Have A Question?
Get in touch!
A ratio is a way of comparing two or more quantities of the same kind. It shows the relative sizes of the quantities. The order of the numbers in a ratio is important. 2:3 is different from 3:2. An example using ratios is this: In a fruit bowl, there are 3 apples and 2 oranges. The ratio of apples to oranges is 3:2.
Ratios can be written in several ways:
Sometimes, you might have a ratio with big numbers like: 482 to 50, perhaps when you’re trying to get the ratio for a certain colour of paint. These numbers are big and harder to manage. So, it’s important we learn how to simplify ratios as it makes it easier to work with.
To simplify a ratio, find the greatest common factor (GCF) of all the numbers in the ratio. The GCF is the largest number that divides evenly into all the numbers. Then, divide each number in the ratio by the GCF.
For example, let’s simplify the ratio 12:18.
Can you see that ratio is easier to process and can be scaled up to larger amounts if needed? It’s a ratio that’s easier to work with in the context of doing further calculations.
Let’s do some easy questions:
Solutions:
Now, let’s do some harder questions, the type of questions that require critical thinking:
Question 1: The Concert Ticket Sales
A concert venue is divided into three sections: A, B, and C. The ratio of the number of tickets sold in section A to section B is 4:5. The ratio of the number of tickets sold in section B to section C is 3:2.
If there were 60 more tickets sold for section B than for section C, how many tickets were sold for section A?
Solution:
Question 2: The Juice Mixture
A juice stall sells a special fruit punch made by mixing apple juice and orange juice in the ratio of 2:3. The stall owner makes a large batch of the punch and uses a total of 30 litres of apple and orange juice combined. He then adds pineapple juice to the mixture. The amount of pineapple juice is double the amount of apple juice used.
How many litres of pineapple juice did the stall owner use in the mixture?
Solution:
Now that you have gone through ratios, we’ll learn about rates. A rate is a special type of ratio that compares two quantities with different units.
One common way to see a rate is when you go into the grocery store or if you do your shopping online, you’ll see it where the product pricing is. They may express the pricing as $2.00/100 grams for a certain type of toothpaste and $4.39/100 grams for another type of toothpaste. These rates allow you to compare what’s more expensive per 100grams of the product. It kind of gives you the ‘real’ cost rather than the overall price.
For example, let’s say one tube of toothpaste costs $4.39 but the tube only contains 100gram. But another tube is larger and has 300grams but costs $6.00. With rates, you’ll see that the larger tube costs $2.00 per 100grams, so it is actually ‘cheaper’ than the first tube, even though the absolute sales value is more. Rates help you make those decisions that are more considered and finely tuned.
There is a type of rate called a unit rate, this is where the second quantity is 1 unit. For example: Speed: kilometres per hour (km/h); Price: dollars per kilogram ($/kg); Heart rate: beats per minute (bpm).
To calculate rates:
Let’s go through an example:
A car travels 200 kilometres in 4 hours. What is its average speed? Speed = Distance / Time = 200 km / 4 hours = 50 km/h
Let’s do some practice questions, this time with application to a scenario, just because rates can be tricky!
Question 1: The Urgent Print Job
A printer prints 120 pages in 3 minutes. The printer has a warm up time of 15 seconds. If a person needs to print only 6 urgent pages, how long will it take from the moment they press "print" until the last page is printed? Express your answer in seconds.
Thought Process:
Answer: It will take 24 seconds to print 6 pages, including the warm-up time.
Question 2: The Worker's Overtime
A worker earns a base wage of $160 for 8 hours of work. For every hour of overtime worked beyond the standard 8 hours, the worker earns 2 times their regular hourly wage. If the worker earned $280 on a particular day, how many hours did the worker work in total that day?
Thought Process:
Answer: The worker worked 11 hours in total that day.
Time-distance problems are fundamentally related to ratios and rates.
Speed is a rate that represents the distance travelled per unit of time (e.g., kilometres per hour). This is essentially a ratio comparing distance and time.
When solving time-distance problems, you're often manipulating the ratio of distance to time (or the rate of speed) using proportions to find a missing value – such as calculating the distance travelled when given the speed and time, or finding the time taken when given the distance and speed.
Let’s see what formula you need to know when solving these types of questions:
Let’s look at an easy question: A car travels at a constant speed of 70 km/h for 3 hours. How far does it travel? The answer: Distance = Speed * Time = 70 km/h * 3 hours = 210 km
Solving distance-time problems requires a few steps and these are:
Let’s do an easy question: A train travels 400 km at a constant speed and takes 5 hours to complete the journey. What is the train's speed? Answer: Speed = Distance / Time = 400 km / 5 hours = 80 km/h
Let’s now do some harder questions to see distance-time problems in applied situations:
Question 1: The Train's Journey
A train travels from City A to City B. The distance between the cities is 480 km. The train travels at a constant speed of 80 km/h for the first 2 hours of the journey. Due to track maintenance, it then reduces its speed to 40 km/h for the remainder of the journey.
What is the total time taken for the train to complete the journey from City A to City B?
Solution:
Question 2: The Runner's Race
A runner is training for a 15 km race. For the first 6 km, the runner maintains a constant speed of 12 km/h. The runner then runs the remaining distance at a constant speed, and they complete the entire 15km in 1 hour and 15 minutes, how fast did the runner run for the remaining part of the race?
Solution:
Now, it's your turn to practice.
The questions in this checkpoint are provided to give you an introduction to possible questions you may see in your exam. Don't worry too much as you'll continue to build your skills throughout the course.
Click on the button below and start your practice questions. We recommend doing untimed mode first, and then, when you're ready, do timed mode.
Every question has a suggested solutions videos after you complete the question. This video explains to you the steps to take to answer the question and provides tips and tricks.
Once you're done with the practice questions, move on to the next checkpoint.
Now, let’s get started on your practice questions.
10 questions
Take a Timed Test Take an Untimed Test