To express the relative sizes, the ratio of one quantity to another is used. Ratios is often expressed in the form of a fraction.

Here,

The first quantity is the numerator

The second quantity is the denominator.

Thus, if u and v are positive quantities, then the ratio of u to v can be written as the fraction “\(\frac{u}{v}\)”.

Also expressed as u : v (u colon v)

Here,

The first quantity is the numerator

The second quantity is the denominator.

Thus, if u and v are positive quantities, then the ratio of u to v can be written as the fraction “\(\frac{u}{v}\)”.

Also expressed as u : v (u colon v)

A ratio is a comparison between two or more quantities (in the form of a quotient).

Here, the number of boys is compared to the number of girls.

Now, it is important to note that, the ratio 2 to 5 does not imply that there are exactly 2 boys and exactly 5 girls at the school.

Ratio express nothing about actual quantities, instead, expresses only about proportion.

**Proportion**:

An equation relating two ratios is known as a proportion

**Example**: \(\frac{15}{10}\) = \(\frac{3}{2}\)

Here, 2 to 5 ratio means in the school, there are 2 boys for every 5 girls.

Also, it is very important to know the order of the numbers.

There are three types of notations to express the ratio i.e.

Here, for example

- 2 to 5 (Used ‘to’ to separate the terms)
- 2 : 5 (Used a colon to separate the terms)
- \(\frac{2}{5}\) (Used a fraction to separate the terms)

In general, if the information that can re-worded in the form

“For every something ‘x’, there is/are something else ‘y’ ….”, then we are typically dealing with a ratio question. So, be sure to keep this in mind.

Most ratio questions fall into two categories i.e.

Equivalent ratios

Portioning

**Equivalent ratios**

1. Should recognize that the ratio 1 : 2 is equivalent to 3 : 6. Both ratios express the same proportions.

Can create equivalent ratios by multiplying or dividing both terms by the same value.

**Examples**:

**1**. Consider \(\frac{2}{7}\)

Multiply and divide 2/7 by 2 then the ratio is \(\frac{4}{14}\)

Hence, \(\frac{2}{7}\) = \(\frac{4}{14}\)

**2**. 3 : 4

Multiply both terms 3 : 14 by 5 then the ratio is 15 : 20

Hence, 3 : 4 = 15 : 20

**Portioning into ratios**

- Add the terms in the ration and let the sum = T
- Divide the total quantity into T equal parts.
- Divide the T equal parts into the target ratio.

**Example**:

Bruce has cookies, which he will give to Kendra and Patty in a 2 to 1 ratio. How many cookies does each person receive?

- Add the terms in the ratio i.e. 2 + 1 = 3

This expresses that

“For every 3 cookies, Kendra receives 2 and patty receive 1”.

“For every 3 bags of cookies, Kendra receives 2 and patty receive 1”. - Divide the cookies into 3 bags
- Divide the bags in a 2 : 1 ratio

This means, Kendra gets 2 bags and Patty gets 1 bag.

Kendra: Patty

10 : 5

Kendra gets 5 cookies and Patty gets 5 cookies.

**Combining ratios**

To solve these, there are there are two different approaches wherever required to combine the information from two ratios.

Approach 1:

Find the equivalent ratios until there are matching terms

Combine the ratios

**Example**: In a certain dog kennel, the ratio of the number of poodles to the number of boxers is 5 to 2, and the ratio of the number of boxers to the number of terriers is 3 to 4. If there are 60 poodles, how many terriers are there?

Given that,

Poodles : boxers = 5 : 2

Boxers : terriers = 3 : 4

Find the equivalent ratios so that ratios have matching terms.

Since both ratios have number of boxers, use this as a bridge to connect the two ratios

As there is no matching in the given ratios

Multiply by 3 → i.e. Poodles : boxers = 5 x 3 : 2 x 3 = 15 : 6

Multiply by 2 → i.e. Boxers : terriers = 3 x 2 : 4 x 2 = 6 : 8

Now, poodles : boxers : terriers = 15 : 6 : 8

If boxers is ignored, then poodles : terriers = 15 : 8

Now, the ratio \(\frac{poodles}{terriers}\) = \(\frac{15}{8}\)

Given that, number of poodles = 60. Replace it.

