How to Solve Ratio Problems with Formulas and Worked Examples
There should be two quantities required to compare them with each other. For example, a : b, they are called terms(a and b). The first term(a) is called antecedent and the second term(b) is called the consequent. Generally, a ratio is expressed in the simplest form.
Two Quantities of Ratio.
Remarks:
The order of the terms in a ratio is important, we can not write a : b to b : a. They are different.
Ratio exists only between quantities of the same kind as well as in the same units.
We can convert them into fractions whenever needed.
Fraction and Ratio.
Types of Ratios
Some of the important ratios are:
Inverse Ratio- The new ratio obtained by reversing the terms of a ratio is called the inverse ratio. For example, the inverse ratio of 2 : 3 will be 3 : 2.
Compound Ratio- The new ratio formed by the product of the previous terms of two or more ratios and the product of the last terms is called a mixed ratio. For example, the mixed ratio of two ratios (a : b) and (c : d) will be (ad : bc). Similarly the mixed ratio of 2 : 3 , 4 : 5 and 6 : 7 will be 2 × 4 × 6 : 3 × 5 × 7 i.e. 48 : 105 or 16 : 35.
Duplicate Ratio- If a new ratio is made by mixing a ratio with the same, then it is called a square ratio. For example, the square ratio of 2 : 3 is 2² : 3³. That is 2 × 2 : 3 × 3 or 4 : 9.
How To Solve Ratio?
When two numbers are divided by each other, the ratio of those numbers is obtained. As a and b are two numbers then a/b will be their ratio. If we want to find the ratio of two numbers 3 and 7, then 3/7 will be their ratio. It will be written as 3:7.
Simple Ratio Problems(Illustrations)
What is the ratio of 2 to 3?
Solution: 2 + 3 = 5. We have 5 parts in the ratio of 2 : 3.
25 cm and 1 m. What is the ratio?
Solution: 1 m = 100 cm.
25 cm and 100 cm ratio of 25/100 = 1 : 4.
The sum of the two numbers is 60 and the difference is 6. What will be the ratio?
Solution: Required ratio of numbers:
(60 + 6) / (60 - 6) = 66/54 = 11/9 or 11 : 9.
Solved Questions
What will be the inverse ratio of 5 : 6?
Solution: The inverse ratio of 5 : 6 is 6 : 5.
What will be the duplicate ratio of 3 : 4?
Solution: The duplicate ratio of 3 : 4 = 3² : 4² or 9 : 16
If a quantity is divided in the ratio of 5 : 7, the larger part is 100. Find the quantity.
Solution: Let the quantity be x.
Then the two parts will be 5x / 7 + 5 and 7x / 7 + 5.
Hence, if the larger part is 100, we get 7x / 5 + 7 = 100.
7x / 12 = 100
7x = 100 × 12
7x = 1200
x = 1200 / 7
x = 171.42
Therefore, the quantity is 171.42.
Learning by Doing
Problems On Ratio:
1. Shilpa earns ₹150 in 4 hours and isha ₹300 in 7 hours. What will be the ratio of their earnings?
Ans: Shilpa earning: 150 X 4 = 600.
Isha earning: 300 X 7 = 2100.
Ratio= 600 : 2100
= 2 : 7
2. The ratio between the speeds of two trains is 4:5. If the second train runs 400 kms. in 6 hours. What will be the speed of the first train?
Ans: The speed of train in 1 hours : 400/6 = 66(approx)
Ratio = 4x/5x=y/66
= 4x X 66 = 5x X y
= 264 x = 5xy
= 264x/5x=y
= 52.8=y
The speed of the first train is 52.8 km/hr.
Summary
The Ratio refers to the comparison of at least two quantities of each other. If a and b are two quantities of the same kind (in the same units), then the fraction a/b is called the ratio of a to b. It is written as a : b. The quantities a and b are called the terms of the ratio, a is called the first term or antecedent and b is called the second term or consequent. The ratio compounded of the two ratios a : b and c : d is ad : bc. A ratio compounded of itself is called its duplicate ratio. a² : b² is the duplicate ratio of a : b. For any ratio a : b, the inverse ratio is b : a.
FAQs on Understanding Ratio Problems in Mathematics
1. What is a ratio in Maths?
A ratio is a comparison of two quantities by division. It shows how many times one value contains or is contained within the other.
- Written as a : b or a/b
- Represents a relationship between quantities of the same kind
- Example: If there are 6 boys and 3 girls, the ratio of boys to girls is 6:3, which simplifies to 2:1
2. How do you simplify a ratio?
To simplify a ratio, divide both terms by their greatest common divisor (GCD).
- Step 1: Find the GCD of both numbers
- Step 2: Divide each term by the GCD
- Example: Simplify 12:18
- GCD of 12 and 18 is 6
- 12 ÷ 6 : 18 ÷ 6 = 2:3
3. What is the formula for dividing a quantity in a given ratio?
To divide a quantity in the ratio a : b, each part equals (a or b / (a + b)) × Total.
- Step 1: Add the ratio terms → a + b
- Step 2: Divide each term by the total parts
- Step 3: Multiply by the given quantity
- Example: Divide 100 in ratio 2:3
- Total parts = 2 + 3 = 5
- First part = (2/5) × 100 = 40
- Second part = (3/5) × 100 = 60
4. What is the difference between ratio and proportion?
A ratio compares two quantities, while a proportion states that two ratios are equal.
- Ratio example: 2:3
- Proportion example: 2:3 = 4:6
- Proportion formula: a/b = c/d
5. How do you solve ratio word problems step by step?
To solve ratio word problems, convert the situation into a ratio and divide accordingly.
- Step 1: Identify the ratio given
- Step 2: Add the ratio terms
- Step 3: Divide the total quantity by the sum
- Step 4: Multiply by each ratio term
- Example: If the ratio is 3:2 and total is 50
- Total parts = 5
- Each part = 50 ÷ 5 = 10
- Shares = 30 and 20
6. How do you find equivalent ratios?
Equivalent ratios are found by multiplying or dividing both terms by the same number.
- Start with a ratio, for example 2:5
- Multiply both terms by 2 → 4:10
- Multiply both terms by 3 → 6:15
7. Can a ratio have three terms?
Yes, a three-term ratio compares three quantities and is written as a : b : c.
- Example: 2:3:5
- Total parts = 2 + 3 + 5 = 10
- If dividing 100, shares are 20, 30, and 50
8. What are common mistakes in ratio problems?
Common mistakes in ratio problems include incorrect simplification and ignoring total parts.
- Not finding the correct GCD when simplifying
- Forgetting to add ratio terms before dividing
- Mixing up the order of quantities
- Treating ratios like subtraction instead of division
9. How is ratio used in real life?
A ratio is used in real life to compare quantities such as speed, recipes, maps, and finance.
- Cooking recipes (2 cups flour : 1 cup sugar)
- Scale drawings and maps (1:100 scale)
- Speed (distance : time)
- Financial analysis and sharing profits
10. How do you convert a ratio into a fraction or percentage?
To convert a ratio into a fraction, write it as a/b, and to convert into a percentage, multiply the fraction by 100.
- Example: 2:5
- Fraction = 2/5
- Percentage = (2/5) × 100 = 40%
