#### Percentages can get surprisingly complicated on the UCAT. Part of that is because they are never straightforward.

While it’s easy enough to work out 50% or 10% of a number, it’s rarely so easy to make the conversion on the UCAT, especially when the percentages given are, say, 31% or 18%.

To make matters worse, the UCAT won’t just test you on the simple process of finding the percentage of a number. Instead, it’ll present problems in such a way that **calculating the percentage is one of the multiple steps to derive your final answer**. Scenarios you can expect in the exam include cases where you calculate in reverse, find a combination of percentages, make comparisons or find a percent change.

In this article, we will explore the key skills needed when dealing with percentage problems in the UCAT quantitative reasoning. These are all basic skills and should be used as a guide to help recognise weak areas and structure targeted study. I’ve done my best to keep this article straight to the point. However, if you would like strategies and techniques to improve your accuracy and speed on this subject do grab our **UCAT Question type strategy** book.

### Skill #1: Finding a Percentage

A **percentage** is a proportion that shows a number as parts per hundred. The symbol ‘%’ means ‘per cent’. 15% means 15 out of every 100. Percentages are just one way of expressing numbers that are part of a whole. These numbers can also be written as fractions or decimals. 50% can also be written as a fraction, , or a decimal, 0.5. They are all exactly the same amount. To calculate the percentage of something within a group, you have to find the proportion and then multiply by 100. Let’s look at the problem below:

During practice, it’s good practice to recognise the number of steps required to solve a problem.

**Solution:**

*Step 1* – Number of books not on sale = 660 – 30 = 630

*Step 2* – Percentage of books not on sale = 630/660*100 = **95.5%**

The above problem is a 2-step calculation, knowing this beforehand can help you manage your time appropriately in the exam.

### Skill #2: Increasing and Decreasing an Amount by a Percentage

In the UCAT you may be required to **increase** or **decrease** an amount by a percentage. In this scenario, first, calculate the percentage of the amount and then either **add** this answer on to increase the value or **subtract** this answer to decrease the value. Let’s look at another problem:

**Solution:**

*Step 1* – 1.13 x 74 = **83.6 kg**

Instead of working out 13% then adding it to 74 kg, I save time by multiplying by 1.13.

### Skill #3: Expressing numbers as a Percentage of Another Number

Expect problems where you have to express one number as a percentage of another. This can get pretty complicated when you have to do multiple calculations to arrive at the final answer. Let’s consider the problem below:

It really doesn’t get much time-consuming than this in the UCAT when it comes to expressing numbers as a percentage of another number. Since this problem requires more the 3 steps to calculate you want to use shortcuts as much as possible, potentially flag the question come back to it later depending on your overall triage strategy.

**Solution:**

*Step 1* – 0.2 x 20 = 4

*Step 2* – 0.4 x 30 = 12

*Step 3* – 0.8 x 10 = 8

*Step 4* – Sum of colour ads = 24

*Step 5* – percentage of total colour ads = 24/60*100 = **40%**

### Skill #4: Percentage Change

Some percentage problems in the UCAT look at the difference between two numbers where you’ll be required to work out the percentage increase or the percentage decrease. This is known as **percentage change**. When dealing with these sort of problems, it can be handy to know the percentage change formula, see below:

Let’s look at a problem where we can apply this formula:

**Solution:**

Percentage change = 20/40*100 = **50%**

Pretty easy right? Unfortunately, questions are never this straightforward in the exam. Before I share a typical UCAT problem, let’s look at a reverse percentage change.

### Skill #5: Reverse Percentage Change

There will be percentage problems in the UCAT where you’ll be required to work backwards using percentage change. This can be tricky at times, especially when they are long text problems. You may find yourself re-reading the question to set *reference points, *i.e. recognise the *old value* and *new value* so you can input them in the percentage change equation. Unfortunately, you will find this can be a waste of time, sometimes its easier to just think logically about the problem and **set a linear equation**. Let’s look at an example:

First, ask yourself how many steps to solve the above problem? Always a good habit to get into when preparing for the QR subtest. There are many ways to solve this problem, but in the UCAT you ideally want to use shortcuts whenever possible to save time. See below the approach I would use:

**Solution:**

Using the *equation method*

80% of *x = *£100

where *x* is the original price.

