Ratio and Proportion: Types, Tricks, Cross Multiplication, Direct & Inverse Variation, & Fast Methods

Ratio and proportion are among the most useful ideas in mathematics. They help us compare quantities, divide things fairly, scale recipes, solve speed problems, understand mixtures, and move into higher math with confidence. Many students first see ratio as “a to b,” and proportion as “two ratios equal each other,” but the topic is much richer than that. When you understand ratio and proportion properly, you begin to see comparison, scaling, and balance in a powerful way.

This topic appears in school exams, competitive exams, business math, science, maps, recipes, partnership problems, and real-life decision making. A strong grip on ratio and proportion saves time and makes many questions easier.

In this article, you will learn ratio and proportion from the very beginning to advanced level, with clear logic, visual steps, short tricks, solved examples, and speed methods used by strong math students.

Definition of Ratio and Proportion

What Is a Ratio?

A ratio compares two or more quantities of the same kind. For example: If there are 2 boys and 3 girls, then the ratio of boys to girls is 2:3. This means for every 2 boys, there are 3 girls or for every three girls, there are two boys. A ratio tells us how much of one thing exists compared to another. Imagine it like a fraction, where there is total portion and divided portion. If you know fraction then good. If not, learn it first. Learn Fractions: Click Here

Ratio Can Be Written in Different Ways: There are normally four different ways to write Ratio. Based on above examples, let’s see.

• 2 to 3
• 2:3
• 40:60 percentage
• 2/3 or $\frac23$. The slash is a symbol of fraction. But a ratio is not fraction. They look similar.

All of these express the same comparison, though the meaning depends on context. In simple words, these four are the same. Ratio is for comparison among quantities and fraction is a part of a whole quantity.

What Is Proportion?

A proportion says that two ratios are equal. In proportion, we talk about equality of two or more ratios.
Example: 2:3 = 4:6
This is a proportion because both ratios have the same value. You can also write it as:
$\frac23 = \frac46$

How do we know that they are equal? Divide right side 4/6 by 2. You will get 2/3 again. Hence, a proportion is a balance between two ratios.

Big Idea: The Comparison

Ratio	Proportion
Ratio compares two or more quantities.	Proportion balances two or more quantities.
It tells, “How much of one thing compared to another?”	It tells, “Two or more ratios are equal.”
Examples: 4:5 6:8 9:11	Examples: 2:3 = 4:6 4:12 = 8:24 5:8 = 10:16
Note: We compare two or more quantities but that doesn’t mean, they are equal.	Note: The two or more ratios must be equal. If they are not equal, we do not call it proportion. e.g. 4:8 $\neq$ 5:9 These two rations are not equal, so not a proportion.

That is the simplest way to remember the difference.

How to Simplify Ratios

Ratio is not the same as addition or subtraction. It is a comparison. To simplify a ratio, divide both terms by their greatest common factor.

Examples:

• 12:18
  Common factor = 6 (A factor which can divide both and is the largest / greatest number.)
  12 ÷ 6 = 2
  18 ÷ 6 = 3
  So:
  12:18 = 2:3.
  
• 25:40
  Common factor = 5
  25 ÷ 5 = 5
  40 ÷ 5 = 8
  So:
  25:40 = 5:8
  
• If a classroom has 10 boys and 15 girls:
  The ratio of boys : girls = 10:15
  Common factor: 5
  10 ÷ 5 = 2
  15 ÷ 5 = 3
  This can be simplified to: 2:3 because both numbers can be divided by 5.

So the ratio is always best written in simplest form unless the question asks otherwise. If we try to check or test whether the ratios are equal to one another then it becomes proportion.
$2:3 = 10:15 \\ 2:3 = 12:18 \\ 5:8 \neq 12:18 \\ 5:8 \neq 10:15 \\ 2:3 = 10:15 = 12:18$

The 1st, 2nd, and 5th are proportion because they are equal, how? Let’s test with a very practical example a pizza.

