay increases too. So, the numbers of items and cost are directly proportional.
Two quantities \(x\) and \(y\) are in direct proportion if the ratio \(x:y\) is always the same.
Indirect proportion
If one quantity decreases as another quantity increases then they are inversely (or indirectly) proportional.
Two quantities \(x\) and \(y\) are indirectly proportional when their product \(xy\) is constant.
Proportion is also referred to as variation. In the next video you will go over many examples to learn to identify direct or indirect proportion.
Check your understanding of direct and indirect proportion by trying the activity: Recognise direct and indirect variation. [Edit: I suggest putting this activity in a separate box with a link] -
When you buy and sell items there are costs or expenses involved.
Example
I sell baskets on marketplace for \($8\) each. If I sell 1 basket will I make a profit on it? To answer this question we must consider how much it costs me to buy the basket. This is called the cost price of a sale. If the cost price of the basket is \($5\) then I will make \($8-$5=$3\). So, I will make a profit on the basket since the selling price is greater than the cost price.
I can also calculate the percentage profit made on the sale of 1 basket as follows: \(\frac{3}{5}\times 100=60%\)
Profit: If the money received is more than the expenses or the initial costs, then you make a profit.
\(Profit = Selling price - cost price\)
Loss: If the money received is less than the expenses or initial costs, then you make a loss.
\(Loss = Cost price - selling price\)
No matter which currency is used the method for calculating profit or loss is the same.
This video will go over a few more examples of profit and loss. Try the activity that follows the video to make sure you understand the concepts of profit and loss.
Try the activity Profit and loss word problems to test your understanding. [Edit: I suggest putting this activity in a separate box with a link] -
There are two different ways in which interest is calculated; simple interest or compound interest. In this section we will go over simple interest.
Simple interest is calculated on only the initial or principal amount invested or borrowed. The interest received or charged for each period will always be the same.
The formula to calculate the accumulated (or final) amount earned using simple interest is:
Example
Determine the value of an investment of \($13 000\) at 12% per year (also called per annum (p.a.)) simple interest for three years.
\( A=13 000(1+0.12 \times3) =$17 680 \)
You will also come across the formula \(SI=P\times r\times t\) to calculate simple interest. The following interactive video will take you through the steps to change the subject of the formula for simple interest using the alternate formula. Make sure you answer the questions on the video.
Exercise: Simple interest
Try this exercise to test that you understand how simple interest works.-
Applications of simple interest
A hire purchase (HP) agreement, also known as an installment plan, is an application of simple interest. In HP agreements a person agrees to buy an item at a certain interest rate over a stated period and will usually pay a deposit to secure the item.
In the video you will learn how to apply the simple interest formula. After you have watched the video try the exercise that follows to test your knowledge.
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