Tips on How to Solve Percentage Problems

How to Solve Percentage Problems

In this Tips on how to Solve Percentage Problems, it consists of

  • What a Percentage problem is and how to solve it.

  • Sample questions and Answers

1. What Is a Percentage?

  • How to Add Percentages
  • How to Subtract Percentages

2. Converting Decimals and Fractions to Percentage Values

3.  How to Work Out the Percentage of a Known Value

  • Question 1: Calculating the Percentage of a Known Value

4. Calculating What Percentage One Number is of Another Number

  • Question 2: Calculating what percentage one number is of another

5. How to Work Out a Percentage Increase

6. How to Work out a Percentage Decrease or Discount

7. Working Backwards from a Percentage

  • Question 5: Working backwards from a percentage

8. Major Errors and Things to Look out For

What Is a Percentage?

A percentage is seen as a given part or amount in every hundred. Percentages are used very widely in both mathematics and everyday situations; Percentages are usually represented by the % symbol.

Here are some common ways that percentages are used in day-to-day life:

Calculating how students failed in a particular course

Working out how much VAT you need to pay on a purchase

Calculating how much to pay as a tithe in your church

How to Solve Percentage Problems

Percentages are widely used in numerical reasoning tests, so it is important that you are able to understand and interpret them effectively.

How to Add Percentages

If you are adding consecutive percentages, unfortunately it is not as simple as adding them together and then applying that number to the problem.

If you need to find the total of £200 + 10% + 15%, the initial thought might be to calculate £200 + 25%.

Instead, you need to calculate them separately and in order.

How to Work Out Percentage Problems [Questions and Answers]

First, add 100 to each percentage and then convert it to make a decimal larger than 1:

10% becomes 110% which is converted to 1.10

15% becomes 115% which is converted to 1.15

The original value is then multiplied by these numbers.

To ensure that the number is manipulated correctly, the multiplication needs to be completed in the order that it is presented in the question:

200 x 1.10 = 220 This is the first step: £200 + 10% 220 X 1.15 = 253

This is the second step: (£200 + 10%) + 15%

Therefore, the answer to £200 + 10% + 15% is 253.

To multiply percentages, you can convert them into decimals, multiply the decimals, convert back into a percentage.

For example, if you are asked to multiply 15% and 40% together, the calculation would look like:

15% = 0.15

40% = 0.40

0.15 x 0.40 = 0.06

0.06 x 100 = 6%

If you prefer to work in fractions, the calculation can also be done that way.

For example, if you need to multiply 10% by 30%, you would convert them into fractions out of 100 then simplify:

10% = 10/100 = 1/10

30% = 30/100 = 3/10 Then multiply each fraction together. There is already a common denominator:

1/10 x 3/10 = 3/100

Then convert the fraction back into a percentage:

3/100 = 3%

How to Subtract Percentages

To subtract one percentage from another, just ignore the percentage signs and treat them like whole numbers.

For example

To subtract 20% from 50%, perform the sum 50 – 20 to get 30. The answer is 30%. If you are subtracting a percentage from a whole number, you first need to convert it to a decimal.

If you are asked to subtract 25% from 45 (for example, when calculating a discount), then you need to start by converting 25% to a decimal, which is 0.25.

To calculate the amount that should be subtracted, multiply the original number by the decimal:

45 x 0.25 = 11.25

Then subtract this amount from the base figure:

45 – 11.25 = 33.75

You can also take the decimal you converted the percentage into, subtract it from 1, then multiply the original number by it:

25% = 0.25

1 – 0.25 = 0.75

0.75 x 45 = 33.75

Converting Decimals and Fractions to Percentage Values

When taking a numerical reasoning test, you may be required to move fluidly between questions using percentages, fractions and decimals. It is very straightforward to convert numbers between these different representations, and these are key techniques to learn.

To translate fractions into percentages, you should divide the bottom number in the fraction by the top number. This will give a decimal figure. Then multiply that decimal figure by 100, to create the percentage. Here’s an example:

I/2 →1 % 2 = 0.5 → 0.5 × 100 = 50%

How to Solve Percentage Problems

To translate decimals into percentages is even easier. Simply multiply the decimal figure by 100 to calculate the percentage:

0.65 → 0.65 × 100 = 65%

Finally, to convert percentages into decimals is also very straightforward. You need to divide the percentage by 100 to calculate the decimal:

37 → 37 × 100 = 0.37

How to Work Out the Percentage of a Known Value

[Total value\100] × desired percentage

This is something that people use all the time. For example, if you went out for dinner, spent £56 and wanted to leave a 10% tip, this is how you would work it out:

(£56/100) x 10 = £5.60

Question 1: Calculating the Percentage of a Known Value

Calculating the percentage of a known value is quite straightforward. In a numerical reasoning test, these questions tend to require you to identify/manipulate the relevant information in order to use the formula. Graphs or tables will often be used to present the information, such as the one below.