Equal Distribution of Pizza (Proportion)	Distribution of Pizza
We have a whole pizza and we want to distribute it based on ratios. We have three example sets, right now.	When we do not want to distribute equally then!
1st) 2:3 = 10:15 First, we cut the pizza into portions. 2 + 3 = 5 portions or 10 + 15 = 25 portions. Trust me choose any, you will overall get the same size of pizza. Suppose Sally and Mark make portions 2 + 3, which means out of 5 portions Sally takes 2 slices and Mark 3. Easy! Now, Sally and Mark make smaller portions of pizza 10 + 15, out of 25. Sally takes 10 and Mark 15. Still the same portion. Do you remember the common factor above? Yes, common factor of 10:15 is 5. It means Sally gets 5 + 5 slices and Mark gets 5 + 5 + 5 slices. A bunch of 5 slices extra than Sally. The same way, 2:3, the common factor is 1. Sally gets 1 + 1 slices and Mark gets 1 + 1 + 1 slices. See the pattern: Sally (1 + 1) or (5 + 5) Mark (1 + 1 + 1) or (5 + 5 + 5)	3rd) $\bold{5:8 \neq 12:18}$ We cut the same pizza for Sally and Mark. Let’s test the left side ratio first. Sally got 5 out of 13 pizza slices. Mark got 8 out of 13 pizza slices. The common factor is 1 that means we distribute slices of pizza one by one. The right side ratio has 12 + 18 = 30 slices. Sally got 12 and Mark got 18 slices. The common factor is a bunch of 6 slices. We distribute 6 slices at once in group. Sally (1 + 1 + 1 + 1 + 1) or (6 + 6) Mark (1 + 1 + 1 + 1 + 1 + 1 + 1 + 1) or (6 + 6 + 6) It is not equal. Do you see the same pattern like the left side, no. Look closely, in common factor 1, Mark got 3 portion extra than Sally and on the other hand in common factor 6, Mark only got 1 portion extra than Sally. Mark: 3 portion on left and 1 portion on right. $3 \neq 1$ Unequal.
2nd) 2:3 = 12:18 Same way like above Sally got 2 and Mark got 3 slices out of 5 slices. In right side ratio, Sally got 12 and Mark got 18 slices out of 30 slices. The pattern: Sally (1 + 1) or (6 + 6) Mark (1 + 1 + 1) or (6 + 6 + 6) Again whether we cut 5 big slices or 30 small slices, the proportion goes the same way.	4th $\bold{5:8 \neq 10:15}$ We do the same way like above. 10 + 15 = 25 slices and common factor is bunch of 5 portion at once. Sally (1 + 1 + 1 + 1 + 1) or (5 + 5) Mark (1 + 1 + 1 + 1 + 1 + 1 + 1 + 1) or (5 + 5 + 5) Again the same unbalanced distribution. Mark: 3 portion on left and 1 portion on right. $3 \neq 1$ Unequal.
3rd) 2:3 = 10:15 = 12:18 Just like the example 1st, but we have an additional 12:18. Let’s test it too. Sally (1 + 1) or (5 + 5) or (6 + 6) Mark (1 + 1 + 1) or (5 + 5 + 5) or (6 + 6 + 6) This is called equal distribution. In all three ratios, Mark gets 1 portion extra every time. The portion really matters here.	If you are struggling, I suggest you to talk to our Expert, consult and have free doubt sessions with them. Go to homepage and look below the expert section.

Important Rule: You simplify ratios just like fractions, but remember ratios compare quantities. The meaning stays the same when both parts are scaled equally.

Equivalent Ratios

Equivalent ratios are ratios that have the same value. They are like proportions. We have learnt them above but we are giving you another perspective of common factors in it. We know the opposite of divide is multiply, right?

Example:

• 2:3 = 4:6 = 6:9 = 8:12
  These all represent the same comparison. You can make equivalent ratios by multiplying both terms by the same number.
  4:6 when 2:3 multiplied by 2
  6:9 when 2:3 multiplied by 3
  8:12 when 2:3 multiplied by 4
  Did you see the pattern 2:3?
  