0.8*x* = 100

*x* = 100/0.8 = **£125**

Expect the examiners to include £120 as one of the options. Or an answer option that is influenced by choosing the wrong reference point. With enough practice, you’ll become aware of some of the tactics examiners use to catch students out.

Now that you are aware of normal percentage change and reverse percentage change concepts, let’s look at a type of problem you can expect in the exam:

Start to think of how many steps you can solve the above problem. You would have noticed you need to do a reverse percentage change then calculate a percentage.

**Solution:**

*Step 1* – Use the equation method to find the cost price

140% of *x* = £28

where *x* is the cost price

1.4*x* = 28

*x* = £20

*Step 2* – Employee discount

70% of *x = y*

where *y* is the employee discounted price

y = 0.7*20 = **£14**

Did you notice I took shortcuts? Instead of working out 40% then subtracting it from £28, I save time by dividing by 1.4. The same method when calculating employee discount, rather than calculating 30% of the cost price then deducting it, I save time by multiplying by 0.7.

### Skill #6: Making Comparisons

Sometimes questions are presented in tabular or text format where you need to make comparisons between different scenarios to solve the problem. See an example below:

**Solution:**

*Step 1* – Roberts new rent

1.031 x 920 =£948.52

*Step 2* – Marcus’ new rent

0.965 x 968 = £934.12

*Step 3* – Rent per annum for both men

Robert’s = 948.52 x 12 = £11,382.24

Marcus’ = 934.12 x 12 = £11,209.44

*Step 4* – Difference in rent per annum = 11,382.24 – 11,209.44 = £**172.80**

This is a 4-step calculation, due to the time constraints in the exam you may not have enough time. I would recommend rounding Marcus’ monthly rent to £970, then estimating the final answer.

### List of popular question-types in the UCAT

Percentage problems can be in either **text** or **tabular** format covering anything from Discounts, Coupons, Compound interest, Income taxes, Proportion in terms of Percentages, Value Added Tax (VAT), Tariffs. During practice, try to identify which **format you struggle with the most** as well as the **percentage concepts you find difficult**. For example, you may struggle with income tax problems in tabular format or compound interest in text format. Recognise what you struggle on and do more targeted practice to improve your ability you solve such problems more accurately in the test.

### Top Tips for Solving Percentage Problems in the UCAT

#### 1. Avoid Common Mistakes

Calculating percentages come with a long series of potential mistakes one can make when dealing with percentage calculations. Here are some of the common mistakes to avoid:

- Choosing the wrong reference point (i.e. values)
- Choosing the wrong denominator and Numerator
- Not knowing when to add or subtract

**2. Always use Shortcuts to derive Answer**

Percentage problems in the UCAT are normally multiple-step calculations, so be sure to use shortcuts whenever possible to save time. Great strategies include setting equations, multiplying decimals and rounding.

#### 3. Do as much mental maths as possible

There are many mental math tricks you can adopt when calculating percentages. There are tricks you can use to work out 5%, 10%, 20%, 30%, 50% and 75%.

#### 4. Use formulas and create equations when possible

Setting equations are great for simplifying word problems and solving reverse percentages. Practice creating equations for complex problems.

#### 5. Always recognise the number of steps before solving the problem

Always try to work out the number of steps to solve a problem. Classify it into 4 categories:

- 1- step – can be solved in one step
- 2-step – Can be solved in two steps
- 3-step – Can be solved in three steps
- 3+ step – Can be solved in more than 3 steps

This can help with deciding if you should attack the problem right away or later (as part of your triage strategy).