Mathematics · Class 6 · Ratio and Proportion · Term 3

Solving Problems with Ratios

Apply your knowledge of ratios to solve real-world problems, such as dividing a quantity into parts according to a given ratio.

TL;DR:Have you ever wondered how to share a pizza fairly with friends, even if everyone isn't getting an equal slice? We'll explore how ratios help us divide things perfectly in real-life situations.

• CBSE Learning Outcomes
• NCERT Class 6: Chapter 12 - Ratio and Proportion

About This Topic

This topic, 'Solving Problems with Ratios', is a crucial application-based extension of the foundational concept of ratios, aligning with the NCERT framework's emphasis on connecting mathematics to daily life. For Class 6 students, this marks a significant step from simply understanding what a ratio is (a comparison of quantities) to using it as a tool for problem-solving. The core pedagogical goal is to help students see a ratio not just as two numbers separated by a colon, but as a representation of 'parts' of a whole. By tackling real-world scenarios like dividing money, mixing ingredients, or comparing statistics, students develop their analytical and logical reasoning skills. The topic builds heavily on their prior knowledge of fractions and the unitary method. The focus should be on helping students deconstruct word problems: identifying the quantities to be compared, setting up the ratio in the correct order, and then using a systematic method, like finding the value of a single part, to arrive at the solution. This lays the groundwork for more complex topics like proportion, percentages, and algebraic thinking in higher classes.

Key Questions

1. Explain how to divide a total amount between two people in a specific ratio.
2. Analyse a problem to set up the correct ratio for comparison.
3. Justify your steps when solving a word problem involving the ratio of ages or ingredients.

Learning Objectives

• Divide a given quantity into two or more parts according to a specified ratio.
• Analyse a word problem to correctly formulate the ratio for comparison.
• Solve real-world problems involving ratios of ingredients, money, or other quantities.
• Justify the method used to solve a ratio problem by explaining the meaning of 'parts' and 'whole'.
• Calculate an unknown quantity when one quantity and the ratio between them are given.

Key Vocabulary

Ratio	A comparison of two or more quantities of the same kind, showing their relative sizes.
Antecedent	The first term or number in a ratio.
Consequent	The second term or number in a ratio.
Simplest Form	A ratio where its terms (antecedent and consequent) have no common factor other than 1.
Proportion	A statement that two ratios are equal. For example, 1:2 = 3:6.

Watch Out for These Misconceptions

Common Misconception

If the ratio of boys to girls is 2:3, students might think there are only 2 boys and 3 girls in total.

What to Teach Instead

Explain that a ratio is in its simplest form. The actual numbers are multiples of the ratio. So, the number of boys is 2x and girls is 3x, where 'x' is a common multiplier.

Common Misconception

When dividing a quantity, students might divide the total amount by each number in the ratio separately.

What to Teach Instead

Clarify that the ratio represents parts of a whole. We must first add the parts of the ratio (e.g., 2+3=5) to find the total number of shares the quantity is being divided into. Then, find the value of one share.

Common Misconception

Students often get the order of the ratio wrong when reading a word problem.

What to Teach Instead

Emphasise that the order matters. The quantity mentioned first in the sentence corresponds to the first number (antecedent) in the ratio, and the second quantity corresponds to the second number (consequent).

Active Learning Ideas

Collaborative Problem-Solving

The Perfect Drink Recipe

Students are given a simple recipe for a drink like 'nimbu pani' in a ratio (e.g., 2 parts lemon juice, 1 part sugar, 5 parts water). They then have to calculate the exact amount of each ingredient needed to make a large jug for the whole class.

25 min·Small Groups

Collaborative Problem-Solving

Pocket Money Division

In pairs, students are given a scenario: 'Divide ₹150 pocket money between two siblings, Rohan and Priya, in the ratio of their ages, 8 years and 7 years.' They must calculate how much money each sibling gets.

20 min·Pairs

Collaborative Problem-Solving

Classroom Blueprint

Students measure the length and breadth of their classroom in metres. They then have to draw a scaled-down version in their notebooks using a simple ratio like 1 metre : 2 cm, applying the ratio to their measurements.

30 min·Individual

Real-World Connections

• Mixing ingredients for recipes, like the ratio of water to concentrate for making juice.
• Using scales on maps where a small distance on the map represents a large actual distance.
• Dividing profits among business partners based on their investment ratio.
• Comparing the performance of two cricket teams based on their win-loss ratio.
• Mixing different colours of paint to achieve a specific shade.

Assessment Ideas

Exit Ticket

Use an 'Exit Slip' with a single word problem, such as 'A ribbon 45 cm long is cut into two pieces in the ratio 4:5. Find the length of each piece.' This quickly assesses understanding of the core procedure.

Quick Check

A chapter-end test with a mix of problems: dividing quantities, finding missing terms, and more complex word problems involving ages or mixtures.

Quick Check

Provide students with a solved problem containing a common error. Ask them to 'Be the Teacher' and identify, explain, and correct the mistake in the solution.

Frequently Asked Questions

What is the difference between a ratio and a fraction?

A fraction always represents a part of a single whole (e.g., 3/4 of a pizza). A ratio is a comparison between two or more quantities, which may or may not be parts of the same whole (e.g., the ratio of cars to bikes on a road).

Why do we add the numbers in a ratio like 3:5 to get 8?

When you are dividing a total amount in a ratio of 3:5, it means the amount is split into 3 parts for one person and 5 parts for another. By adding 3 + 5, you find the total number of equal parts (8) the amount is being divided into. This helps you find the value of a single part.

Can a ratio be written for quantities with different units?

No, to form a ratio, the quantities must be in the same unit. For example, to find the ratio of 50 paise to 2 rupees, you must first convert both to paise (50:200) or both to rupees (0.5:2) before simplifying.