• 3:5 = 6:10 = 9:15
  6:10 when 3:5 multiplied by 2
  9:15 when 3:5 multiplied both by 3:

This is very useful in proportions and scaling.

Ratios as Sharing

Ratios are often used to divide amounts fairly. Sharing means when we want to distribute portions, use ratios to divide it accurately.
Example: Share 60 candies to Mana and Maya in the ratio 2:3.
Step 1: Add the parts
2 + 3 = 5 parts
Step 2: Divide total by total parts
60 ÷ 5 = 12 per part
Step 3: Multiply each ratio part
2 × 12 = 24
3 × 12 = 36
Answer: Mana gets 24 candies and Maya 36 candies.

Ratio and Fraction

A ratio and a fraction are closely related. You can say that a ratio is a bigger form of fraction. A fraction is portion of a ratio. 2:3 can be written as $\frac23$. But they are not always used in the same way.

• Ratio compares one quantity to another.
• Fraction represents part of a whole quantity.

Example: 2 + 3 = 5.
If 2 out of 5 students are boys, then boys as a fraction of the class is 2/5.
If boys and girls are 2 and 3 respectively, then boys : girls = 2:3.

Ratio and Division

A ratio can also be understood through division. Like we have tested above by multiplying common factors, the same way, we did by dividing at first, remember!
Example: If 8 pencils are divided among 4 students, each gets 2.
The ratio 8:4 simplifies to 2:1. The common factor is 2 here. This is why ratio and division are deeply connected.

How to Solve Proportion: All Methods and Rule

We already know what proportion is. There are many methods and rule used to solve Proportion. One by one we will learn all, but what we have learnt above so far, you should understand the concepts and core to it.

Cross Multiplication

As the name suggest, we multiply the fractions inside the ratio. The cross refers as left side denominator multiplied to right side numerator. The same way, right side denominator with left side numerator.
The Rule: If a:b = c:d Then: a × d = b × c.

Examples:

• 2:3 = 4:6
  Look fractions: $\frac23 = \frac46$
  Cross multiplication: $\frac23 \times \frac64 = \frac{12}{12}$
  This is called cross multiplication.
  2 × 6 = 12
  3 × 4 = 12
  Since both sides are equal, the proportion is correct.
  
• Sometimes, the both side may not be equal then we do not call it proportion but still we use Cross Multiplication to solve. 3:6 = 4:5
  We should always simplify if we see opportunity. The common factor is 3.
  $3 \div 3 = 1\\ 6 \div 3 = 2$
  Fractions: $\frac12 = \frac45$
  Cross-multiplication: $\frac12 \times \frac54 = \frac58$
  1 × 5 = 5
  2 × 4 = 8
  Since both sides are not equal, this is not a proportion.

This rule is one of the most important shortcuts in ratio and proportion.

Direct Proportion

In direct proportion, when one quantity increases, the other also increases in the same ratio. In simple words, both numerator and denominator increases in direct proportion.

Examples:

• If 2 notebooks cost 10 rupees, then 4 notebooks cost 20 rupees.
  At first we had this 2:10 then we simply double it. Multiplying the ratio by 2, we get 4:10.
  $\frac{2}{10} \times \frac22 = \frac{4}{10}$ But we are doing direct proportion and we must check constant.
  
  Test: $2:10 = \frac{2}{10} = \frac15$
  
  $4:20 = \frac{4}{20} = \frac15$. Notice the constant 1/5 in both. It is confirmed Direct Proportion.
  
• More workers usually mean more output if speed stays the same. Suppose, we had 6 workers and they produce 30 items in hour. What if we hire 18 workers, how many items we can produce in an hour.
  The original: 6:30.
  Increase in workers from 6 to 18. The common factor here is 3 times.
  6 × 3 = 18.
  Imply the common factor to denominator. 30 × 3 = 90.
  The new ratio is 18:90.
  Test: $6:30 = \frac{6}{30} = \frac15$
  
  $18:90 = \frac{18}{90} = \frac15$
  This is direct proportion. You will see different problems like that in exams, so be prepared.