Table 1: Numbers of testees taking an online test using gadgets at different times of day

Time Ipads laptops Desktop
8:00 – 9:00 850 1100 650
10:00 – 11:00 800 2350 1400
12:00 – 1:00 750 1870 980
2:00 – 3:00 600 1500 790

Question 1a. What percentage of test takers that use gadget like iPad to take their online test and those that use ipads before 2:00?

To solve this question, you need to:

First add together all the test takers that use ipad (850 + 800 + 750 + 600 = 3,000)

Then add together all the test takers that use ipad before 2:00 (850 + 800 + 750 = 2,400)

Then use the formula: (2,400/3,000) x 100 = 80%

Question 1b. What percentage of the total test takers that use laptop?

To solve this question, you need to:

First add together all of the all the test takers shown on the table: (850 + 800 + 750 + 600 + 1,100 + 2,350 + 1,870 + 1,500 + 620 + 1,400 + 980 + 790 = 13,610)

Then add together all test takers (620 + 1,400 + 980 + 790 = 3,790)

Then use the formula: (3,790/13,610) x 100 = 27.8%

Question 1c. What percentage of test takers taking their online between 12:00 to 1:00 were using laptops?

To solve this question, you need to:

First add together all of the test takers taking their online test between 12:00 to 1:00: (750 + 1,870 + 980 = 3,600)

Then calculate the test takers using laptops between 12:00 to 1:00, the answer is 1,870.

Then use the formula: (1,870/3,600) x 100 = 52%

Calculating What Percentage One Number is of Another Number

This allows you to work out what percentage of the total a particular value contributes. It is easiest to explain how to do this with an example.

Assumed you had 3 unripe bananas (this is value A) and 5 ripe bananas (this is value B). Together they give 8 apples, which is 100% of the apples you have.

3 unripe bananas + 5 ripe bananas = 8 bananas in total

Value A                     Value B        = 100 %

To find the percentage of the unripe bananas you need to use the following formula:

% A

How to solve the problem above

Therefore,

To be sure of the correct answer, add the two percentages together from the answer you got which will give you 100%, otherwise your answer is incorrect.

Question 2: Calculating what percentage one number is of another

In school bus of Mayday College, there are 12 girls and 8 boys. What percentage of the class are girls?

To solve this, follow following steps:

Work out how many children there are in the class in total: 12 + 8 = 20

Use the formula:   60%

So, the answer is 60%.

How to Work Out a Percentage Increase

Percentage increases are useful for understanding, for example, what a gas price hike means for you in financial terms, or to understand what a percentage rise in inflation equates to.

The formula you need to use to calculate this is:

Increase %

Example: if a family originally paid $1,000 per year for their Gas, and their bill was increasing by $500, this is a 50% increase. The formula to calculate this is :  50%

Using percentages like this is also useful for comparing changes to different numbers, see example below:

Energy Bills Yearly From 2016 – 2018

Names 2016 2017 2018
George $650 $825 $920
Alice $800 $870 $1000
Brian $575 $650 $700

Question 3a: By what percentage did George’s energy bill increase between 2016 and 2018?

To work out this question, you need to:

First, work out how much George’s energy bill increased between 2017 and 2018: $650– $920 = $270

Then use this formula: $270/$650 x 100 = 41.5%

Therefore, George’s energy bill increased by 41.5%

Question 3b: Whose energy bill increased by the greatest percentage between 2017 and 2018?

First, solve how much each individual’s energy bills increased between 2017 and 2018:

George: $825 – $920 = $95

Alice: $870 – $1000 = $130

Brian: $650 – $700 = $50

Then use the formula to work out the percentage increase of each:

George: $95/$825 x 100 = 11.5%

Alice: $130/$870 x 100 = 9.4%

Brian: $50/$650 x 100 = 7.7%

Therefore, George’s energy bill increased by the greatest percentage between 2015 and 2016.

Question 3c. who paid the lowest percentage increase in energy bills by any of the individual between 2016 and 2017?

