Ratio and proportion find applications in solving many daily life problems for example while we are comparing altitudes, weights, length, and time or if we are negotiating with company transactions, also while adding elements in cooking, and much more. Ratios refer to the quantitative relation between two numbers or amounts or quantities. It shows the number of times one value contains the other value or is contained within the other value. Proportion simply implies that one ratio is equal to the other. We are going to learn the key concepts of ratio and proportion along with the various types of questions, types of proportion, tips and tricks with formulas, etc. We have also added a few solved examples, which candidates will find beneficial in their exam preparation.

What is Ratio and Proportion in Maths

Ratios are used for comparing two quantities of an identical style whereas when two or more such ratios are identical, they are declared to be in proportion.

Maths Notes Free PDFs

Ratios are used when we are required to express one number as a fraction of another. If we have two quantities, say x and y, then the ratio of x to y is calculated as x/y and is written as x:y. The first term of the ratio is called antecedent and the second term is called the consequent.

Compound ratio is the ratio obtained if two or more ratios are given and the antecedent of one is multiplied by the antecedent of others and consequences are multiplied by the consequences of others. Compounded ratio of the ratios (a : b), (c : d), (e : f) will be (ace : bdf)

Proportion is an equation that specifies that the two given ratios are identical to one another. We can say that the proportion states the equivalency of the two fractions or the ratios i.e. Equivalent Ratios. Proportions are represented by the symbol (::) or equal to (=).

That is the proportion is signified by double colons. For example, ratio 6 : 8 is the same as ratio 3 : 4. This can be written as 6 : 8 :: 3 : 4.

Product of means = Product of extremes

Thus, a : b :: c : d ⇔ (b × c) = (a × d)

A ratio is a way to compare two things of the same kind, like comparing the number of boys to girls in a class or the amount of sugar to water in a juice recipe.

We use division to find a ratio. For example, if there are 2 apples and 5 oranges, we say the ratio is 2 to 5. This means for every 2 apples, there are 5 oranges.

We can write a ratio in three ways:

• 2 to 5
   
• 2:5 (using the colon symbol)
   
• 2/5 (like a fraction)
   

But remember! You can only compare things using a ratio when they are in the same unit. For example, you can compare 5 meters to 10 meters, but not 5 meters to 10 liters.

Ratios are very useful in real life – in recipes, shopping discounts, sharing things equally, and even in sports scores!

So, a ratio tells us how many times one quantity is compared to another. It helps us understand relationships between two amounts easily and clearly.

Types of Proportion

The proportion can be categorized into the following types:

• Direct Proportion
• Inverse Proportion
• Continued Proportion

Direct Proportion: In proportion, if two sets of given numerals are rising or falling in the same ratio, then the ratios are considered to be directly proportional to each other. That is we can say that direct proportion illustrates the relationship between two portions wherein the gains in one there is a growth in the other quantity too. Likewise, if one quantity drops, the other also decreases.

Therefore, if “X” and “Y” are two quantities, then the direction proportion is composed as 

x∝y.

Inverse Proportion: The inverse proportion as the name outlines are in contrast to the direct one; where the relationship between two quantities is defined such that growth in one leads to a decline in the other quantity. Likewise, if there is a drop in one portion, there is an expansion in the other portion.

Accordingly, the inverse proportion of two quantities, say “p” and “q” is represented by 

p∝(1q).

Continued Proportion: If we assume two ratios to be p: q and r: s and we are interested in determining the continued proportion for the given ratio. Then we transform the means to a single term/digit. That is we will find the LCM of means.

• For the provided ratio, the LCM of q & r will be qr.
• The next step is to multiply the first ratio by r and the second ratio by q as shown:
• First ratio- rp:rq
• Second ratio- rq:qs
• Thus, the continued proportion can be documented in the form of \(rp : qr : qs[/ latex]

Difference Between Ratio and Proportion

The difference between ratio and proportion is given in the table below:

Ratio and Proportion Summary

What is a Ratio?

A ratio is a way to compare two or more quantities of the same kind.
Example: If you have 2 apples and 3 oranges, you can write their ratio as 2:3. This means for every 2 apples, there are 3 oranges.

What is Proportion?