Direct Proportion Formula

The formula is about constant growth on ratio. Just like above, we found a common factor 2. If x and y are in direct proportion then x/y = constant. The constant is an output that comes out when the x and y divides.

Example: 2/10 = 4/20 = 6/30
The common factor of growth here is 2, then 3.
$\frac{2}{10} \times \frac22 = \frac{4}{20} \\ \frac{2}{10} \times \frac33 = \frac{6}{30}$
Let’s check constant:
$2:10 = \frac{2}{10} = \frac15 \\ \\ 4:20 = \frac{4}{20} = \frac15 \\ \\ 6:30 = \frac{6}{30} = \frac15$

All these ratios are equal because both numerator and denominator grows by a constant.

Inverse Proportion

In inverse proportion, when one quantity increases, the other decreases. In simple words if denominator increases then numerator must be decreasing. The same way if numerator increases then denominator must be decreasing.

Remember one important thing, it is a proportion, so the multiplication of the same ratio’s numerator and denominator must be equal to other ratios.

Examples:

• If 4 workers finish a job in 6 days, then 8 workers may finish it in 3 days, assuming equal efficiency.
  4:6 = 8:3
  Let’s figure out the ratios are equal, means proportion or not.
  In inverse, we need one increasing and another decreasing but the common factor must be the same. The common factor is 2.
  $\frac{4 \times 2}{6 \div 2} = \frac83$.
  Test the constant:
  $4:6 = 4 \times 6 = 24 \\ 8:3 = 8 \times 3 = 24$
  This is a inverse proportion because the factor is the same for both and one increases and another decreases.

Inverse Proportion Formula

If x and y are in inverse proportion: x × y = constant. In direct proportion the x and y of the same ratio divides but in inverse we multiply x and y of the ratio and test the constant with others.

Example: 2:12, 4:6, and 8:3. Are these ratios inverse proportion or not?
The common factor here is 2.
$\frac{2 \times 2}{12 \div 2} = \frac46 \\ \\ \frac{8 \div 2}{3 \times 2} = \frac46$
Let’s test the constant:
2 × 12 = 24
4 × 6 = 24
8 × 3 = 24

So the product stays the same. The ratios are equal which means proportion.

Types of Ratios

We have learnt proportions and a little about ratio as well but there are many types of ratios out there. Let’s learn them all one-by-one.

Simple Ratio

As the name suggests, it is a normal ratio where there are two quantities. The one, we have done so far.
Examples:

• 4:6
• 3:7
• 2:8
• 1:9
• The Problem: A prize of Rs. 4,500 is shared between two friends, Ralph and Ban, in a ratio of 4:5. How much money does each person receive?
  Solution: Total parts: 4 + 5 = 9.
  Find the value of single part: 4500 ÷ 9 = 500.
  Calculate each person’s share:
  Ralph share (4 parts): 4 × 500 = 2000
  Ban share (5 parts): 5 × 500 = 2500.
  Total: 2000 + 2500 = 4500.

Compound Ratio

Compound means the sum of two or more values. Here we use Compound Ratio then a ratio formed by combining two or more ratios.
Example: A Basic Compound ratio of 3:2, 4:7, and 1:9.
To solve a basic compound ratio, we multiply all the numerators together and form a new one. The same way, we multiply all denominator.
$\frac{3\times4\times1}{2\times4\times9} = \frac{12}{72}$. This is a compound. We have discussed before that if there is a possibility to simplify then do it. Notice the compound value’s largest common factor is 12.
So: $\frac{12\div12}{72\div12} = \frac16$ .
Every time you follow the rule and multiply the tops with tops and bottom with bottom. It is called Basic Compound Ratio.