To work this out, you need to:

First, solve how much each individual’s energy bills increased between 2016 and 2017:

George: $825 – $650 = $175

Alice: $870 – $800 = $70

Brian: $650 – $575 = $75

Then use the formula to work out the percentage increase of each:

George: $175/$650x 100 = 29.9%

Alice: $70/$800 x 100 = 8.8%

Brian: $75/$575 x 100 = 13%

The lowest percentage increase in energy bills incurred by any of the individual between 2016 and 2017 is Alice.

How to Work out a Percentage Decrease or Discount

Percentage decrease or discount is part of our daily lives because most people are familiar with percentage decreases or discounts, as these are used in shop sales or online store. For example, if you want to buy a shirt and you see that the shop is offering a 20% discount, then you know that you can achieve a substantial saving over the full price of £1000.

There are two ways of calculating a percentage discount.

Firstly, you can calculate the discount and then subtract this from the starting price. To do this you would use the following steps:

Step1: Convert the percentage discount to decimal: 20% = 0.2

Step 2: Multiply the starting number by the decimal: £1000 x 0.2 = 200

Step 3: Therefore, the discount is 200, then you subtract the discount from the starting price to give you the new price: £1000 – 200 = 800

Secondly, you to directly calculate what the new price would be (rather than the discount). To do this you would need to use the following steps:

Step1: Subtract the percentage discount from 100%: 100 – 20 = 80%

Step 2: Convert 80% to decimal: 0.8

Step 3: To find the new price, multiply the starting number by the decimal: £1000 x 0.8 = £800

Mia and Thompson both want to buy a new textbook. They negotiate hard and both manage to secure a discount. Mia eventually manages to secure a 10% discount on her textbook, which originally cost £300, and Thompson secures a 15% discount on his own textbook with an original price of £500.

Question 4a: Who paid the least for their new textbooks?

To answer this question, you need to solve how much each textbook cost. Using the steps above:

Mia: 100% – 10% = 90% → 90% = 0.9

£300 x 0.9 = £270

Thompson: 100% – 14% = 86%, 86% = 0.86, £300 x 0.86 = £258

Therefore, Thompson paid the least for his new textbook.

Question 4b: How much was the discount that Mia managed to secure?

To answer this question, you need to solve through the following steps:

Convert the % to a decimal: 10% = 0.1

Multiply the starting price by 0.1: £300 x 0.1 = £30

However, Mia managed to secure a discount of £30.

Working Backwards from a Percentage

Sometimes, you may be presented with a figure that is greater than 100%.

For example, if Brian bought a car for £250,000 in 2015, its value may increase over the years and in 5 years’ time, it is worth 20% more than he paid for it.

The new value could be calculated by:

Converting the percentage to a decimal: 120% = 1.2

Multiplying the starting value by 1.2: £250,000 x 1.2 = £300,000

Therefore, in 2020 his car was worth £300,000.

Question 5: Working backwards from a percentage

Andrew has £5900 in his savings account. This is 25% more than he initially invested. How much was his initial investment?

To answer this, you need to go through the following steps:

£5900 = 125%

5900/125 = £47.2

£47.2x 100 = £4720

Therefore, Andrew previously invested £4720 in his savings account.

Major Errors and Things to Look out For

Follow these tips below that will aid you getting most question on percentages questions correctly:

Make sure that if you are converting decimals to percentages (or vice versa) that you get the decimal point in the right place. Often the multiple-choice answers to numerical reasoning tests will include incorrect answers with exactly this error, so if you have made this mistake there may well be an answer waiting to catch you out.

When comparing percentages make sure that you have a common baseline (otherwise the percentages will be unrelated to one another).

One area that often catches people out is year-on-year percentage increases. For example: Tiana has £10 and each year this increases by 5%. How much will she have after 3 years? Some people can be tempted to add together the 5% for the 3 years i.e., 15% and multiply the £10 by 15% giving £11.5. This is incorrect. The correct way of approaching questions like this is to remember that EACH year the initial £10 increased by 5%. So, at the end of year 1, Freya would have £10 x 1.05 = £10.5. At the end of year 2, she would have £10.5 x 1.05 = 11.025, and so on. It is important to add in each of these steps to arrive at the correct answer.

