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**An In-Depth Study Guide for Numerical Reasoning Ratios Questions**

Learn quickly take in and understand ratio maths questions, avoid pit falls and use short cuts to solve faster and more accurately than your competitors.

**Content Table:**

- Free Numerical Ratios Test
- Ways of expressing ratios
- Free Tests
- Calculating ratios
- Sample questions
- Tops solving tips
- Useful Resources

In maths, a ratio represents a comparison between two or more quantities, indicating how their sizes relates to one another.

In a sense, ratios are merely a different way expressing 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.

To do that, you are required to quickly navigate between ratios, fractions, decimals and percentages, converting between the different representations effectively.

** Note:** Ratios do not represent definite amounts, but a relation between the amounts.

When expressing ratios in words, we usually use the word ‘"to’", stating that "the ratio is "‘a’" to ‘"b’", where "‘a’" and ‘"b’" represent numbers.

**For example:** in a basket with one apple and two oranges the ratio of apples to oranges is 1 to 2, which means that for every apple in the basket we will find two oranges in that basket.

**Typical presentation:** When expressing ratio mathematically, we usually write the numbers being compared with a colon sign between them: ‘"a:b’".

**For example:** in a basket with one apple and two oranges the ratio of apples to oranges is 1:2, which means that for every apple in the basket we will find 2 oranges in that basket.

**Other types of presentations:** Since a ratio is the quotient of the first quantity divided by the second quantity.

This value tells us how many of the first quantity can fit into the second quantity. Therefore, ratios can appear as fractions, decimals and even percentages, expressing how many of the first quantity can fit into the second quantity.

**Fractions:**

**For example:** In a basket with one apple and two oranges the fractional value of the ratio of apples to oranges is 1/2 , which means that 2 times the number of apples in the basket, will equal to the number of oranges in that basket.

Notice that when writing the ratio of apples to oranges in the form of a fraction, the **first item** in the ratio (in this case - apples) will appear as the **numerator** and the second item in the ratio (in this case - oranges) will appear as the **denominator**.

**Decimals:**

**For example:** In a basket with one apple and two oranges the decimal value of the ratio of apples to oranges is 0.5, which means that for every orange in the basket, there is half an apple in that basket.

**Percentage:**

**For example:** In a basket with one apple and two oranges the percentage value of the ratio of apples to oranges is 50%, which means that for whatever number of oranges in the basket, the number of apples in that basket is 50% the number of oranges (in other words the number of apples in the basket is **half** the number of oranges in the basket).

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Notice when reading and writing ratios, you must be precise with the order of words and numbers: the first item to appear in the question must correspond to the first number (from the left) to appear in the question.

The second item to appear in the question must correspond to the second number (from the left) to appear in the question etc.

That way, when writing the numerical representation of the ratio between the items, the numbers will be written according to the **order of items** you are asked about.

**Example Question:**

**There are seven shirts and eleven trousers in the closet.**

**Question:** What is the shirts to trousers ratio?

Answer: The ratio of shirts to trousers is 7:11.

**Question:** What is the trousers to shirts ratio?

Answer: The ratio of trousers to shirts is 11: 7.

There are three main types of ratios you may encounter while solving numerical questions:

- The relation between two or more quantities of the
**same**category/unite.

**For example**, the ratio of boys to girls in a classroom is 2:3. Both boys and girls belong to the same category known as ‘people’. This ratio expresses the idea that for every 2 boys in the classroom there are 3 girls present in that classroom.

The relation between two or more quantities of** different **categories/unites.For example, the ratio of distance, measured in km, to time measured in hours, results in speed, which is the quantity that measures distance over time spent and whose units are ** ^{km}/_{hour}**.

This ratio expresses the idea that for every hour, X number of kms are travelled.

- The relation between a category’s sub segment of its total sum.

**For example**, the ratio of females living in New York City, to the entire New York City population is 100:194.

This ratio explains the comparative relation of females to the total NYC population (both males and females). It expresses the idea that for any number of people living in NYC, 100/194 of that population are females.

Let's say that there are 194,000,000 people living in NYC, then 100,000,000 of them are females.

Learning how to calculate ratios from other numerical data is crucial for the understanding of ratios.

**For example:** There are seven shirts and four trousers in the cupboard. Calculate the ratio of shirts to clothes.

Here you are asked to perform 2 sets of calculations;

- First, to calculate the
**total number of clothes**: 7 + 4 = 11 – cloths items (both shirts and trousers). - Second, to express the ratio of
**shirts to clothes**, paying attention to the items and numbers orders. As there are**seven shirts**out of a total of**eleven items of clothing**the ratio of**shirts to clothes**is**7:11**. - Ratios questions require you to use various math operations such as basic arithmetic, fractions, decimals and percentages and to quickly and accurately convert between them, using a calculator at times. Therefore, for further calculator skill mastering - feel prepared by reading how to use a calculator.