Compound Ratio with Inverse: We have just finished an inverse. You should know the meaning of the word “Inverse.” It means opposite or reverse. We know the rule of compound from basic, we just need to inverse it now.
Example: 4:5 and 3:9.
It would be better to simplify 3:9 first by dividing 3.
$\frac{3\div3}{9\div3} = \frac13$.
Now let’s inverse it and solve: $\frac45 = \frac13$
$\frac45 \times \frac31 = \frac{12}{5}$.
This is called inverse compound ratio. Just follow the rule and steps always.

Compound Ratio with Unknown Variable: When we have given two or more compound ratios and we are asked find a unknown variable means value. We use this.
Example: If the compound ratio of 5:8 and 3:7 is 45:x, find the value of x.
Note: The question did not say compound ratio with inverse, so we will use basic compound.
$\frac{5\times3}{8\times7} = \frac{15}{56}$.
Set it equal to asked ratio: $\frac{15}{56} = \frac{45}{x}$
Cross multiply to solve any problems like this.
$15\times x = 56\times45 \\ 15x = 2520 \\ x = \frac{2520}{15} \\ \\ x = 168.$
The ratio is 45:168. But here is simple hack for you. Notice 15 and 45. You can always simplify the problems to save some time.
$\frac{15}{56} = \frac{45}{x} \\ \\ x = \frac{56 \times 45}{15}$
Divide 45 by 15, you get: 3
$x = 56 \times 3 \\ x = 168.$
Easy right?

Word Problem Example: In a factory, the working hours are reduced in the ratio of 9:8, and the wages per hour are increased in the ratio of 4:5. What is the compound ratio that determines the change in the total weekly wages of a worker?
Solution: Write the ratios: Time change = 9:8, Wage change = 4:5
Total wages are calculated by multiplying Time by the Wage rate. Therefore, we compound these two ratios.
Multiply antecedents / numerator: 9 × 4 = 36
Multiply consequents / denominator: 8 × 5 = 40
The compound ratio is 36:40.
Simplify by dividing 4, the ratio you get is 9:10.
This means for every ratio 9 reduced in working hours increased ratio 10 in wages per hour.

Continued Ratio

A continued ratio compares three or more quantities of the same kind, expressed in the form a : b : c. To solve these problems, you link overlapping ratios by finding the Least Common Multiple (LCM) to create a single, unified ratio. You do remember LCM? If not, Learn LCM – Click Here.

Examples:

• The Problem: Unifying Overlapping Ratios
  In a bag, the ratio of Red to Green marbles is 3:5, and the ratio of Green to Blue marbles is 4:3. What is the combined continued ratio of Red : Green : Blue?
  The Solution: Write down the given ratios:
  Red : Green = 3:5
  Green : Blue = 4:3
  Find the LCM of the overlapping term (Blue), which is LCM (5, 4) = 20.
  $\begin{array}{|c|l|r|} \hline Calculations & Factors & Values \\ \hline 4÷2 = 2 & 2 & \bold{5, 4} \\ \hline 2÷2 = 1 & 2 & 5, 2 \\ \hline 5÷5 = 1 & 5 & 5, 1 \\ \hline \ \bold{LCM = 20} & ~ & 1, 1 \\ \hline \end{array}$
  2 × 2 × 5 = 20.
  We always solve this way: Follow the rule
  Ratio 1 is 3:5 and Ration 2 is 4:3 where Green comes twice 5 for first and 4 for second. We divide it to 20 then multiply the number with Ratio 1 and 2 to get Unified Overlapping Ratio.
  Take 5 from Ratio 1 for: 20 ÷ 5 = 4.
  Multiply with: 3 × 4 = 12 and 5 × 4 = 20.
  New Ratio 1 is 12:20.
  Take 4 from Ratio 2 for: 20 ÷ 4 = 5
  Multiply with: 4 × 5 = 20 and 3 × 5 = 15.
  New Ratio 2 is 20:15.
  See that in both ratio, now Green is 20.
  Combine them into a single continued ratio Red : Green : Blue = 12 : 20 : 15.
  