Another common error is around percentage increases. For example, the price of a loaf of bread increases by 10%. After the increase the price was £1.10, how much did the bread cost before the increase. A really common error is for people to try and solve this type of question by calculating: £1.10 x 0.9 = £0.99. This is incorrect. Remember, that £1.10 = 110%, therefore you must use this calculation: (£1.10/110) x 100 = £1.00

Avoid using the % button on your calculator unless you are really confident in what you are doing. It might seem like a sensible short cut but it can lead to you making basic errors.

Visit testpremier.com if you would like to practice more percentage questions to boost your confidence.

RATIOS AND PROPORTION

 Table of contents:

1. Meaning of Ratio and it Formular

2. Meaning of Proportion and it Formular

3. Types of Proportion

  • Direct Proportion
  • Inverse Proportion
  • Continued Proportion

4. Important Properties Proportion

5. Sample Questions

Meaning of Ratio and it Formular

In maths, a ratio represents a comparison between two or more numbers by division of the same kind, for instance the ratio of a to is written as a:b = a/b. In ratio, (a) can be said to be the first term or antecedent while (b) is called the second term or consequent

For example: The ratio of 3:5 can be represented as 3/5, therefore the 3 → the first term or antecedent and 5 → the second term or consequent.

Another instance, the ratio of number of boys in a class to the number of girls is 3:5. Here, 3 and 5 are not taken as the exact count of the students but a multiple of them, which means the number of boys can be 3 or 6 or 9…etc and the number of girls is 5 or 10 or 15… etc. It also means that in every eight students, there are three boys and five girls.

Ratios are usually express in fractions, decimals and even percentages. Ratios are widely used in numerical reasoning tests. Therefore, mastering ratios is extremely important for succeeding on numerical reasoning tests.

Meaning of Proportion and it Formular

A lot of questions on ratio are solved by using proportion. Proportion means comparison of two ratios. If a:b = c:d, then a, b, c, d are said to be in proportion and written as a:b :: c:d or a/b = c/d. a, d are called the product of extremes and b, c are called the product of means.

For a proportion a:b = c:d, product of means = product of extremes → b*c = a*d the

For example: In a mixture of 45 litres, the ratio of sugar solution to salt solution is 1:2. What is the amount of sugar solution to be added if the ratio has to be 2:1?

Number of litres of sugar solution in the mixture = (1/(1+2)) *45 = 15 litres.

So, 45-15 = 30 litres of salt solution is present in it.

Let the quantity of sugar solution to be added be x litres.

Setting up the proportion,

sugar solution / salt solution = (15+x)/30 = 2/1 → x = 45.

Therefore, 45 litres of sugar solution has to be added to bring it to the ratio 2:1.  The proportion can be classified into the following categories, such as:

Types of Proportion

Direct Proportion

Inverse Proportion

Continued Proportion

Direct Proportion

The direct proportion describes the relationship between two quantities, in which the increases in one quantity, there is an increase in the other quantity also. Similarly, if one quantity decreases, the other quantity also decreases. Hence, if “a” and “b” are two quantities, then the direction proportion is written as a∝b.

Inverse Proportion

The inverse proportion describes the relationship between two quantities in which an increase in one quantity leads to a decrease in the other quantity. Similarly, if there is a decrease in one quantity, there is an increase in the other quantity. Therefore, the inverse proportion of two quantities, say “a” and “b” is represented by a∝(1/b).

Continued Proportion

Consider two ratios to be a: b and c: d. Then in order to find the continued proportion for the two given ratio terms, we convert the means to a single term/number. This would, in general, be the LCM of means.

For the given ratio, the LCM of b & c will be bc. Thus, multiplying the first ratio by c and the second ratio by b, we have:

First ratio- ca:bc

Second ratio- bc: bd

Therefore, the continued proportion can be written in the form of ca: bc: bd.

 Important Properties Proportion

Addendo – If a : b = c : d, then a + c : b + d

Subtrahendo – If a : b = c : d, then a – c : b – d

Dividendo – If a : b = c : d, then a – b : b = c – d : d

Componendo – If a : b = c : d, then a + b : b = c+d : d

Alternendo – If a : b = c : d, then a : c = b: d

Invertendo – If a : b = c : d, then b : a = d : c

Componendo and dividendo – If a : b = c : d, then a + b : a – b = c + d : c – d

Sample Questions

For every three boys in the class there are five girls.

What is the ratio of girls to boys?

Solution:

From the order of words in the sentence which is, for every 3 boys in the class there are 5 girls. The ratio of boys to girls will be 3:5, Likewise for every 5 girls in the class there are 3 boys, therefore, the ratio of girls to boys will be 5:3

Write two equivalent ratios of 6:9.