Let T be the number of terriers

\(\frac{60}{T}\) = \(\frac{15}{8}\)

By cross multiplying,

15 T = 60 * 8

T = \(\frac{ (60 * 8)}{15}\)

T = 32

Therefore, there are 32 terriers.

Approach 2:

Solve the one ratio

Then apply the results to the other ratio.

By using the same example mentioned above,

Given that poodles/boxers = \(\frac{5}{2}\)

Also given that number of poodles = 60

Let B be the number of boxers

\(\frac{60}{B}\) = \(\frac{5}{2}\)

By cross multiplying

5 B = 60 *2

B = \(\frac{120}{5}\)

B = 24

Given that boxers/ terriers = \(\frac{3}{4}\)

Let T be the number of terriers

\(\frac{24}{T}\) = \(\frac{3}{4}\)

By cross multiplying

T = 32

Therefore, number of terriers = 32

- Ratio = \(\frac{of}{to}\)

\(\frac{12}{20}\) = \(\frac{6}{10}\) = \(\frac{3}{5}\) = 3 :: 5 = Oranges :: Apples

**2. The ratio of boys to girls is 3 to 4. If there are 135 boys, how many girls are there?**

**Solution**:

- Use cross multiplication

\(\frac{3}{4}\) = \(\frac{135}{girls}\)

(4 x 135) = (3 x girls) = 540

\(\frac{540}{ 3}\) = girls

girls = 180

There are 180 girls.

**3. In a certain year, California extracted \(\frac{2}{7}\)th and Texas extracted \(\frac{1}{7}\)th of all the Uranium ore extracted in the United States. If all the other states combined extracted 28 million tons that year, how many million tons did Texas extract that year?**

- A. 12

B. 7

C. 14

D. 5

E. 8

**Solution**:

- Together, California and Texas extracted \(\frac{2}{7} + \frac{1}{7}\) = \(\frac{3}{7}\)th of all the Uranium ore extracted in the country.

Therefore, all the other states extracted 1 – \(\frac{3}{7}\) = \(\frac{4}{7}\)th of all the Uranium ore extracted in the country.

The combined production of all the other states = 28 million tons.

i.e., \(\frac{4}{7}\)th of total production = 28 million tons.

So, total production =28 * \(\frac{4}{7}\) = 49 million tons.

Texas extracted \(\frac{1}{7}\)th of the total = \(\frac{1}{7}\)* 49 = 7 million tons.

The correct Answer is 7 million tons and the correct Choice is (B).

**4. If the ratio of the ages of two friends A and B is in the ratio 3 : 5 and that of B and C is 3 : 5 and the sum of their ages is 147, then how old is B?**

**Solution**:

- The ratio of the ages of A and B is 3 : 5.

The ratio of the ages of B and C is 3 : 5.

B’s age is the common link to both this ratio. Therefore, if we make the numerical value of the ratio of B’s age in both the ratios same, then we can compare the ages of all 3 in a single ratio.

This can be done by getting the value of B in both ratios to be the LCM of 3 and 5 i.e., 15.

The first ratio between A and B will, therefore, be 9 : 15 and

The second ratio between B and C will be 15: 25.

Now combining the two ratios, we get A : B : C = 9 : 15 : 25.

Let their ages be 9x, 15x and 25x.

Then, the sum of their ages will be 9x + 15x + 25x = 49x

The question states that the sum of their ages is 147.

i.e., 49x = 147 or x = 3.

Therefore, B’s age = 15x = 15*3 = 45

**5. If 1 : 2 = 9 : x, then find the value of x?**

**Solution**:

- 1 : 2 = 9 : x

Rewrite the given equation i.e.

\(\frac{1}{2}\) = \(\frac{9}{x}\)

Apply If \(\frac{a}{b}\) = \(\frac{c}{d}\) then ad = bc

1 * x = 2 * 9

x = 18

So, the value of x is 18