A proportion shows that two ratios are equal.
You can write it in two ways:

• a : b = c : d
   

• a : b :: c : d
  This means the comparison between a and b is the same as the comparison between c and d.

Important Rules about Ratio and Proportion

• If you multiply or divide both parts of a ratio by the same number, the ratio stays the same.
  Example: 2:3 is the same as 4:6 because both numbers are multiplied by 2.

Continued Proportion

• For three numbers to be in continued proportion:
  The first is to the second, as the second is to the third.
  Example: 2:4:8
  → 2:4 = 1:2 and 4:8 = 1:2, so they are in continued proportion.
   

• For four numbers to be in continued proportion:
  The first is to the second, as the third is to the fourth.
  Example: 3:6 and 9:18 → both equal 1:2, so they are in proportion.

Important Properties of Proportion

When two ratios are equal, we say they are in proportion. Let’s say:

If a : b = c : d, then the following tricks (properties) can help solve problems faster:

1. Addendo

If a : b = c : d,
then (a + c) : (b + d)
You can add both parts of the ratios on top and bottom.

2. Subtrahendo

If a : b = c : d,
then (a – c) : (b – d)
You can subtract both parts of the ratios from top and bottom.

3. Dividendo

If a : b = c : d,
then (a – b) : b = (c – d) : d
Subtract top from bottom, then compare with the bottom.

4. Componendo

If a : b = c : d,
then (a + b) : b = (c + d) : d
Add top and bottom, then compare with the bottom.

5. Alternendo

If a : b = c : d,
then a : c = b : d
You can swap across the ratio.

6. Invertendo

If a : b = c : d,
then b : a = d : c
You can flip both sides of the ratio.

7. Componendo and Dividendo

If a : b = c : d,
then (a + b) : (a – b) = (c + d) : (c – d)
Add and subtract the terms, then compare.

Important Notes on Ratio and Proportion 

• We can compare two things only when they have the same type of units (like kg with kg, meters with meters, etc.).
   
• Two ratios are in proportion when they are equal.
  For example, 2:4 and 3:6 are in proportion because both are equal when simplified.
   
• To check if two ratios are equal, you can use the cross multiplication method.
  Example: For 2:3 and 4:6, multiply 2 × 6 and 3 × 4. If both are the same, the ratios are in proportion.
   
• If you multiply or divide both parts of a ratio by the same number, the value of the ratio does not change.
  Example: 3:5 is the same as 6:10 (both multiplied by 2).
   
• When three numbers are in continued proportion, it means the first is to the second as the second is to the third.
  Example: a:b = b:c
   
• For four numbers in continued proportion, the first is to the second as the third is to the fourth.
  Example: a:b = c:d

Ratio and Proportion Tricks 

Ratio and Proportion Examples

Example 1: If A : B = 2 : 3 and B : C = 5 : 7 then what is the ratio A : B : C ?

Solution: A : B = 2 : 3 B : C = 5 : 7

Multiply by 3/5 so as to make the ratio term of B Common, B : C = 5 × 3/5 : 7 × 3/5

⇒ B : C = 3 : 21/5

A : B : C = 2 : 3 : 21/5

=2 × 5 : 3 × 5 : 21/5 × 5

Hence, A : B : C = 10 : 15 : 21

Example 2: What is the equivalent compound ratio of 17 : 23 ∷ 115 : 153 ∷ 18 : 25

Solution: We know, compound ratio of the ratios (a : b), (c : d), (e : f) will be (ace : bdf) Thus, the compound ratio of (17 : 23), (115 : 153), (18 : 25) = (17 × 115 × 18) / (23 × 153 × 25) = 2 : 5

Example 3: If 3 : 27 ∷ 5 : ?

Solution: If 3 : 27 ∷ 5 : ?

3/27 = 5/?

? = 5 × 27/3

? = 45

Example 4: Find the mean proportional between 14 & 15?

Solution: Mean proportional = √(ab)

⇒ √(14 × 15)

⇒ 14.5

So, the mean proportional of 14 and 15 = 14.5

Example 5: Mean proportion of 4 and 36 is a and third proportional of 18 and a is b. Find the fourth proportion of b, 12, 14.