• The Problem: Dividing a Total Amount
  A sum of $4,200 is divided among four partners—Arnold, Becca, Carla, and Dean—in the continued ratio of 2 : 3 : 4 : 5. How much does each person receive?
  The Solution: Let the individual shares be 2x, 3x, 4x, and 5x.
  Total amount: 2x + 3x + 4x + 5x = 4,200
  $14x = 4200 \\ \\ x = \frac{4200}{14} \\ \\ x = 300.$
  Fill the x into individuals:
  Arnold: 2x = 2 × 300 = 600
  Becca: 3x = 3 × 300 = 900
  Carla: 4x = 4 × 300 = 1200
  Dean: 5x = 5 × 300 = 1500
  Let’s see the total to test its accuracy! 600 + 900 + 1200 + 1500 = 4500.
  
• The Problem: Solving Using a Known Variable
  The wealth of five siblings—Rothschild, Rockefeller, Windsor, Saudi, and Medici—are in the continued ratio of 9 : 7 : 5 : 3 : 1. If Saudi has 21 Trillion, find others wealth. (Assume others in Trillion as well)
  The Solution: Let the wealth be 9x, 7x, 5x, 3x, and 1x.
  Since we have given only Saudi’s wealth then: 3x = 21
  $3x = 21 \\ \\ x = \frac{21}{3} \\ \\ x = 7.$
  Fill the x into individuals:
  Rothschild: 9x = 9 × 7 = 63
  Rockefeller: 7x = 7 × 7 = 49
  Windsor: 5x = 5 × 7 = 35
  Saudi: 3x = 3 × 7 = 21
  Saudi: 1x = 1 × 7 = 7
  This is how we do it when the total is not given, the question sometimes gives individual worth. The whole process is somewhat same like above.

Part-to-Part Ratio

A part-to-part ratio compares different sections of a whole to one another (e.g., comparing apples to oranges) rather than comparing a single part to the entire total. They are not difficult at all. We have done many examples of these on top.

Examples:

• The Problem: Basic Scaling Problem
  The ratio of black marbles to blue marbles in a bag is 5:8. If there are 30 black marbles, how many blue marbles are there?
  The Solution: Set up ratio equation $\frac{Black}{Blue} = \frac58 = \frac{30}{x}$
  We need to find the x = Blue.
  Only way to solve is to do Cross-Multiply:
  $\frac{5}{8} = \frac{30}{x} \\ \\ x = \frac{30 \times 8}{5}$
  Divide 30 by 5, you get: 6
  $x = 6 \times 8 \\ x = 48.$
  So the actual marbles are Black : Blue = 30 : 48.
  
• The Problem: Sharing Based on a Total Amount
  The ages of two brothers, Amit and Shah, are in the ratio of 7:2. If the sum of their ages is 72, how old is each brother?
  The Solution: Total of Ratio: 7 + 2 = 9
  Divide total age to individual portions: 72 ÷ 9 = 8.
  Multiply the individual portion to the ratio:
  Amit (7 portion): 7 × 8 = 56
  Shah (2 portion): 2 × 8 = 16.
  Test it 56 + 16 = 72. Do you remember seeing problem like that above, right? The only change in this topic is One Ratio.
  
• The Problem: Finding the Difference Between Parts
  The ratio of girls to boys in a fight club is 4:7. There are 27 more boys than girls. How many total students are in the club and also tell boys and girls?
  The Solution: Always find the difference in ratio first: 7 – 4 = 3.
  Divide the difference against more boys: 27 ÷ 3 = 9.
  Multiply the result with girls to boys ratios total: 7 + 4 = 11.
  11 × 9 = 99.
  There are total 99 students.
  Individual student: 99 ÷ 11 = 9.
  Girls (4 portion): 4 × 9 = 36
  Boys (7 portion): 7 × 9 = 63.
  Confirm the answers: 63 – 36 = 27 more boys.

Part-to-Whole Ratio

A part-to-whole ratio compares a specific subgroup (the part) to the entire group (the whole). These ratios can be written as fractions, decimals, or in “A to B” format. In simple words, look the name “from a part of ratio to whole ratio” meaning from a portion of ratio to find total value of ratio.