For every one boy in the class there are three girls. **What is the ratio of girls to boys?**

- 1:3
- 3:1
- 1:4
- 4:1

The correct answer is **B** (3:1)

The order of words in the sentence is very important: for every one boy there are 3 girls, thus the ratio of boys to girls is 1:3. However, for every three girls there is one boy, thus the ratio of **girls to boys** is 3:1.

Mike owns a collection of toy cars. 3/8 of the cars are green and the rest are pink.

What is the ratio of green to pink cars?

- 3:5
- 3:8
- 5:8
- 8:5

The correct answer is **A** (3:5)

3/8 of the total number of cars are green. This means that out of a set of 8 cars in total, 3 are green. Since the cars are either green or pink, the remaining 5 cars are pink. Thus, the ratio of green to pink cars is: 3:5.

Peter and Paul are roommates.

Each of them pays the rent, according to their room’s size.

Peter’s room is twice as big as Paul’s.

**How much rent does Peter pay, if the total rent is £1200?**

- 600
- 800
- 400
- 500

The correct answer is **B** (800)

The ratio of Peter's room size to Paul's room size is 2:1. According to the data, the same ratio applies to rent payment: Peter pays twice as much as Paul. We can represent this using an equation: 2X + X =1200

*Peter pays 2X while Paul pays X and together the payment equals 1200 (total sum of rent). *

If we develop this equation we get: 3X = 1200 => X = 400

This means, that Paul pays 400 and Peter pays 800.

Scott and Rachel are enthusiastic car collectors. The cars they own are either German or Japanese made cars.

The German to Japanese ratio in Scott's collection is 5:2 in favour of the Germans.

The German to Japanese ratio in Rachel's collection is 4:3 in favour of the Germans

The number of Japanese cars Scott owns is identical to the number of Japanese cars Rachel owns.

**What is the ratio between the total amount of cars (German and Japanese) Scott has and the total amount of cars Rachel has?**

- 15:8
- 9:7
- 1:1
- 3:2

The correct answer is** D **(3:2)

A good way to tackle this question will be to use a ratio table:

German | Japanese | |

Scott | 5 | 2 |

Rachel | 4 | 3 |

In the question we are given 2 sets of ratios between German and Japanese cars in each collection. Each collection has its own row in the table.

Since we know the number of Japanese cars is identical in both collections we will modify the table so it will indicate this equality in a manner that is easier to manipulate and preferably using the minimum common product. We will expand Scott's ratio by 3 and Rachel's ratio by 2:

German | Japanese | |

Scott | 15 | 6 |

Rachel | 8 | 6 |

Thus, we can see that the ratio between Scott's total amount of cars and Rachel's total amount of cars is 21:14, simplify the ratio, dividing it by 7 => **3:2**

__Shortcut way:__

Rachel =R

Scott = S

For Japanese cars: 2S=3R

Since you are told that the number of Scott's Japanese cars equals to the number of Rachel's Japanese cars, then the Scott to Rachel Japanese cars ratio also represent the ratio between Scott's total number of cars and Rachel's total number of cars:

Therefore: S/R=3/2

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

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

- 5:29
- 2:7.6
- 4.3:5.8
- 7.6:2
- 11.6:2

The correct answer is **A** (5:29)

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

Roger wanted his garden to have a ratio of T:2R:4B. However, Roger’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 Roger’s garden is T:2R:8.6B. Therefore, the ratio of roses to plants in Roger’s garden is 2:(1+2+8.6) = 2:11.6 = 10:58 = 5:29.

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**Familiarisation:**Familiarise and rehearse maths operations such as basic arithmetic operations, fractions, decimals, percentages etc.**Read each question carefully.****Mark differently:**Mark the items in the question and their corresponding amounts in different colours to differentiate between them.**Figure out what you are asked about.****Pay attention to the order:**The order of items and their corresponding numbers, as presented in the question, may differ from the order of items you are asked to write in ratios.**Numerator and denominator:**When asked to write a ratio of x to y in the form of a fraction, the**first item**in the ratio (in this case - x) will appear as the**numerator**and the**second item**in the ratio (in this case - y) will appear as the**denominator**.**Estimation and Elimination:**At times a full and accurate calculation is not needed in order to solve the question correctly. Observe the question's data as well as the answer choices, and use the time saving technique of estimation and elimination when possible, to reach the correct answer.

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