We’re now fully in exam season. I hope your revision is going well. In this blog, I will explain 5 of the most common mistakes I’ve seen in foundation tier GCSE maths. I will explain why people get it wrong so often, how you avoid this and what the correct solution is.
1. Rounding to decimal places
Question:
Round 3.468 to 2 decimal places
Incorrect solution:
3.57
The reason students make this mistake is they round both the first and second decimal places. They see the 4 has a 6 to its right and therefore round this up to a 5. However, you shouldn’t do this – when you round to 2 decimal places, you should only round the second decimal number. All the numbers before this should stay the same.
Correct solution:
3.47
You should round the 6 up to a 7, because it has an 8 to its right. But the 4 in the first decimal place should stay the same.
To practice more questions like this, try our rounding worksheet.
2. Distance, speed and time
Question:
Robert drove at 30 miles per hour for 40 minutes.
How many miles did he travel?
Incorrect solution:
Distance = Speed × time
30 × 40 = 1,200 miles
This common mistake should be obvious to spot. Do you really think it’s possible that Robert drove 1,200 miles in 40 minutes? Unless Robert is driving a spaceship in a sci-fi movie! For context, that is even further than the distance from Bristol to Rome – a journey that would take a plane over 2 hours. Clearly something has gone wrong.
The problem with this question is that the speed has been given in miles per hour but the time has been given in minutes. So, when we multiply 30 by 40, we are actually working out how far Robert has driven travelling at 30 miles per minute. We first need to convert the time unit into hours. Then we can perform the calculation. Correct solution below…
Correct solution:
Converting 40 minutes into hours:
40 ÷ 60 = 2/3 hours
Distance = Speed × Time
30 × 2/3 = 20 miles
To practice more questions like this, try our worksheet.
If you need more help with distance, speed and time questions, you can read my blog on this topic.
3. Reverse percentages
Question:
The price of a coat increased by 20%.
After the increase, the price became £84.
What was the price of the coat before the increase?
Incorrect solution:
£84 × 0.8 = £67.20
This is a very common mistake. It’s such a simple question, but it is so often answered incorrectly.
By answering it this way, what you are actually doing is taking 20% from £84. But the problem is that 20% of £84 is not going to be the same as 20% of the original amount – as the original amount is smaller than £84.
You can avoid making this mistake by checking your answer. If the answer of £67.20 is correct, then when you add 20% to it you should get £84, right?
67.20 × 1.2 = £80.64 (multiplying by 1.2 will increase a number by 20%)
This proves the answer is wrong.
What you actually need to do is divide by 1.2, because the original value was multiplied by 1.2 so you need to reverse this calculation to get back to the original value.
Correct solution:
£84 ÷ 1.2 = £70
This question is a mixture of interest and depreciation and percentage changes. I haven’t written any specific questions like this, but following both the links will give you practice of the skills needed to answer this question. There is a CorbettMaths worksheet on this topic though, which will give you good practice of these types of question: reverse percentages. It also relies heavily on the use of multipliers, so keep your eyes peeled for a blog on multipliers in the coming months – it’s in the works.
4. Venn diagrams
Question:
A group of 50 people were asked if they own a cat or a dog.
16 people said they own a cat.
9 people said the own a cat and a dog.
12 people said they own neither.
Complete the Venn diagram.
Incorrect solution:
The Venn diagram isn’t filled in completely incorrectly. The two numbers in red are the incorrect numbers.
The mistake that’s been made here is that the question says “16 people said they own a cat”. It doesn’t say “16 people said they own a cat only”. This is a very subtle change but it makes a huge difference.
When you put the 16 where it is, you are saying that 16 people own a cat but not a dog. The information given in the question simply says that 16 people own a cat – it doesn’t mention whether those people also own a dog. So what this incorrect Venn diagram is saying is that 25 people own a cat, because the 9 people in the middle also own a cat.
Correct solution:
What we instead need to do is understand that everything in the Cat circle (the circle on the left) adds up to 16. This includes the middle. To work out what goes in place of the underlined number in the above incorrect solution, we need to subtract 9 from 16 to find out how many people only own a cat.
16 – 9 = 7
So, 7 people own a cat but not a dog:
To complete the Venn diagram, we add up all the numbers currently on the diagram and subtract this from 50.
7 + 9 + 12 = 28
50 – 28 = 22
So 22 people own a dog but not a cat.
The correct Venn diagram:
A resounding victory for the dogs then! I’m definitely a dog person, so maybe wrote this question in a slightly biased fashion. If you were wondering why the picture accompanying this blog is of a dog, this is why! There are only so many maths-related stock photos you can use. That’s my first dog, Jessie, who unfortunately passed away in 2021 after a great 10 years.
This Venn diagram mistake is probably the question answered incorrectly most often out of all of the mistakes discussed in this blog. To practice more questions like this, try our worksheet.
5. Finding the mean from a frequency table
Question:
The frequency table below shows the number of gold stars the students in a class received last week.
| Gold stars received | Frequency |
| 1 | 11 |
| 2 | 8 |
| 3 | 4 |
| 4 | 4 |
| 5 | 3 |
Work out the mean number of gold stars received last week.
Incorrect solution:
Work out total number of gold stars received:
1 × 11 = 11
2 × 8 = 16
3 × 4 = 12
4 × 4 = 16
5 × 3 = 15
11 + 16 + 12 + 16 + 15 = 70
70 ÷ 5 = 14
This one can be very easily avoided by doing a very simple sense check. In the frequency table, the maximum number of gold stars received was 5. So how can the mean number of gold stars received be 14? In fact, how can it possibly be anything larger than 5? This makes no sense, so clearly something is wrong.
The mistake made is not with calculating the 70 gold stars – this was done perfectly.
The mistake was dividing it by 5. I think students just see that there are 5 rows of data and, on auto-pilot, divide by this. But this is incorrect.
Correct solution:
To calculate the mean, you would need to divide the total number of gold stars received by the total number of students.
There are not 5 students. To work out the total number of students, you add up the numbers in the frequency column of the table:
11 + 8 + 4 + 4 + 3 = 30
So to calculate the mean, we divide 70 by 30…
70 ÷ 30 = 2.33 (to 2 decimal places)
A simple check confirms that this answer makes sense, as it is roughly in the middle of the maximum number of gold stars received (5) and the minimum number of gold stars received (1).
To practice more questions like this, try our worksheet.
I hope you’ve found this useful and that you won’t make these mistakes when you do your exams!
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