Solution:

Given Ratio: 6:9, which is equal to 6/9.

Multiplying or dividing the same numbers on both numerator and denominator, we will get the equivalent ratio.

⇒(6×3)/(9×3) = 18/27 = 18:27

⇒(6÷3)/(9÷3) = 2/3 = 2: 3

Therefore, the two equivalent ratios of 6:9 are 2:3 and 18:27

A solution is 4 parts water, 3 parts wine, and 1 part honey. If a container of this solution contains 3 gallons of water, how much total solution is there in it?

Solution:

To start with, notice that there is a 3:4 ratio between the water in your container and the water specified by the mix of the components.  Given that there are 8 total parts in your solution, this means that you can set up this equation:

=

Multiplying both sides by 8, you get:

x=6

There are 6 total gallons of solution.

If the sum of a, b and c is 400, and a is 1/3 b and b is 1/4 c, what is the value of a?

Solution:

For this type of problem, build an equation that represents the relationships between the quantities and solve for the quantity you need. The problem states that b=1/4c and a = 1/3b. Thus, a = 1/3(1/4)c, or 1/12 of c. Now put everything in terms of c, thus 1/12c + 1/4c +c = 400. Now comes the tricky step–combine like terms and create the improper fraction (1/12c + 3/12c + 12/12c = 16/12c). Reduce the fraction to 4/3. So, 400 is 4/3 of c. Thus, c is 3/4 of 400, or 300. a is 1/12 of 300, or 25.

A ratio is a mathematical expression written in the form of a:b, where a and b are any integers. It expresses a fraction. For example. 2:3 = ⅔.

Solution:

If a:b::c:d is a proportion, then;

a/b=c/d

ad=bc

Example: If ⅔=4/6, then,

2 x 6 = 3 x 4

12 = 12

The dimensions of the rectangular field are given. The length and breadth of the rectangular field are 50 meters and 15 meters. What is the ratio of the length and breadth of the field?

Solution:

Length of the rectangular field = 50 m

Breadth of the rectangular field = 15 m

Hence, the ratio of length to breadth = 50: 15

⇒ 50: 15 = 10: 3.

Thus, the ratio of length and breadth of the rectangular field is 10:3.

Max wants his garden to be one-part tulips, two parts roses and four parts bushes.

If Max’s Garden had 115% more bushes than he wanted, what would be the ratio of roses to plants in Max’s Garden?

Solution:

Let T represent tulips, R represent roses and B represent bushes.

Max wanted his garden to have a ratio of T:2R:4B. However, Max’s garden actually has 115% MORE bushes, which means the actual number of bushes is 4B + (1.15*4)B = 4B + 4.6B = 8.6B.

Thus, the actual ratio of Max’s Garden is T:2R:8.6B. Therefore, the ratio of roses to plants in Max’s Garden is 2:(1+2+8.6) = 2:11.6 = 10:58 = 5:29.

There exists 45 people in an office. Out of which female employees are 25 and the remaining are male employees. Find the ratio of

a] The count of females to males.

b] The count of males to females.

Solution:

Count of females = 25

Total count of employees = 45

Count of males = 45 – 25 = 20

The ratio of the count of females to the count of males

= 25 : 20

= 5 : 4

The count of males to the count of females

= 20 : 25

= 4 : 5

Fred and Jason are roommates. Each of them pays the rent, according to their room’s size. Fred’s room is twice as big as Jason’s. How much rent does Fred pay, if the total rent is £1800?

Solution:

The ratio of Fred’s room size to Jason’s room size is 2:1. According to the data, the same ratio applies to rent payment: Fred pays twice as much as Jason. We can represent this using an equation: 2X + X =1800

Fred pays 2X while Jason pays X and together the payment equals 1800 (total sum of rent).

If we develop this equation we get:  3X = 1800  => X = 600

This means, that Jason pays 400 and Fred pays 1000.

Check whether the following statements are true or false.

a] 12 : 18 = 28 : 56

b] 25 people : 130 people = 15kg : 78kg

Solution:

a] 12 : 18 = 28 : 56

The given statement is false.

12 : 18 = 12 / 18 = 2 / 3 = 2 : 3

28 : 56 = 28 / 56 = 1 / 2 = 1 : 2

They are unequal.

b] 25 persons : 130 persons = 15kg : 78kg

The given statement is true.

25 people : 130 people = 5: 26

15kg : 78kg = 5: 26

They are equal.