Examples:

• Problem 1: Finding the Whole
  A local animal shelter determines that 40% of all cats in the facility are calico. If there are 16 calico cats, what is the total number of cats in the shelter?
  Solution: Convert percentage to fraction: 40% of 100% always.
  $\frac{40}{100} = \frac25$
  Set up equation: The part (16 calico cats) related to the whole (x quantity) as 2 related to 5.
  Because 2 : 3 means 2 + 3 = 5. We have done this multiple times so far, right?
  $\frac25 = \frac{16}{x}$
  Cross-multiply: $x = \frac{16 \times 5}{2}$
  Divide 16 by 2 = 8.
  x = 8 × 5
  x = 40.
  There are 40 total cats in the shelter out of those 40 cats, 16 are calico and 24 are others.
  See, how we found the total value out of a part.
  16:24 = 16 + 24 = 40.
  
• Problem 3: Converting Part-to-Part into Part-to-Whole
  In a group of kids, the ratio of 8 year old to 13 year old is 9 : 11. If there are 60 children in total, how many of them are 13 year old?
  Solution: Ratio total: 9 + 11 = 20. (The ratio gives us is a Part-to-part but then we combine it to find the Part-to-whole.)
  13 year old ratio to total: 11:20
  $\frac{11}{20}$
  Find quantity by multiplying it with total children: $\frac{11}{20} \times 60$
  Divide 60 by 20 = 3.
  11 × 3 = 33.
  There are 33 children of 13 year age in the group.

*Fact: A Part-to-Part is 9 : 11 and Part-to-Whole is 11:20 or 9:20. These are different and must not be confused.

Solving Ratio Problems: Many Examples Used in Exams

You will see many real life math problems faced by students in their study and exams. Learn all at once.

Examples:

• Simplify 18:24
  GCF = 6 (Highest Common Factor = HCF)
  18 ÷ 6 = 3
  24 ÷ 6 = 4
  Answer: 3:4
  
• If the ratio of pens to pencils is 5:7 and there are 35 pencils, how many pens are there?
  Step 1: Ratio parts = 5 + 7 = 12
  Step 2: 35 pencils represent 7 parts
  Step 3: One part = 35 ÷ 7 = 5
  Step 4: Pens = 5 parts = 5 × 5 = 25
  Answer: 25 pens
  
• Divide 84 in the ratio 5:2
  Step 1: Add parts
  5 + 2 = 7
  Step 2: Find one part
  84 ÷ 7 = 12
  Step 3: Multiply
  5 × 12 = 60
  2 × 12 = 24
  Answer: 60 and 24
  
• Fast Shortcut Example
  If total is T and ratio is a:b:
  First sum = a + b
  One part = T ÷ (a + b)
  Shares = a × one part and b × one part
  
• Divide 180 in ratio 2:3:4
  Step 1: Add parts
  2 + 3 + 4 = 9
  Step 2: One part
  180 ÷ 9 = 20
  Step 3: Multiply each part
  2 × 20 = 40
  3 × 20 = 60
  4 × 20 = 80
  Answer: 40, 60, 80
  
• In partnership, money or profit is divided according to investment ratio.
  A invests 2000, B invests 3000.
  Ratio = 2000:3000 = 2:3
  If profit is 5000:
  Total parts = 2 + 3 = 5
  One part = 5000 ÷ 5 = 1000
  A gets: 2 × 1000 = 2000
  B gets: 3 × 1000 = 3000.
  
• If speed is constant, distance and time are directly proportional.
  If a car travels 60 km in 2 hours, how far in 5 hours?
  Speed: 60 ÷ 2 = 30 km/h
  Distance in 5 hours: 30 × 5 = 150 km
  This uses proportion.
  
• A juice mix uses water and syrup in ratio 4:1.
  If total mixture is 25 liters:
  Total parts = 4 + 1 = 5
  One part = 25 ÷ 5 = 5
  Water = 4 × 5 = 20 liters
  Syrup = 1 × 5 = 5 liters
  This is a classic ratio application.
  