Solution: Given,

Mean proportional of 4 and 36 = a

⇒ a2 = 4 × 36

⇒ a = 12

Third proportional of 18 and 12 = b

⇒ 122 = 18 × b

⇒ b = 8

Fourth proportional of 8, 12 and 14

⇒ 8/12 = 14/?

⇒ ? = 21

Example 6: A bag has coins of Rs. 1, 50 Paise and 25 Paise in ratio of 5 : 9 : 4. What is the worth of the bag if the total number of coins in the bags is 72?

Solution: ⇒ Number of Rs. 1 Coins = 5/18 × 72 = 20

⇒ Number of 50 Paise coins = 9/18 × 72 = 36

⇒ Number of 25 Paise coins = 4/18 × 72 = 16

⇒ Total worth of the bag = (20 × 1) + (0.5 × 36) + (0.25 × 16) = 20 + 18 + 4 = Rs. 42

Example 7: If 18 : 13.5 : : 16 : x and (x + y) : y : : 18 : 10, then what is the value of y?

Solution: 18 : 13.5 : : 16 : x x = (16 × 13.5)/18 x = 12

Now,

(x + y) : y : : 18 : 10

(12 + y) : y : : 9 : 5 5(12 + y) = 9y

60 + 5y = 9y

4y = 60

y = 15

Example 8: There are a certain number of Rs.10, Rs.20 and Rs.50 notes available in a box. The ratio of the number of notes of Rs.10, Rs.20 and Rs.50 is 3 ∶ 4 ∶ 6. The total amount available in a box is Rs.2460. The amount of Rs.10 and Rs.50 in a box is –

Solution: Let the number of notes of Rs.10, Rs.20 and Rs.50 be 3a, 4a and 6a respectively. Given,

⇒ 10 × 3a + 20 × 4a + 50 × 6a = 2460

⇒ 410a = 2460

⇒ a = 6

Number of notes of Rs.10 = 3 × 6 = 18

Number of Notes of Rs.20 = 4 × 6 = 24 Number of notes of Rs.50 = 6 × 6 = 36

Required amount = 10 × 18 + 50 × 36 = Rs.1980

Example 9: Mr. Raj divides Rs. 1573 such that 4 times the 1st share, thrice the 2nd share and twice the third share amount to the same. Then the value of the 2nd share is:

Solution: Given: Total amount = Rs. 1573

Calculation: Let the share of A, B and C is 4A : 3B : 2C. A : B : C = 1/4 : 1/3 : 1/2 = 3 : 4 : 6

The value of the 2nd share = (4/13) × 1573 = Rs. 484

Example 10: Wayne wants to use Nitrogen, Potassium, and Phosphorus in his field as fertilizers. When any of them is mixed in the field, their quantity reduces by 1 kg every day due to chemical reactions. He mixed Nitrogen, Potassium, and Phosphorus on 7th November, 9th November, and 15th November, respectively. He spent equal amounts on buying each of the three. What should be the ratio of prices of Nitrogen, Potassium, and Phosphorus, so that there is an equal quantity of each of them in the field on 16th November, and that quantity is 11 kg?

Solution: Given: Quantities of Nitrogen, Potassium and Phosphorus in the field on 16th November are 11 kg.

Concept used: If equal amounts are spent on buying the components, the ratio of their prices will be inverse of ratios of their quantities.

Calculation: Nitrogen was mixed on 7th November.

Quantity of Nitrogen when it was mixed = 11 + (16 – 7) = 20 kg

Potassium was mixed on 9th November.

Quantity of Potassium when it was mixed = 11 + (16 – 9) = 18 kg

Phosphorus was mixed on 15th November.

Quantity of Phosphorus when it was mixed = 11 + (16 – 15) = 12 kg

Expenditure = quantity bought × Price per unit

⇒ Ratio of their prices = (1/20):(1/18):(1/12) = 9:10:15

∴ The required ratio is 9 ∶ 10 ∶ 15.

We hope you found this article regarding Ratio and Proportion informative and helpful, and please do not hesitate to contact us for any doubts or queries regarding the same. You can also download the Testbook App, which is absolutely free and start preparing for any government competitive examination by taking the mock tests before the examination to boost your preparation. For better practice, solve the below provided previous year papers and mock tests for each of the given entrance exam: