Ratio Proportion Ex 12.1: Master Ratios in Class 6 Maths

Welcome, young mathematicians! In Chapter 12 of your Class 6 Maths NCERT textbook, we embark on an exciting journey into the world of 'Ratio and Proportion'. Specifically, Exercise 12.1 focuses on understanding and calculating ratios. Have you ever wondered how to compare the number of boys to girls in your class, or the amount of sugar to flour in a recipe? Ratios are the perfect tool for this! They help us compare two quantities of the same kind. By the end of this page, you'll not only understand what a ratio is but also how to write it, simplify it, and apply it to solve real-life problems, just like those in your NCERT Exercise 12.1. Get ready to compare and conquer!

Understanding What a Ratio Is

A ratio is a way to compare two quantities of the same kind by division. Think of it as telling you 'how many times' one quantity contains another, or 'what fraction' one quantity is of another. For example, if you have 3 red apples and 5 green apples, the ratio of red apples to green apples is 3 to 5. We can write this ratio in a few ways:

  1. Using a colon: 3 : 5
  2. As a fraction: 3/5
  3. Using the word 'to': 3 to 5

It's super important that the quantities you are comparing are of the same kind and in the same units. You can't directly compare 3 apples to 5 oranges in a ratio of 'fruit type' unless you specify 'apples to oranges'. Also, you can't compare 2 meters to 50 centimeters directly without converting one to the other (e.g., 200 cm to 50 cm). Ratios have no units themselves because the units cancel out during comparison.

How to Simplify Ratios to Their Simplest Form

  1. Step 1: Write the Ratio as a Fraction — If your ratio is given as 'a : b', write it as a fraction 'a/b'. For example, if the ratio is 10 : 15, write it as 10/15.
  2. Step 2: Find the Highest Common Factor (HCF) — Find the HCF of the numerator (the first number in the ratio) and the denominator (the second number in the ratio). For 10/15, the factors of 10 are 1, 2, 5, 10. The factors of 15 are 1, 3, 5, 15. The HCF is 5.
  3. Step 3: Divide Both Numbers by the HCF — Divide both parts of the ratio (the numerator and denominator of the fraction) by their HCF. For 10/15, divide both by 5: 10 ÷ 5 = 2 and 15 ÷ 5 = 3. So, the simplified fraction is 2/3.
  4. Step 4: Write the Simplified Ratio — Write the simplified fraction back in ratio form. So, 2/3 becomes 2 : 3. This is the simplest form because 2 and 3 have no common factors other than 1.

Worked Examples: Writing and Simplifying Ratios

  • Example 1: Boys and Girls in a Class In a class of 30 students, there are 18 girls and 12 boys. Find the ratio of: (a) Number of girls to the number of boys. (b) Number of boys to the total number of students. Solution: (a) Number of girls = 18, Number of boys = 12 Ratio of girls to boys = 18 : 12 To simplify: Write as 18/12. HCF of 18 and 12 is 6. Divide both by 6: (18 ÷ 6) / (12 ÷ 6) = 3/2 So, the ratio is 3 : 2. (b) Number of boys = 12, Total number of students = 30 Ratio of boys to total students = 12 : 30 To simplify: Write as 12/30. HCF of 12 and 30 is 6. Divide both by 6: (12 ÷ 6) / (30 ÷ 6) = 2/5 So, the ratio is 2 : 5.
  • Example 2: Comparing Lengths Find the ratio of 40 cm to 2 meters. Solution: Here, the units are different (cm and meters). We need to convert them to the same unit. We know that 1 meter = 100 cm. So, 2 meters = 2 * 100 cm = 200 cm. Now, we compare 40 cm to 200 cm. Ratio = 40 : 200 To simplify: Write as 40/200. HCF of 40 and 200 is 40. Divide both by 40: (40 ÷ 40) / (200 ÷ 40) = 1/5 So, the ratio is 1 : 5.

Important Tips for Ratios in Exercise 12.1

When solving problems related to ratios, especially like those in NCERT Exercise 12.1, keep these crucial points in mind:

  1. Same Units are a MUST! Always ensure the quantities you are comparing are expressed in the same units. If you're comparing kilograms and grams, convert one to match the other before forming the ratio. Forgetting this is a very common mistake!
  2. Order Matters! The order in which quantities are mentioned in the problem is the order they should appear in the ratio. 'Ratio of A to B' is written as A : B, which is different from B : A.
  3. Always Simplify! Unless specifically asked not to, always express your final ratio in its simplest form. This means dividing both terms by their Highest Common Factor (HCF) until they have no common factors other than 1.

Practice Questions with Solutions

  • Q: Find the ratio of 24 to 36. A: Step 1: Write the ratio as a fraction: 24/36. Step 2: Find the HCF of 24 and 36. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. The HCF is 12. Step 3: Divide both numbers by the HCF: 24 ÷ 12 = 2 and 36 ÷ 12 = 3. Step 4: Write the simplified ratio: 2 : 3. Final answer: 2 : 3
  • Q: In a school, there are 80 teachers and 2400 students. Find the ratio of the number of teachers to the number of students. A: Step 1: Identify the quantities: Teachers = 80, Students = 2400. Step 2: Form the ratio: Teachers : Students = 80 : 2400. Step 3: Write as a fraction: 80/2400. Step 4: Simplify the fraction. We can cancel a zero from both: 8/240. Step 5: Find the HCF of 8 and 240. The HCF is 8. Step 6: Divide both by 8: 8 ÷ 8 = 1 and 240 ÷ 8 = 30. Step 7: Write the simplified ratio: 1 : 30. Final answer: 1 : 30
  • Q: A pencil is 15 cm long and a pen is 1.5 dm long. Find the ratio of the length of the pencil to the length of the pen. A: Step 1: Identify the quantities and their units: Pencil length = 15 cm, Pen length = 1.5 dm. Step 2: Convert units to be the same. We know 1 dm = 10 cm. So, 1.5 dm = 1.5 × 10 cm = 15 cm. Step 3: Form the ratio: Pencil length : Pen length = 15 cm : 15 cm. Step 4: Write as a fraction: 15/15. Step 5: Simplify: 15 ÷ 15 = 1 and 15 ÷ 15 = 1. Step 6: Write the simplified ratio: 1 : 1. Final answer: 1 : 1
  • Q: What is the ratio of 75 paise to ₹3? A: Step 1: Identify quantities and units: 75 paise, ₹3. Step 2: Convert units to be the same. We know ₹1 = 100 paise. So, ₹3 = 3 × 100 paise = 300 paise. Step 3: Form the ratio: 75 paise : 300 paise. Step 4: Write as a fraction: 75/300. Step 5: Simplify. Both are divisible by 25: 75 ÷ 25 = 3 and 300 ÷ 25 = 12. So, we get 3/12. Step 6: Simplify further. Both 3 and 12 are divisible by 3: 3 ÷ 3 = 1 and 12 ÷ 3 = 4. So, we get 1/4. Step 7: Write the simplified ratio: 1 : 4. Final answer: 1 : 4

Frequently Asked Questions

What is a ratio in simple terms?

A ratio is a way to compare two or more quantities of the same kind. It tells us how much of one quantity there is in comparison to another, often expressed as 'a to b', 'a:b', or a/b.

Why do quantities need to be in the same units for a ratio?

Quantities must be in the same units because ratios compare 'like with like'. If units are different (like meters and centimeters), the comparison wouldn't be meaningful until they are converted to a common unit, allowing for a fair comparison by division.

Is 3:5 the same as 5:3?

No, 3:5 is not the same as 5:3. The order of numbers in a ratio is very important. 3:5 means the first quantity is 3 parts when the second quantity is 5 parts, whereas 5:3 means the first quantity is 5 parts when the second is 3 parts.