Ratio And Proportion Class 6 Notes

Key Points to Remember

• A ratio compares two quantities of the same kind and in the same units by division.
• The order of terms in a ratio is crucial. For example, 2:3 is not the same as 3:2.
• Ratios have no units as the units cancel out during comparison.
• A ratio is usually expressed in its simplest form, like a fraction.
• Equivalent ratios are obtained by multiplying or dividing both terms of the ratio by the same non-zero number.
• Proportion is an equality of two ratios. If a:b = c:d, then a, b, c, d are in proportion.
• In a proportion a:b :: c:d, 'a' and 'd' are extremes, and 'b' and 'c' are means.
• The fundamental property of proportion is: Product of Means = Product of Extremes (b x c = a x d).
• The Unitary Method is used to find the value of a required number of units by first finding the value of a single unit.

Glossary of Terms

Ratio

A comparison of two quantities of the same kind, expressed as a fraction or using a colon (e.g., a:b).

Antecedent

The first term of a ratio (the numerator if expressed as a fraction).

Consequent

The second term of a ratio (the denominator if expressed as a fraction).

Equivalent Ratios

Ratios that represent the same relationship between quantities, even if the numbers are different (e.g., 1:2 and 2:4).

Proportion

A statement that two ratios are equal. It indicates that four quantities are related such that the ratio of the first two is equal to the ratio of the last two.

Extremes

In a proportion a:b :: c:d, the first and fourth terms (a and d) are called the extremes.

Means

In a proportion a:b :: c:d, the second and third terms (b and c) are called the means.

Unitary Method

A technique to solve problems by first finding the value of a single unit and then using that to calculate the value of the required number of units.

Understanding Ratio and Proportion

At its core, ratio is a simple yet powerful way to compare two quantities. When we say the ratio of boys to girls in a class is 2:3, it means for every 2 boys, there are 3 girls. This comparison is always done between quantities of the same kind and usually in the same units. If units are different, they must be converted before forming a ratio. For instance, to find the ratio of 50 cm to 2 m, we first convert 2 m to 200 cm, making the ratio 50 cm : 200 cm, which simplifies to 1:4. The first term of the ratio is called the antecedent, and the second term is the consequent. Ratios are typically expressed in their simplest form, much like fractions.

Moving on, proportion takes the concept of ratio a step further. A proportion is essentially an equality between two ratios. If two ratios, say a:b and c:d, are equal, then we say that a, b, c, and d are in proportion. This is often written as a:b :: c:d, where '::' signifies 'is in proportion to' or 'is equal to'. In any proportion a:b :: c:d, the terms 'a' and 'd' are known as the extremes (outer terms), while 'b' and 'c' are known as the means (middle terms). A crucial property of proportions is that the product of the means is always equal to the product of the extremes. That is, b × c = a × d. This property is incredibly useful for solving problems where one term of a proportion is unknown. Mastering this distinction and the fundamental property is key to solving a variety of problems involving comparisons and relationships between quantities.

Worked Examples

• {"title":"Example 1: Finding a Ratio in Simplest Form","markdown":"Problem: Find the ratio of 45 minutes to 3 hours.\n\nSolution:\n1. Convert units to be the same: 3 hours = 3 × 60 minutes = 180 minutes.\n2. Form the ratio: 45 minutes : 180 minutes.\n3. Simplify the ratio: 45/180 = 1/4.\nAnswer: The ratio is 1:4."}
• {"title":"Example 2: Checking for Proportion","markdown":"Problem: Are 30, 40, 45, 60 in proportion?\n\nSolution:\n1. Form two ratios: 30:40 and 45:60.\n2. Simplify each ratio:\n 30:40 = 30/40 = 3/4\n 45:60 = 45/60 = (3 × 15) / (4 × 15) = 3/4\n3. Since both ratios are equal (3/4 = 3/4), the numbers are in proportion.\nAnswer: Yes, 30, 40, 45, 60 are in proportion."}
• {"title":"Example 3: Unitary Method","markdown":"Problem: If the cost of 5 pens is ₹50, what is the cost of 12 pens?\n\nSolution:\n1. Cost of 5 pens = ₹50.\n2. Cost of 1 pen = ₹50 / 5 = ₹10 (Unitary Method).\n3. Cost of 12 pens = 12 × ₹10 = ₹120.\nAnswer: The cost of 12 pens is ₹120."}

Ratio vs. Proportion

Exam Tip: Units and Simplification

Always pay close attention to the units of the quantities you are comparing when forming a ratio. They must be the same. If they are different (e.g., meters and centimeters, hours and minutes), convert one quantity so both are in the same unit before calculating the ratio. Once you have the ratio, always remember to express it in its simplest form unless explicitly asked otherwise. This often involves finding the Greatest Common Factor (GCF) of the antecedent and consequent and dividing both by it. Incorrect units or unsimplified ratios are common pitfalls that can lead to lost marks!

Practice Questions with Solutions

• Q: What is the ratio of 200 grams to 2 kilograms in simplest form? A: First, convert 2 kg to grams: 2 kg = 2000 g. The ratio is 200:2000, which simplifies to 1:10.
• Q: If 4, 8, x, 16 are in proportion, find the value of x. A: In a proportion, the product of means equals the product of extremes. So, 8 x = 4 16. This means 8x = 64, so x = 8.
• Q: Is the ratio 15:20 equivalent to 3:4? Why or why not? A: Yes, because 15:20 simplifies to (15÷5):(20÷5) = 3:4. Both ratios represent the same relationship.
• Q: A car travels 150 km in 3 hours. How far will it travel in 7 hours at the same speed? A: Using the unitary method, in 1 hour the car travels 150 km / 3 = 50 km. In 7 hours, it will travel 7 * 50 km = 350 km.

Frequently Asked Questions