• Map scales are ratios too.
  *Fun Fact: 1 cm on a map = 5 km in real life
  This means the ratio is: 1 km in cm = 1,00,000
  So: 1 cm : 1 Km
  1:500,000 if both are converted into the same unit
  Scaling is one of the most practical uses of ratio.
  
• The unitary method is a powerful technique.
  If 6 notebooks cost 90 rupees, how much do 10 notebooks cost?
  Step 1: Find cost of 1 notebook
  90 ÷ 6 = 15
  Step 2: Cost of 10 notebooks
  15 × 10 = 150
  Answer: 150 rupees
  This method is simple and very fast for many proportion problems.
  
• x:5 = 12:20
  Write as fraction: x/5 = 12/20
  Cross multiply: 20x = 60
  x = 3
  So: x = 3
  Cross multiplication is one of the fastest algebraic methods in ratio and proportion.
  
• If a:b = c:d
  Then:
  a/d? No. The correct rule is:
  a × d = b × c
  Do not mix it up.
  
• Three numbers a, b, c are in continued proportion if:
  a:b = b:c
  This means b is the geometric mean of a and c.
  2, 4, 8 are in continued proportion because: 2:4 = 4:8.
  This topic appears in advanced math and algebra.
  
• If a and c are two numbers, their mean proportion is b such that:
  a:b = b:c
  Then: b² = ac
  So: b = √(ac)
  Question: Mean proportion between 4 and 9:
  b = √(4 × 9) = √36 = 6
  So 6 is the mean proportion.
  
• To compare ratios, convert them to the same form.
  2:3 and 4:5
  Convert:
  2/3 ≈ 0.666
  4/5 = 0.8
  So: 4:5 is larger
  
  Another example: 3:4 and 6:8
  3/4 = 0.75
  6/8 = 0.75
  So they are equal.
  
• Ratios can often be turned into percentages.
  2:8
  This means 2 out of 10 parts.
  2/10 = 20%
  So 2:8 can be thought of as 20% to 80%. This is useful in statistics and data interpretation.

The Most Important Ratio Tricks

Here are the strongest shortcuts used by fast students.

1. Always simplify ratios first when possible.
2. For sharing problems, add parts first.
3. For equal sharing, one part = total ÷ sum of ratio parts.
4. For proportion, use cross multiplication.
5. For direct proportion, use equal ratios.
6. For inverse proportion, use equal products.
7. For comparing ratios, convert to fractions or decimals.
8. For ratio in money, work with the same units.
9. For mixtures, use total parts carefully.
10. For multiple quantities, keep the ratio ordered clearly.

Conclusion

Ratio and proportion are not just topics for exams. They are the language of comparison, scaling, balance, and fair division. Once you understand how ratios simplify, how proportions balance, how cross multiplication works, and when to use direct or inverse proportion, the whole topic becomes much more manageable. The fastest students do not guess. They recognize structure, convert ratios carefully, and use the right shortcut at the right time.

Common Mistakes in Ratio and Proportion

1. Forgetting to simplify the ratio
2. Confusing ratio with subtraction
3. Adding ratio numbers directly when sharing
4. Mixing units, like cm and m, without converting
5. Using direct proportion when the situation is inverse
6. Cross multiplying incorrectly
7. Reversing the order of the ratio
8. Forgetting that ratio compares quantities of the same kind

Example: 3:4 is not the same as 4:3. Order matters.

Ratio and proportion appear in real life:

• Recipes
• Map scales
• Speed and distance
• Currency conversion
• Business partnership
• Mixtures and blends
• Classroom gender ratios
• Population comparisons

Appear in Advanced Math

• Probability
• Probability comparisons
• Geometry similarity
• Scale factors
• Rates and speed
• Algebraic equations
• Geometry diagrams
• Data modeling
• Mixture problems

This topic is everywhere once you start looking for it. Master ratio and proportion well, and many other topics become easier, including fractions, percentage, speed, mixture, algebra, and geometry.